KILLED proof of input_gi47Cpy7Rw.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).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), 20 ms] (12) CpxTypedWeightedCompleteTrs (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 17 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 5 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 573 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 92 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 144 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 385 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 126 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 233 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 96 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 203 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 57 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 1032 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 390 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 1086 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 433 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 1333 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 375 ms] (66) CpxRNTS (67) CompletionProof [UPPER BOUND(ID), 0 ms] (68) CpxTypedWeightedCompleteTrs (69) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 20 ms] (70) CpxRNTS (71) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (72) CdtProblem (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (74) CdtProblem (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (76) CdtProblem (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (78) CdtProblem (79) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms] (80) CdtProblem (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (82) CdtProblem (83) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 18 ms] (84) CdtProblem (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 257 ms] (86) CdtProblem (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (88) CdtProblem (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (90) CdtProblem (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (92) CdtProblem (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (94) CdtProblem (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (96) CdtProblem (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (98) CdtProblem (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 88 ms] (100) CdtProblem (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 89 ms] (102) CdtProblem (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (104) CdtProblem (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (106) CdtProblem (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 115 ms] (108) CdtProblem (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms] (110) CdtProblem (111) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (112) CdtProblem (113) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 119 ms] (114) CdtProblem (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 118 ms] (116) CdtProblem (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (118) CdtProblem (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (120) CdtProblem (121) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 146 ms] (122) CdtProblem (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (124) CdtProblem (125) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (126) CdtProblem (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (128) CdtProblem (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (130) CdtProblem (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (132) CdtProblem (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (134) CdtProblem (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (136) CdtProblem (137) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (138) CdtProblem (139) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 188 ms] (140) CdtProblem (141) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 196 ms] (142) CdtProblem (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (144) CdtProblem (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (146) CdtProblem (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 221 ms] (148) CdtProblem (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 201 ms] (150) CdtProblem (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (152) CdtProblem (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (154) CdtProblem (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 247 ms] (156) CdtProblem (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (158) CdtProblem (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (160) CdtProblem (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (162) CdtProblem (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms] (164) CdtProblem (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (166) CdtProblem (167) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 9 ms] (168) CdtProblem (169) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 270 ms] (170) CdtProblem (171) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 312 ms] (172) CdtProblem (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms] (174) CdtProblem (175) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms] (176) CdtProblem (177) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 286 ms] (178) CdtProblem (179) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 338 ms] (180) CdtProblem (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 337 ms] (182) CdtProblem (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (184) CdtProblem (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (186) CdtProblem (187) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (188) CdtProblem (189) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (190) CdtProblem (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms] (192) CdtProblem (193) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 378 ms] (194) CdtProblem (195) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 380 ms] (196) CdtProblem (197) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 403 ms] (198) CdtProblem (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (200) CdtProblem (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (202) CdtProblem (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (204) CdtProblem (205) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (206) CdtProblem (207) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 419 ms] (208) CdtProblem (209) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (210) CdtProblem (211) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (212) CdtProblem (213) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 536 ms] (214) CdtProblem (215) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 506 ms] (216) CdtProblem (217) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 24 ms] (218) CdtProblem (219) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (220) CdtProblem (221) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 478 ms] (222) CdtProblem (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CdtProblem (225) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (226) CdtProblem (227) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (228) CdtProblem (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms] (230) CdtProblem (231) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 38 ms] (232) CdtProblem (233) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (234) CdtProblem (235) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 458 ms] (236) CdtProblem (237) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 446 ms] (238) CdtProblem (239) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (240) CdtProblem (241) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 13 ms] (242) CdtProblem (243) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 731 ms] (244) CdtProblem (245) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 721 ms] (246) CdtProblem (247) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 696 ms] (248) CdtProblem (249) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (250) CdtProblem (251) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (252) CdtProblem (253) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (254) CdtProblem (255) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (256) CdtProblem (257) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 26 ms] (258) CdtProblem (259) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 529 ms] (260) CdtProblem (261) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 610 ms] (262) CdtProblem (263) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 608 ms] (264) CdtProblem (265) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (266) CdtProblem (267) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms] (268) CdtProblem (269) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms] (270) CdtProblem (271) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (272) 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(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0 length(cons(x, xs)) -> s(length(xs)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) get(n, nil) -> 0 get(n, cons(x, nil)) -> x get(0, cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0') -> true ge(0', s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0' length(cons(x, xs)) -> s(length(xs)) half(0') -> 0' half(s(0')) -> 0' half(s(s(x))) -> s(half(x)) get(n, nil) -> 0' get(n, cons(x, nil)) -> x get(0', cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) RelTrsToTrsProof (UPPER BOUND(ID)) transformed relative TRS to TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: qsort(xs) -> qs(half(length(xs)), xs) qs(n, nil) -> nil qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) filterlow(n, nil) -> nil filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) if1(true, n, x, xs) -> filterlow(n, xs) if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) filterhigh(n, nil) -> nil filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) if2(true, n, x, xs) -> filterhigh(n, xs) if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) ge(x, 0) -> true ge(0, s(x)) -> false ge(s(x), s(y)) -> ge(x, y) append(nil, ys) -> ys append(cons(x, xs), ys) -> cons(x, append(xs, ys)) length(nil) -> 0 length(cons(x, xs)) -> s(length(xs)) half(0) -> 0 half(s(0)) -> 0 half(s(s(x))) -> s(half(x)) get(n, nil) -> 0 get(n, cons(x, nil)) -> x get(0, cons(x, cons(y, xs))) -> x get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) S is empty. Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (5) RelTrsToWeightedTrsProof (UPPER BOUND(ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (6) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF). The TRS R consists of the following rules: qsort(xs) -> qs(half(length(xs)), xs) [1] qs(n, nil) -> nil [1] qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] length(nil) -> 0 [1] length(cons(x, xs)) -> s(length(xs)) [1] half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] get(n, nil) -> 0 [1] get(n, cons(x, nil)) -> x [1] get(0, cons(x, cons(y, xs))) -> x [1] get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) [1] Rewrite Strategy: INNERMOST ---------------------------------------- (7) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (8) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: qsort(xs) -> qs(half(length(xs)), xs) [1] qs(n, nil) -> nil [1] qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] length(nil) -> 0 [1] length(cons(x, xs)) -> s(length(xs)) [1] half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] get(n, nil) -> 0 [1] get(n, cons(x, nil)) -> x [1] get(0, cons(x, cons(y, xs))) -> x [1] get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) [1] The TRS has the following type information: qsort :: nil:cons:ys -> nil:cons:ys qs :: 0:s -> nil:cons:ys -> nil:cons:ys half :: 0:s -> 0:s length :: nil:cons:ys -> 0:s 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 get :: 0:s -> nil:cons:ys -> 0:s filterhigh :: 0:s -> nil:cons:ys -> nil:cons:ys if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys ge :: 0:s -> 0:s -> true:false true :: true:false false :: true:false if2 :: true:false -> 0:s -> 0:s -> nil:cons:ys -> nil:cons:ys 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys Rewrite Strategy: INNERMOST ---------------------------------------- (9) CompletionProof (UPPER BOUND(ID)) The transformation into a RNTS is sound, since: (a) The obligation is a constructor system where every type has a constant constructor, (b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols: qsort_1 (c) The following functions are completely defined: half_1 length_1 ge_2 qs_2 filterlow_2 get_2 filterhigh_2 if1_4 if2_4 append_2 Due to the following rules being added: length(v0) -> 0 [0] qs(v0, v1) -> null_qs [0] filterlow(v0, v1) -> null_filterlow [0] get(v0, v1) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] And the following fresh constants: null_qs, 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(xs) -> qs(half(length(xs)), xs) [1] qs(n, nil) -> nil [1] qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] length(nil) -> 0 [1] length(cons(x, xs)) -> s(length(xs)) [1] half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] get(n, nil) -> 0 [1] get(n, cons(x, nil)) -> x [1] get(0, cons(x, cons(y, xs))) -> x [1] get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) [1] length(v0) -> 0 [0] qs(v0, v1) -> null_qs [0] filterlow(v0, v1) -> null_filterlow [0] get(v0, v1) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append qs :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append half :: 0:s -> 0:s length :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> 0:s nil :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append cons :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append append :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append filterlow :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append get :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> 0:s filterhigh :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs: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_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_qs :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_filterlow :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_filterhigh :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_append :: nil:cons:ys:null_qs: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) -> qs(half(0), nil) [2] qsort(cons(x', xs')) -> qs(half(s(length(xs'))), cons(x', xs')) [2] qsort(xs) -> qs(half(0), xs) [1] qs(n, nil) -> nil [1] qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(0, xs)) -> if1(true, n, 0, xs) [2] filterlow(0, cons(s(x1), xs)) -> if1(false, 0, s(x1), xs) [2] filterlow(s(x2), cons(s(y8), xs)) -> if1(ge(x2, y8), s(x2), s(y8), 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(x3), cons(0, xs)) -> if2(false, s(x3), 0, xs) [2] filterhigh(s(y9), cons(s(x4), xs)) -> if2(ge(x4, y9), s(y9), s(x4), 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] length(nil) -> 0 [1] length(cons(x, xs)) -> s(length(xs)) [1] half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] get(n, nil) -> 0 [1] get(n, cons(x, nil)) -> x [1] get(0, cons(x, cons(y, xs))) -> x [1] get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) [1] length(v0) -> 0 [0] qs(v0, v1) -> null_qs [0] filterlow(v0, v1) -> null_filterlow [0] get(v0, v1) -> 0 [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append qs :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append half :: 0:s -> 0:s length :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> 0:s nil :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append cons :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append append :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append filterlow :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append get :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> 0:s filterhigh :: 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append if1 :: true:false -> 0:s -> 0:s -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs: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_qs:null_filterlow:null_filterhigh:null_append -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append 0 :: 0:s s :: 0:s -> 0:s ys :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_qs :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_filterlow :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_filterhigh :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append null_append :: nil:cons:ys:null_qs: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_qs => 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(x4, y9), 1 + y9, 1 + x4, xs) :|: y9 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs, z = 1 + y9 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 + x3, 0, xs) :|: xs >= 0, z' = 1 + 0 + xs, z = 1 + x3, x3 >= 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') -{ 2 }-> if1(ge(x2, y8), 1 + x2, 1 + y8, xs) :|: xs >= 0, y8 >= 0, z = 1 + x2, x2 >= 0, z' = 1 + (1 + y8) + xs 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 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: n >= 0, z = n, x >= 0, z' = 1 + x + 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(n, 1 + y + xs) :|: n >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 1 + n get(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 get(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(n), filterlow(get(n, 1 + x + xs), 1 + x + xs)), 1 + get(n, 1 + x + xs) + qs(half(n), filterhigh(get(n, 1 + x + xs), 1 + x + xs))) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 qs(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 qsort(z) -{ 1 }-> qs(half(0), xs) :|: xs >= 0, z = xs qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' ---------------------------------------- (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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(z - 1, 1 + y + xs) :|: z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(z), filterlow(get(z, 1 + x + xs), 1 + x + xs)), 1 + get(z, 1 + x + xs) + qs(half(z), filterhigh(get(z, 1 + x + xs), 1 + x + xs))) :|: z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { get } { length } { append } { ge } { half } { filterhigh, if2 } { filterlow, if1 } { qs } { 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(z - 1, 1 + y + xs) :|: z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(z), filterlow(get(z, 1 + x + xs), 1 + x + xs)), 1 + get(z, 1 + x + xs) + qs(half(z), filterhigh(get(z, 1 + x + xs), 1 + x + xs))) :|: z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {get}, {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(z - 1, 1 + y + xs) :|: z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(z), filterlow(get(z, 1 + x + xs), 1 + x + xs)), 1 + get(z, 1 + x + xs) + qs(half(z), filterhigh(get(z, 1 + x + xs), 1 + x + xs))) :|: z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {get}, {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: get after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z' ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(z - 1, 1 + y + xs) :|: z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(z), filterlow(get(z, 1 + x + xs), 1 + x + xs)), 1 + get(z, 1 + x + xs) + qs(half(z), filterhigh(get(z, 1 + x + xs), 1 + x + xs))) :|: z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {get}, {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: ?, size: O(n^1) [z'] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: get after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(z - 1, 1 + y + xs) :|: z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(z), filterlow(get(z, 1 + x + xs), 1 + x + xs)), 1 + get(z, 1 + x + xs) + qs(half(z), filterhigh(get(z, 1 + x + xs), 1 + x + xs))) :|: z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] ---------------------------------------- (27) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (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') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {length}, {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: ?, size: O(n^1) [z] ---------------------------------------- (29) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: length after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 2 }-> qs(half(1 + length(xs')), 1 + x' + xs') :|: x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (31) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (32) Obligation: Complexity RNTS consisting of the following rules: 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(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] ---------------------------------------- (33) 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 ---------------------------------------- (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') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {append}, {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (35) 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 ---------------------------------------- (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') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ---------------------------------------- (39) 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 ---------------------------------------- (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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {ge}, {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: ?, size: O(1) [1] ---------------------------------------- (41) 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' ---------------------------------------- (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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 2 }-> if2(ge(x4, z - 1), 1 + (z - 1), 1 + x4, xs) :|: z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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, y8), 1 + (z - 1), 1 + y8, xs) :|: xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: 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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {half}, {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: ?, size: O(n^1) [z] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: half after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 10 + 3*x + 3*xs }-> append(qs(half(z), filterlow(s, 1 + x + xs)), 1 + s' + qs(half(z), filterhigh(s'', 1 + x + xs))) :|: s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 1 }-> qs(half(0), z) :|: z >= 0 qsort(z) -{ 2 }-> qs(half(0), 0) :|: z = 0 qsort(z) -{ 3 + xs' }-> qs(half(1 + s2), 1 + x' + xs') :|: s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: 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 + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 14 + 3*x + 3*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, filterhigh(s'', 1 + x + xs))) :|: s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] ---------------------------------------- (51) 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 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 14 + 3*x + 3*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, filterhigh(s'', 1 + x + xs))) :|: s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterhigh,if2}, {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] filterhigh: runtime: ?, size: O(n^1) [z'] if2: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (53) 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 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append(z, z') -{ 2 + xs }-> 1 + x + s4 :|: s4 >= 0, s4 <= xs + 1, z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 3 + z }-> if2(s6, 1 + (z - 1), 1 + x4, xs) :|: s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs 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 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 14 + 3*x + 3*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, filterhigh(s'', 1 + x + xs))) :|: s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] ---------------------------------------- (55) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (56) Obligation: Complexity RNTS consisting of the following rules: 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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) -{ 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 29 + 4*s'' + s''*x + s''*xs + 7*x + 7*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, s14)) :|: s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] ---------------------------------------- (57) 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 ---------------------------------------- (58) 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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) -{ 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 29 + 4*s'' + s''*x + s''*xs + 7*x + 7*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, s14)) :|: s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {filterlow,if1}, {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] filterlow: runtime: ?, size: O(n^1) [z'] if1: runtime: ?, size: O(n^1) [1 + z'' + z1] ---------------------------------------- (59) 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 ---------------------------------------- (60) 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 4 + y8 }-> if1(s5, 1 + (z - 1), 1 + y8, xs) :|: s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 2 }-> if1(1, z, 0, z' - 1) :|: z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 2 }-> if1(0, 0, 1 + x1, xs) :|: xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 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) -{ 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 29 + 4*s'' + s''*x + s''*xs + 7*x + 7*xs + 2*z }-> append(qs(s11, filterlow(s, 1 + x + xs)), 1 + s' + qs(s12, s14)) :|: s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] 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] ---------------------------------------- (61) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (62) 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s21 :|: s21 >= 0, s21 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s22 :|: s22 >= 0, s22 <= 1 + x1 + xs + 1, xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 19 + 7*xs + y8 }-> s23 :|: s23 >= 0, s23 <= 1 + y8 + xs + 1, s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s24 :|: s24 >= 0, s24 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s25 :|: s25 >= 0, s25 <= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 50 + 4*s'' + s''*x + s''*xs + 14*x + 14*xs + 2*z }-> append(qs(s11, s20), 1 + s' + qs(s12, s14)) :|: s20 >= 0, s20 <= 1 + x + xs, s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] 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] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: qs after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (64) 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s21 :|: s21 >= 0, s21 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s22 :|: s22 >= 0, s22 <= 1 + x1 + xs + 1, xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 19 + 7*xs + y8 }-> s23 :|: s23 >= 0, s23 <= 1 + y8 + xs + 1, s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s24 :|: s24 >= 0, s24 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s25 :|: s25 >= 0, s25 <= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 50 + 4*s'' + s''*x + s''*xs + 14*x + 14*xs + 2*z }-> append(qs(s11, s20), 1 + s' + qs(s12, s14)) :|: s20 >= 0, s20 <= 1 + x + xs, s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] 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] qs: runtime: ?, size: INF ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: qs after applying outer abstraction to obtain an ITS, resulting in: INF with polynomial bound: ? ---------------------------------------- (66) 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 + s4 :|: s4 >= 0, s4 <= 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 + x4 + xs + 1, s6 >= 0, s6 <= 1, z - 1 >= 0, xs >= 0, x4 >= 0, z' = 1 + (1 + x4) + xs filterhigh(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 filterlow(z, z') -{ 10 + 7*z' }-> s21 :|: s21 >= 0, s21 <= 0 + (z' - 1) + 1, z >= 0, z' - 1 >= 0 filterlow(z, z') -{ 17 + 7*xs }-> s22 :|: s22 >= 0, s22 <= 1 + x1 + xs + 1, xs >= 0, x1 >= 0, z = 0, z' = 1 + (1 + x1) + xs filterlow(z, z') -{ 19 + 7*xs + y8 }-> s23 :|: s23 >= 0, s23 <= 1 + y8 + xs + 1, s5 >= 0, s5 <= 1, xs >= 0, y8 >= 0, z - 1 >= 0, z' = 1 + (1 + y8) + xs filterlow(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 ge(z, z') -{ 2 + z' }-> s7 :|: s7 >= 0, s7 <= 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 get(z, z') -{ 4 + xs + y }-> s1 :|: s1 >= 0, s1 <= 1 + y + xs, z - 1 >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 get(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 get(z, z') -{ 1 }-> z' - 1 :|: z >= 0, z' - 1 >= 0 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 1 + z }-> 1 + s13 :|: s13 >= 0, s13 <= z - 2, z - 2 >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> s24 :|: s24 >= 0, s24 <= z1, z' >= 0, z1 >= 0, z = 1, z'' >= 0 if1(z, z', z'', z1) -{ 15 + 7*z1 }-> 1 + z'' + s25 :|: s25 >= 0, s25 <= 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 length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: z >= 0 length(z) -{ 2 + xs }-> 1 + s3 :|: s3 >= 0, s3 <= xs, z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 50 + 4*s'' + s''*x + s''*xs + 14*x + 14*xs + 2*z }-> append(qs(s11, s20), 1 + s' + qs(s12, s14)) :|: s20 >= 0, s20 <= 1 + x + xs, s14 >= 0, s14 <= 1 + x + xs, s11 >= 0, s11 <= z, s12 >= 0, s12 <= z, s >= 0, s <= 1 + x + xs, s' >= 0, s' <= 1 + x + xs, s'' >= 0, s'' <= 1 + x + xs, z >= 0, xs >= 0, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0 qs(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 qsort(z) -{ 3 }-> qs(s10, z) :|: s10 >= 0, s10 <= 0, z >= 0 qsort(z) -{ 4 }-> qs(s8, 0) :|: s8 >= 0, s8 <= 0, z = 0 qsort(z) -{ 6 + s2 + xs' }-> qs(s9, 1 + x' + xs') :|: s9 >= 0, s9 <= 1 + s2, s2 >= 0, s2 <= xs', x' >= 0, xs' >= 0, z = 1 + x' + xs' Function symbols to be analyzed: {qs}, {qsort} Previous analysis results are: get: runtime: O(n^1) [2 + z'], size: O(n^1) [z'] length: runtime: O(n^1) [1 + z], size: O(n^1) [z] append: runtime: O(n^1) [1 + z], size: O(n^1) [1 + z] ge: runtime: O(n^1) [2 + z'], size: O(1) [1] half: runtime: O(n^1) [2 + z], size: O(n^1) [z] 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] 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] qs: runtime: INF, size: INF ---------------------------------------- (67) 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: qs(v0, v1) -> null_qs [0] filterlow(v0, v1) -> null_filterlow [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] length(v0) -> null_length [0] get(v0, v1) -> null_get [0] ge(v0, v1) -> null_ge [0] half(v0) -> null_half [0] if1(v0, v1, v2, v3) -> null_if1 [0] if2(v0, v1, v2, v3) -> null_if2 [0] And the following fresh constants: null_qs, null_filterlow, null_filterhigh, null_append, null_length, null_get, null_ge, null_half, null_if1, null_if2 ---------------------------------------- (68) 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(xs) -> qs(half(length(xs)), xs) [1] qs(n, nil) -> nil [1] qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs))))) [1] filterlow(n, nil) -> nil [1] filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs) [1] if1(true, n, x, xs) -> filterlow(n, xs) [1] if1(false, n, x, xs) -> cons(x, filterlow(n, xs)) [1] filterhigh(n, nil) -> nil [1] filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs) [1] if2(true, n, x, xs) -> filterhigh(n, xs) [1] if2(false, n, x, xs) -> cons(x, filterhigh(n, xs)) [1] ge(x, 0) -> true [1] ge(0, s(x)) -> false [1] ge(s(x), s(y)) -> ge(x, y) [1] append(nil, ys) -> ys [1] append(cons(x, xs), ys) -> cons(x, append(xs, ys)) [1] length(nil) -> 0 [1] length(cons(x, xs)) -> s(length(xs)) [1] half(0) -> 0 [1] half(s(0)) -> 0 [1] half(s(s(x))) -> s(half(x)) [1] get(n, nil) -> 0 [1] get(n, cons(x, nil)) -> x [1] get(0, cons(x, cons(y, xs))) -> x [1] get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs)) [1] qs(v0, v1) -> null_qs [0] filterlow(v0, v1) -> null_filterlow [0] filterhigh(v0, v1) -> null_filterhigh [0] append(v0, v1) -> null_append [0] length(v0) -> null_length [0] get(v0, v1) -> null_get [0] ge(v0, v1) -> null_ge [0] half(v0) -> null_half [0] if1(v0, v1, v2, v3) -> null_if1 [0] if2(v0, v1, v2, v3) -> null_if2 [0] The TRS has the following type information: qsort :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 qs :: 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 half :: 0:s:null_length:null_get:null_half -> 0:s:null_length:null_get:null_half length :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> 0:s:null_length:null_get:null_half nil :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 cons :: 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 append :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 filterlow :: 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 get :: 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> 0:s:null_length:null_get:null_half filterhigh :: 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 if1 :: true:false:null_ge -> 0:s:null_length:null_get:null_half -> 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 ge :: 0:s:null_length:null_get:null_half -> 0:s:null_length:null_get:null_half -> true:false:null_ge true :: true:false:null_ge false :: true:false:null_ge if2 :: true:false:null_ge -> 0:s:null_length:null_get:null_half -> 0:s:null_length:null_get:null_half -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 -> nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 0 :: 0:s:null_length:null_get:null_half s :: 0:s:null_length:null_get:null_half -> 0:s:null_length:null_get:null_half ys :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_qs :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_filterlow :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_filterhigh :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_append :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_length :: 0:s:null_length:null_get:null_half null_get :: 0:s:null_length:null_get:null_half null_ge :: true:false:null_ge null_half :: 0:s:null_length:null_get:null_half null_if1 :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 null_if2 :: nil:cons:ys:null_qs:null_filterlow:null_filterhigh:null_append:null_if1:null_if2 Rewrite Strategy: INNERMOST ---------------------------------------- (69) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 0 true => 2 false => 1 0 => 0 ys => 1 null_qs => 0 null_filterlow => 0 null_filterhigh => 0 null_append => 0 null_length => 0 null_get => 0 null_ge => 0 null_half => 0 null_if1 => 0 null_if2 => 0 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: append(z, z') -{ 1 }-> 1 :|: z' = 1, z = 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append(z, z') -{ 1 }-> 1 + x + append(xs, 1) :|: z = 1 + x + xs, xs >= 0, x >= 0, z' = 1 filterhigh(z, z') -{ 1 }-> if2(ge(x, n), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 filterhigh(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterhigh(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 filterlow(z, z') -{ 1 }-> if1(ge(n, x), n, x, xs) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 filterlow(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 filterlow(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 ge(z, z') -{ 1 }-> ge(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x ge(z, z') -{ 1 }-> 2 :|: x >= 0, z = x, z' = 0 ge(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0 ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 get(z, z') -{ 1 }-> x :|: n >= 0, z = n, x >= 0, z' = 1 + x + 0 get(z, z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 0 get(z, z') -{ 1 }-> get(n, 1 + y + xs) :|: n >= 0, xs >= 0, z' = 1 + x + (1 + y + xs), x >= 0, y >= 0, z = 1 + n get(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 get(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 half(z) -{ 1 }-> 0 :|: z = 0 half(z) -{ 1 }-> 0 :|: z = 1 + 0 half(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x) if1(z, z', z'', z1) -{ 1 }-> filterlow(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs if1(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if1(z, z', z'', z1) -{ 1 }-> 1 + x + filterlow(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs if2(z, z', z'', z1) -{ 1 }-> filterhigh(n, xs) :|: z = 2, n >= 0, xs >= 0, x >= 0, z' = n, z'' = x, z1 = xs if2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 if2(z, z', z'', z1) -{ 1 }-> 1 + x + filterhigh(n, xs) :|: n >= 0, xs >= 0, z = 1, x >= 0, z' = n, z'' = x, z1 = xs length(z) -{ 1 }-> 0 :|: z = 0 length(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 length(z) -{ 1 }-> 1 + length(xs) :|: z = 1 + x + xs, xs >= 0, x >= 0 qs(z, z') -{ 1 }-> append(qs(half(n), filterlow(get(n, 1 + x + xs), 1 + x + xs)), 1 + get(n, 1 + x + xs) + qs(half(n), filterhigh(get(n, 1 + x + xs), 1 + x + xs))) :|: n >= 0, xs >= 0, z = n, z' = 1 + x + xs, x >= 0 qs(z, z') -{ 1 }-> 0 :|: n >= 0, z = n, z' = 0 qs(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 qsort(z) -{ 1 }-> qs(half(length(xs)), xs) :|: xs >= 0, z = xs Only complete derivations are relevant for the runtime complexity. ---------------------------------------- (71) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (72) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) QS(z0, nil) -> c1 QS(z0, cons(z1, z2)) -> c2(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(z0, nil) -> c7 FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c11 FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(z0, 0) -> c15 GE(0, s(z0)) -> c16 GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(nil, ys) -> c18 APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(nil) -> c20 LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(0) -> c22 HALF(s(0)) -> c23 HALF(s(s(z0))) -> c24(HALF(z0)) GET(z0, nil) -> c25 GET(z0, cons(z1, nil)) -> c26 GET(0, cons(z0, cons(z1, z2))) -> c27 GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) S tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) QS(z0, nil) -> c1 QS(z0, cons(z1, z2)) -> c2(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(z0, nil) -> c7 FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, nil) -> c11 FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(z0, 0) -> c15 GE(0, s(z0)) -> c16 GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(nil, ys) -> c18 APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(nil) -> c20 LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(0) -> c22 HALF(s(0)) -> c23 HALF(s(s(z0))) -> c24(HALF(z0)) GET(z0, nil) -> c25 GET(z0, cons(z1, nil)) -> c26 GET(0, cons(z0, cons(z1, z2))) -> c27 GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) K tuples:none Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: QSORT_1, QS_2, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2 Compound Symbols: c_3, c1, c2_3, c3_4, c4_2, c5_3, c6_4, c7, c8_2, c9_1, c10_1, c11, c12_2, c13_1, c14_1, c15, c16, c17_1, c18, c19_1, c20, c21_1, c22, c23, c24_1, c25, c26, c27, c28_1 ---------------------------------------- (73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 12 trailing nodes: LENGTH(nil) -> c20 QS(z0, nil) -> c1 FILTERHIGH(z0, nil) -> c11 GE(0, s(z0)) -> c16 FILTERLOW(z0, nil) -> c7 GE(z0, 0) -> c15 GET(0, cons(z0, cons(z1, z2))) -> c27 GET(z0, nil) -> c25 HALF(s(0)) -> c23 APPEND(nil, ys) -> c18 HALF(0) -> c22 GET(z0, cons(z1, nil)) -> c26 ---------------------------------------- (74) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) QS(z0, cons(z1, z2)) -> c2(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) S tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) QS(z0, cons(z1, z2)) -> c2(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(APPEND(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))), QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) K tuples:none Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: QSORT_1, QS_2, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2 Compound Symbols: c_3, c2_3, c3_4, c4_2, c5_3, c6_4, c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1 ---------------------------------------- (75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (76) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) S tuples: QSORT(z0) -> c(QS(half(length(z0)), z0), HALF(length(z0)), LENGTH(z0)) FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) K tuples:none Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: QSORT_1, FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2 Compound Symbols: c_3, c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3 ---------------------------------------- (77) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (78) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) QSORT(z0) -> c1(LENGTH(z0)) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) QSORT(z0) -> c1(LENGTH(z0)) K tuples:none Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1 Compound Symbols: c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1 ---------------------------------------- (79) CdtLeafRemovalProof (ComplexityIfPolyImplication) Removed 1 leading nodes: QSORT(z0) -> c1(LENGTH(z0)) ---------------------------------------- (80) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) K tuples:none Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1 Compound Symbols: c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1 ---------------------------------------- (81) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) ---------------------------------------- (82) Obligation: Complexity Dependency Tuples Problem Rules: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) get(z0, nil) -> 0 get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) Defined Rule Symbols: qsort_1, qs_2, filterlow_2, if1_4, filterhigh_2, if2_4, ge_2, append_2, length_1, half_1, get_2 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1 Compound Symbols: c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1 ---------------------------------------- (83) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: qsort(z0) -> qs(half(length(z0)), z0) qs(z0, nil) -> nil qs(z0, cons(z1, z2)) -> append(qs(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), cons(get(z0, cons(z1, z2)), qs(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))))) append(nil, ys) -> ys append(cons(z0, z1), ys) -> cons(z0, append(z1, ys)) get(z0, nil) -> 0 ---------------------------------------- (84) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1 Compound Symbols: c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1 ---------------------------------------- (85) 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) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) We considered the (Usable) Rules:none And the Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_1 + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = x_1 + x_2 POL(get(x_1, x_2)) = x_1 POL(half(x_1)) = [1] 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(length(x_1)) = [1] + x_1 POL(nil) = [1] POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (86) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) S tuples: FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: FILTERLOW_2, IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1 Compound Symbols: c8_2, c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1 ---------------------------------------- (87) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(z0, cons(z1, z2)) -> c8(IF1(ge(z0, z1), z0, z1, z2), GE(z0, z1)) by FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) ---------------------------------------- (88) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2), GE(z0, 0)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2), GE(0, s(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2 Compound Symbols: c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2 ---------------------------------------- (89) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (90) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, FILTERHIGH_2, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2 Compound Symbols: c9_1, c10_1, c12_2, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1 ---------------------------------------- (91) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(z0, cons(z1, z2)) -> c12(IF2(ge(z1, z0), z0, z1, z2), GE(z1, z0)) by FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) ---------------------------------------- (92) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2), GE(z0, 0)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2), GE(0, s(z0))) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2 ---------------------------------------- (93) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (94) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c2_2, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1 ---------------------------------------- (95) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, z2)) -> c2(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) by QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(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) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2 ---------------------------------------- (97) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (98) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1 ---------------------------------------- (99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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] POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = x_2 POL(get(x_1, x_2)) = x_1 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = [1] POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (100) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1 ---------------------------------------- (101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = x_1 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = [1] POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (102) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c3_3, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1 ---------------------------------------- (103) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, z2)) -> c3(QS(half(z0), filterlow(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERLOW(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) by QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, 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) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3 ---------------------------------------- (105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (106) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2 ---------------------------------------- (107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (108) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c5_2, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2 ---------------------------------------- (109) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, z2)) -> c5(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), HALF(z0)) by QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) ---------------------------------------- (110) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2 ---------------------------------------- (111) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (112) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1 ---------------------------------------- (113) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (114) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1 ---------------------------------------- (115) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (116) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c6_3, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1 ---------------------------------------- (117) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, z2)) -> c6(QS(half(z0), filterhigh(get(z0, cons(z1, z2)), cons(z1, z2))), FILTERHIGH(get(z0, cons(z1, z2)), cons(z1, z2)), GET(z0, cons(z1, z2))) by QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) ---------------------------------------- (118) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3 ---------------------------------------- (119) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 3 trailing tuple parts ---------------------------------------- (120) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2 ---------------------------------------- (121) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (122) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2 ---------------------------------------- (123) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(z0) -> c1(QS(half(length(z0)), z0)) by QSORT(nil) -> c1(QS(half(0), nil)) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, 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) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(nil) -> c1(QS(half(0), nil)) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(QS(half(length(z0)), z0)) QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2 ---------------------------------------- (125) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QSORT(nil) -> c1(QS(half(0), nil)) ---------------------------------------- (126) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QSORT(z0) -> c1(HALF(length(z0))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, QSORT_1, FILTERLOW_2, FILTERHIGH_2 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c1_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2 ---------------------------------------- (127) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(z0) -> c1(HALF(length(z0))) by QSORT(nil) -> c1(HALF(0)) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) ---------------------------------------- (128) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(nil) -> c1(HALF(0)) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: QSORT(z0) -> c1(HALF(length(z0))) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1 ---------------------------------------- (129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QSORT(nil) -> c1(HALF(0)) ---------------------------------------- (130) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_2, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1 ---------------------------------------- (131) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERLOW(s(z0), cons(s(z1), x2)) -> c8(IF1(ge(z0, z1), s(z0), s(z1), x2), GE(s(z0), s(z1))) by FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) ---------------------------------------- (132) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_2, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2 ---------------------------------------- (133) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace FILTERHIGH(s(z1), cons(s(z0), x2)) -> c12(IF2(ge(z0, z1), s(z1), s(z0), x2), GE(s(z0), s(z1))) by FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) ---------------------------------------- (134) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (135) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(x0, cons(z1, z2)) -> c2(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) by QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) ---------------------------------------- (136) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (137) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (138) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (139) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (140) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (141) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) 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) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 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(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_2 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(ge(x_1, x_2)) = [1] POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (142) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (143) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, nil)) -> c2(QS(half(z0), filterlow(z1, cons(z1, nil))), HALF(z0)) by QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) ---------------------------------------- (144) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (146) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (147) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (148) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (149) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (150) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (151) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) by QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) ---------------------------------------- (152) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (154) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (155) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (156) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_2, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2 ---------------------------------------- (157) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) by QS(s(s(x0)), cons(z1, z2)) -> c2(QS(s(half(x0)), if1(ge(get(s(s(x0)), cons(z1, z2)), z1), get(s(s(x0)), cons(z1, z2)), z1, z2)), HALF(s(s(x0)))) QS(s(s(x0)), cons(z1, nil)) -> c2(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), HALF(s(s(x0)))) QS(s(s(x0)), cons(z1, cons(z2, z3))) -> c2(QS(s(half(x0)), filterlow(get(s(x0), cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(s(x0)))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) ---------------------------------------- (158) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2 ---------------------------------------- (159) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2))))) by QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) ---------------------------------------- (160) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2 ---------------------------------------- (161) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(x1, x2)) -> c2(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2)))) by QS(0, cons(z1, z2)) -> c2(QS(0, if1(ge(get(0, cons(z1, z2)), z1), get(0, cons(z1, z2)), z1, z2))) QS(0, cons(z1, nil)) -> c2(QS(0, filterlow(z1, cons(z1, nil)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(0, filterlow(z0, cons(z0, cons(z1, z2))))) ---------------------------------------- (162) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c2_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2 ---------------------------------------- (163) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) by QS(s(0), cons(z1, z2)) -> c2(QS(0, if1(ge(get(s(0), cons(z1, z2)), z1), get(s(0), cons(z1, z2)), z1, z2))) QS(s(0), cons(z1, nil)) -> c2(QS(0, filterlow(z1, cons(z1, nil)))) QS(s(0), cons(z1, cons(z2, z3))) -> c2(QS(0, filterlow(get(0, cons(z2, z3)), cons(z1, cons(z2, z3))))) ---------------------------------------- (164) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (165) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(x0, cons(z1, z2)) -> c3(QS(half(x0), if1(ge(get(x0, cons(z1, z2)), z1), get(x0, cons(z1, z2)), z1, z2)), FILTERLOW(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) by QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) ---------------------------------------- (166) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (167) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (168) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (169) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) 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) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 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(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_2 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(ge(x_1, x_2)) = [1] POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = x_1 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (170) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (171) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (172) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (173) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), filterlow(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) by QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c3(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) ---------------------------------------- (174) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c3(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c3(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1 ---------------------------------------- (175) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (176) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (177) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 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(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (178) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (179) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (180) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (182) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (183) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) by QS(s(0), cons(z1, z2)) -> c3(QS(0, if1(ge(get(s(0), cons(z1, z2)), z1), get(s(0), cons(z1, z2)), z1, z2)), FILTERLOW(get(s(0), cons(z1, z2)), cons(z1, z2)), GET(s(0), cons(z1, z2))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil)), GET(s(0), cons(z1, nil))) QS(s(0), cons(z1, cons(z2, z3))) -> c3(QS(0, filterlow(get(0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(0), cons(z1, cons(z2, z3)))) ---------------------------------------- (184) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil)), GET(s(0), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (185) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (186) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_3, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c_1 ---------------------------------------- (187) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) by QS(s(s(x0)), cons(z1, z2)) -> c3(QS(s(half(x0)), if1(ge(get(s(s(x0)), cons(z1, z2)), z1), get(s(s(x0)), cons(z1, z2)), z1, z2)), FILTERLOW(get(s(s(x0)), cons(z1, z2)), cons(z1, z2)), GET(s(s(x0)), cons(z1, z2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil)), GET(s(s(x0)), cons(z1, nil))) QS(s(s(x0)), cons(z1, cons(z2, z3))) -> c3(QS(s(half(x0)), filterlow(get(s(x0), cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERLOW(get(s(s(x0)), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(s(x0)), cons(z1, cons(z2, z3)))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) ---------------------------------------- (188) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil)), GET(s(s(x0)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1 ---------------------------------------- (189) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (190) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1 ---------------------------------------- (191) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, nil)) -> c3(QS(half(z0), filterlow(z1, cons(z1, nil))), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) by QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c3(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) ---------------------------------------- (192) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c3(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1, c3_1 ---------------------------------------- (193) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 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(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (194) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c3(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1, c3_1 ---------------------------------------- (195) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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] POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (196) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c3(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), FILTERLOW(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1, c3_1 ---------------------------------------- (197) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (198) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1, c3_1 ---------------------------------------- (199) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) by QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) ---------------------------------------- (200) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c3_2, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c_1, c3_1 ---------------------------------------- (201) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(x1, x2)) -> c3(QS(0, filterlow(get(0, cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) by QS(0, cons(z1, z2)) -> c3(QS(0, if1(ge(get(0, cons(z1, z2)), z1), get(0, cons(z1, z2)), z1, z2)), FILTERLOW(get(0, cons(z1, z2)), cons(z1, z2))) QS(0, cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(0, filterlow(z0, cons(z0, cons(z1, z2)))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) ---------------------------------------- (202) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (203) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(x0, cons(z1, z2)) -> c5(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), HALF(x0)) by QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) ---------------------------------------- (204) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), HALF(0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), HALF(0)) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), HALF(s(0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (205) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 4 trailing tuple parts ---------------------------------------- (206) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (207) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (208) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (209) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, nil)) -> c5(QS(half(z0), filterhigh(z1, cons(z1, nil))), HALF(z0)) by QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) ---------------------------------------- (210) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(0)) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil))), HALF(s(0))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (211) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (212) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (213) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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] POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (214) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (215) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (216) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (217) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(z0))) by QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) ---------------------------------------- (218) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(0))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (219) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 2 trailing tuple parts ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (221) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (222) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_2, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1 ---------------------------------------- (223) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) by QS(s(s(x0)), cons(z1, z2)) -> c5(QS(s(half(x0)), if2(ge(z1, get(s(s(x0)), cons(z1, z2))), get(s(s(x0)), cons(z1, z2)), z1, z2)), HALF(s(s(x0)))) QS(s(s(x0)), cons(z1, nil)) -> c5(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), HALF(s(s(x0)))) QS(s(s(x0)), cons(z1, cons(z2, z3))) -> c5(QS(s(half(x0)), filterhigh(get(s(x0), cons(z2, z3)), cons(z1, cons(z2, z3)))), HALF(s(s(x0)))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) ---------------------------------------- (224) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2 ---------------------------------------- (225) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2))))) by QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) ---------------------------------------- (226) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2 ---------------------------------------- (227) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(x1, x2)) -> c5(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2)))) by QS(0, cons(z1, z2)) -> c5(QS(0, if2(ge(z1, get(0, cons(z1, z2))), get(0, cons(z1, z2)), z1, z2))) QS(0, cons(z1, nil)) -> c5(QS(0, filterhigh(z1, cons(z1, nil)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(0, filterhigh(z0, cons(z0, cons(z1, z2))))) ---------------------------------------- (228) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c5_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2 ---------------------------------------- (229) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) by QS(s(0), cons(z1, z2)) -> c5(QS(0, if2(ge(z1, get(s(0), cons(z1, z2))), get(s(0), cons(z1, z2)), z1, z2))) QS(s(0), cons(z1, nil)) -> c5(QS(0, filterhigh(z1, cons(z1, nil)))) QS(s(0), cons(z1, cons(z2, z3))) -> c5(QS(0, filterhigh(get(0, cons(z2, z3)), cons(z1, cons(z2, z3))))) ---------------------------------------- (230) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (231) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(x0, cons(z1, z2)) -> c6(QS(half(x0), if2(ge(z1, get(x0, cons(z1, z2))), get(x0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(x0, cons(z1, z2)), cons(z1, z2)), GET(x0, cons(z1, z2))) by QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) ---------------------------------------- (232) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil)), GET(z0, cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2))), GET(0, cons(z0, cons(z1, z2)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2)), GET(0, cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (233) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (234) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (235) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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] POL(false) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (236) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (237) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = [1] + x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (238) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (239) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), filterhigh(get(z0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) by QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c6(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) ---------------------------------------- (240) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c6(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c6(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(x0), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1 ---------------------------------------- (241) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID)) Split RHS of tuples not part of any SCC ---------------------------------------- (242) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (243) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 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(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (244) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (245) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (246) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (247) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) 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(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (248) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (249) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) by QS(s(0), cons(z1, z2)) -> c6(QS(0, if2(ge(z1, get(s(0), cons(z1, z2))), get(s(0), cons(z1, z2)), z1, z2)), FILTERHIGH(get(s(0), cons(z1, z2)), cons(z1, z2)), GET(s(0), cons(z1, z2))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil)), GET(s(0), cons(z1, nil))) QS(s(0), cons(z1, cons(z2, z3))) -> c6(QS(0, filterhigh(get(0, cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(0), cons(z1, cons(z2, z3)))) ---------------------------------------- (250) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil)), GET(s(0), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (251) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (252) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_3, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c7_1 ---------------------------------------- (253) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) by QS(s(s(x0)), cons(z1, z2)) -> c6(QS(s(half(x0)), if2(ge(z1, get(s(s(x0)), cons(z1, z2))), get(s(s(x0)), cons(z1, z2)), z1, z2)), FILTERHIGH(get(s(s(x0)), cons(z1, z2)), cons(z1, z2)), GET(s(s(x0)), cons(z1, z2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil)), GET(s(s(x0)), cons(z1, nil))) QS(s(s(x0)), cons(z1, cons(z2, z3))) -> c6(QS(s(half(x0)), filterhigh(get(s(x0), cons(z2, z3)), cons(z1, cons(z2, z3)))), FILTERHIGH(get(s(s(x0)), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(s(x0)), cons(z1, cons(z2, z3)))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) ---------------------------------------- (254) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil)), GET(s(s(x0)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1 ---------------------------------------- (255) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing tuple parts ---------------------------------------- (256) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1 ---------------------------------------- (257) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(z0, cons(z1, nil)) -> c6(QS(half(z0), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) by QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c6(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) ---------------------------------------- (258) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c6(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1, c6_1 ---------------------------------------- (259) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) We considered the (Usable) Rules:none And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 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(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = [1] POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = [1] POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = 0 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (260) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(0), cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c6(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1, c6_1 ---------------------------------------- (261) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = 0 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = x_2 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = 0 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = [1] + x_1 POL(nil) = 0 POL(s(x_1)) = [1] POL(true) = 0 POL(ys) = 0 ---------------------------------------- (262) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(s(s(z0)), cons(x1, nil)) -> c6(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(s(s(z0)), cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1, c6_1 ---------------------------------------- (263) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) We considered the (Usable) Rules: half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) half(0) -> 0 length(cons(z0, z1)) -> s(length(z1)) length(nil) -> 0 And the Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) The order we found is given by the following interpretation: Polynomial interpretation : POL(0) = 0 POL(APPEND(x_1, x_2)) = x_1 POL(FILTERHIGH(x_1, x_2)) = 0 POL(FILTERLOW(x_1, x_2)) = 0 POL(GE(x_1, x_2)) = 0 POL(GET(x_1, x_2)) = 0 POL(HALF(x_1)) = 0 POL(IF1(x_1, x_2, x_3, x_4)) = 0 POL(IF2(x_1, x_2, x_3, x_4)) = 0 POL(LENGTH(x_1)) = x_1 POL(QS(x_1, x_2)) = x_1 POL(QSORT(x_1)) = [1] + x_1 POL(c(x_1)) = x_1 POL(c1(x_1)) = x_1 POL(c10(x_1)) = x_1 POL(c12(x_1)) = x_1 POL(c12(x_1, x_2)) = x_1 + x_2 POL(c13(x_1)) = x_1 POL(c14(x_1)) = x_1 POL(c17(x_1)) = x_1 POL(c19(x_1)) = x_1 POL(c2(x_1)) = x_1 POL(c2(x_1, x_2)) = x_1 + x_2 POL(c21(x_1)) = x_1 POL(c24(x_1)) = x_1 POL(c28(x_1)) = x_1 POL(c3(x_1)) = x_1 POL(c3(x_1, x_2)) = x_1 + x_2 POL(c3(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c4(x_1)) = x_1 POL(c5(x_1)) = x_1 POL(c5(x_1, x_2)) = x_1 + x_2 POL(c6(x_1)) = x_1 POL(c6(x_1, x_2)) = x_1 + x_2 POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(c7(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) = 0 POL(filterhigh(x_1, x_2)) = [1] + x_2 POL(filterlow(x_1, x_2)) = [1] POL(ge(x_1, x_2)) = 0 POL(get(x_1, x_2)) = 0 POL(half(x_1)) = x_1 POL(if1(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(if2(x_1, x_2, x_3, x_4)) = [1] + x_3 + x_4 POL(length(x_1)) = x_1 POL(nil) = 0 POL(s(x_1)) = [1] + x_1 POL(true) = 0 POL(ys) = 0 ---------------------------------------- (264) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1, c6_1 ---------------------------------------- (265) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) by QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) ---------------------------------------- (266) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c6_2, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c7_1, c6_1 ---------------------------------------- (267) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QS(0, cons(x1, x2)) -> c6(QS(0, filterhigh(get(0, cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) by QS(0, cons(z1, z2)) -> c6(QS(0, if2(ge(z1, get(0, cons(z1, z2))), get(0, cons(z1, z2)), z1, z2)), FILTERHIGH(get(0, cons(z1, z2)), cons(z1, z2))) QS(0, cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(0, filterhigh(z0, cons(z0, cons(z1, z2)))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) ---------------------------------------- (268) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c6_2, c7_1, c6_1 ---------------------------------------- (269) CdtNarrowingProof (BOTH BOUNDS(ID, ID)) Use narrowing to replace QSORT(cons(z0, z1)) -> c1(HALF(s(length(z1)))) by QSORT(cons(x0, nil)) -> c1(HALF(s(0))) QSORT(cons(x0, cons(z0, z1))) -> c1(HALF(s(s(length(z1))))) ---------------------------------------- (270) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, nil)) -> c1(HALF(s(0))) QSORT(cons(x0, cons(z0, z1))) -> c1(HALF(s(s(length(z1))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, filterlow(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), filterlow(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2)))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, filterhigh(get(s(0), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), filterhigh(get(s(s(z0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c6_2, c7_1, c6_1 ---------------------------------------- (271) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 1 trailing nodes: QSORT(cons(x0, nil)) -> c1(HALF(s(0))) ---------------------------------------- (272) Obligation: Complexity Dependency Tuples Problem Rules: ge(z0, 0) -> true ge(0, s(z0)) -> false ge(s(z0), s(z1)) -> ge(z0, z1) half(0) -> 0 half(s(0)) -> 0 half(s(s(z0))) -> s(half(z0)) filterlow(z0, cons(z1, z2)) -> if1(ge(z0, z1), z0, z1, z2) filterlow(z0, nil) -> nil get(z0, cons(z1, nil)) -> z1 get(0, cons(z0, cons(z1, z2))) -> z0 get(s(z0), cons(z1, cons(z2, z3))) -> get(z0, cons(z2, z3)) 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)) length(nil) -> 0 length(cons(z0, z1)) -> s(length(z1)) Tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) QSORT(cons(z0, z1)) -> c1(QS(half(s(length(z1))), cons(z0, z1))) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c2(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c2(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c2(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c3(QS(s(0), filterlow(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c3(QS(s(s(half(z0))), filterlow(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERLOW(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(s(0)), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), HALF(s(s(0)))) QS(s(s(s(0))), cons(x1, x2)) -> c5(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(0))))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c5(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), HALF(s(s(s(s(z0)))))) QS(s(s(x0)), cons(x1, x2)) -> c5(HALF(s(s(x0)))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(0)), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(0)), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(0)), cons(x1, x2))) QS(s(s(s(0))), cons(x1, x2)) -> c6(QS(s(0), filterhigh(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(0))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(0))), cons(x1, x2))) QS(s(s(s(s(z0)))), cons(x1, x2)) -> c6(QS(s(s(half(z0))), filterhigh(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2))), FILTERHIGH(get(s(s(s(s(z0)))), cons(x1, x2)), cons(x1, x2)), GET(s(s(s(s(z0)))), cons(x1, x2))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QSORT(cons(x0, cons(z0, z1))) -> c1(HALF(s(s(length(z1))))) S tuples: IF1(true, z0, z1, z2) -> c9(FILTERLOW(z0, z2)) IF1(false, z0, z1, z2) -> c10(FILTERLOW(z0, z2)) IF2(true, z0, z1, z2) -> c13(FILTERHIGH(z0, z2)) IF2(false, z0, z1, z2) -> c14(FILTERHIGH(z0, z2)) GE(s(z0), s(z1)) -> c17(GE(z0, z1)) HALF(s(s(z0))) -> c24(HALF(z0)) GET(s(z0), cons(z1, cons(z2, z3))) -> c28(GET(z0, cons(z2, z3))) FILTERLOW(z0, cons(0, x2)) -> c8(IF1(true, z0, 0, x2)) FILTERLOW(0, cons(s(z0), x2)) -> c8(IF1(false, 0, s(z0), x2)) FILTERHIGH(0, cons(z0, x2)) -> c12(IF2(true, 0, z0, x2)) FILTERHIGH(s(z0), cons(0, x2)) -> c12(IF2(false, s(z0), 0, x2)) FILTERLOW(s(z0), cons(s(0), x2)) -> c8(IF1(true, s(z0), s(0), x2), GE(s(z0), s(0))) FILTERLOW(s(0), cons(s(s(z0)), x2)) -> c8(IF1(false, s(0), s(s(z0)), x2), GE(s(0), s(s(z0)))) FILTERLOW(s(s(z0)), cons(s(s(z1)), x2)) -> c8(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)) -> c8(GE(s(x0), s(x1))) FILTERHIGH(s(0), cons(s(z0), x2)) -> c12(IF2(true, s(0), s(z0), x2), GE(s(z0), s(0))) FILTERHIGH(s(s(z0)), cons(s(0), x2)) -> c12(IF2(false, s(s(z0)), s(0), x2), GE(s(0), s(s(z0)))) FILTERHIGH(s(s(z1)), cons(s(s(z0)), x2)) -> c12(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)) -> c12(GE(s(x1), s(x0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c2(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c2(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c2(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c2(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c2(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c2(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c2(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c2(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c2(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c2(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c2(QS(0, filterlow(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(s(z0), cons(z1, cons(z2, z3))), z1), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c3(QS(half(s(z0)), if1(ge(get(z0, cons(z2, z3)), z1), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERLOW(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(get(z0, cons(z1, nil)), z1), z1, z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(get(0, cons(z0, cons(z1, z2))), z0), z0, z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c3(QS(half(z0), if1(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERLOW(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c3(QS(half(0), if1(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERLOW(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c3(QS(0, if1(ge(get(0, cons(x1, x2)), x1), get(0, cons(x1, x2)), x1, x2)), FILTERLOW(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c3(QS(half(s(x0)), if1(ge(get(x0, cons(x2, x3)), z1), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERLOW(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c3(QS(half(s(z0)), filterlow(z1, cons(x1, cons(z1, nil)))), FILTERLOW(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c3(QS(half(s(0)), filterlow(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERLOW(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c3(QS(half(s(s(z0))), filterlow(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERLOW(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c3(QS(half(x0), if1(ge(z0, z0), z0, z0, nil)), FILTERLOW(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c3(QS(0, filterlow(x1, cons(x1, nil))), FILTERLOW(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c3(QS(half(0), if1(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERLOW(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c3(QS(0, filterlow(x0, cons(x0, cons(x1, x2)))), FILTERLOW(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), HALF(s(z0))) QS(z0, cons(z1, nil)) -> c5(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), HALF(z0)) QS(s(z0), cons(z1, cons(z2, z3))) -> c5(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), HALF(s(z0))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2)))) QS(0, cons(z0, cons(z1, z2))) -> c5(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2))) QS(x0, cons(z0, nil)) -> c5(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), HALF(x0)) QS(0, cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(x0), cons(z1, cons(x2, x3))) -> c5(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), HALF(s(x0))) QS(s(z0), cons(x1, cons(z1, nil))) -> c5(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), HALF(s(z0))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c5(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c5(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2)))))) QS(0, cons(z0, cons(x1, x2))) -> c5(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c5(QS(0, filterhigh(x0, cons(x0, cons(x1, x2))))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(s(z0), cons(z1, cons(z2, z3)))), get(z0, cons(z2, z3)), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(s(z0), cons(z1, cons(z2, z3))) -> c6(QS(half(s(z0)), if2(ge(z1, get(z0, cons(z2, z3))), get(s(z0), cons(z1, cons(z2, z3))), z1, cons(z2, z3))), FILTERHIGH(get(s(z0), cons(z1, cons(z2, z3))), cons(z1, cons(z2, z3))), GET(s(z0), cons(z1, cons(z2, z3)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, get(z0, cons(z1, nil))), z1, z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, get(0, cons(z0, cons(z1, z2)))), z0, z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(z0, cons(z1, nil)) -> c6(QS(half(z0), if2(ge(z1, z1), get(z0, cons(z1, nil)), z1, nil)), FILTERHIGH(get(z0, cons(z1, nil)), cons(z1, nil))) QS(0, cons(z0, cons(z1, z2))) -> c6(QS(half(0), if2(ge(z0, z0), get(0, cons(z0, cons(z1, z2))), z0, cons(z1, z2))), FILTERHIGH(get(0, cons(z0, cons(z1, z2))), cons(z0, cons(z1, z2)))) QS(0, cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(0, cons(x1, x2))), get(0, cons(x1, x2)), x1, x2)), FILTERHIGH(get(0, cons(x1, x2)), cons(x1, x2))) QS(s(x0), cons(z1, cons(x2, x3))) -> c6(QS(half(s(x0)), if2(ge(z1, get(x0, cons(x2, x3))), get(x0, cons(x2, x3)), z1, cons(x2, x3))), FILTERHIGH(get(s(x0), cons(z1, cons(x2, x3))), cons(z1, cons(x2, x3))), GET(s(x0), cons(z1, cons(x2, x3)))) QS(s(z0), cons(x1, cons(z1, nil))) -> c6(QS(half(s(z0)), filterhigh(z1, cons(x1, cons(z1, nil)))), FILTERHIGH(get(s(z0), cons(x1, cons(z1, nil))), cons(x1, cons(z1, nil))), GET(s(z0), cons(x1, cons(z1, nil)))) QS(s(0), cons(x1, cons(z0, cons(z1, z2)))) -> c6(QS(half(s(0)), filterhigh(z0, cons(x1, cons(z0, cons(z1, z2))))), FILTERHIGH(get(s(0), cons(x1, cons(z0, cons(z1, z2)))), cons(x1, cons(z0, cons(z1, z2)))), GET(s(0), cons(x1, cons(z0, cons(z1, z2))))) QS(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))) -> c6(QS(half(s(s(z0))), filterhigh(get(z0, cons(z2, z3)), cons(x1, cons(z1, cons(z2, z3))))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(z1, cons(z2, z3)))), cons(x1, cons(z1, cons(z2, z3)))), GET(s(s(z0)), cons(x1, cons(z1, cons(z2, z3))))) QS(x0, cons(z0, nil)) -> c6(QS(half(x0), if2(ge(z0, z0), z0, z0, nil)), FILTERHIGH(get(x0, cons(z0, nil)), cons(z0, nil))) QS(0, cons(x1, nil)) -> c6(QS(0, filterhigh(x1, cons(x1, nil))), FILTERHIGH(get(0, cons(x1, nil)), cons(x1, nil))) QS(0, cons(z0, cons(x1, x2))) -> c6(QS(half(0), if2(ge(z0, z0), z0, z0, cons(x1, x2))), FILTERHIGH(get(0, cons(z0, cons(x1, x2))), cons(z0, cons(x1, x2)))) QS(0, cons(x0, cons(x1, x2))) -> c6(QS(0, filterhigh(x0, cons(x0, cons(x1, x2)))), FILTERHIGH(get(0, cons(x0, cons(x1, x2))), cons(x0, cons(x1, x2)))) K tuples: APPEND(cons(z0, z1), ys) -> c19(APPEND(z1, ys)) LENGTH(cons(z0, z1)) -> c21(LENGTH(z1)) QS(z0, cons(z1, z2)) -> c4(GET(z0, cons(z1, z2))) QS(s(s(z0)), cons(x1, x2)) -> c2(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c2(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2))) QS(x0, cons(0, x2)) -> c2(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), HALF(x0)) QS(s(0), cons(x1, nil)) -> c2(QS(0, filterlow(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c2(QS(s(half(z0)), filterlow(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c2(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c2(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(x0, cons(0, x2)) -> c3(QS(half(x0), if1(true, get(x0, cons(0, x2)), 0, x2)), FILTERLOW(get(x0, cons(0, x2)), cons(0, x2)), GET(x0, cons(0, x2))) QS(s(0), cons(x1, x2)) -> c3(QS(0, if1(ge(get(s(0), cons(x1, x2)), x1), get(s(0), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c3(QS(s(half(z0)), if1(ge(get(s(s(z0)), cons(x1, x2)), x1), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERLOW(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(FILTERLOW(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c3(QS(0, filterlow(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c3(QS(s(half(z0)), filterlow(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERLOW(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c3(FILTERLOW(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c3(QS(0, filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c3(QS(s(half(x0)), filterlow(z1, cons(z1, nil))), FILTERLOW(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) QS(s(s(z0)), cons(x1, x2)) -> c5(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), HALF(s(s(z0)))) QS(s(0), cons(x1, x2)) -> c5(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2))) QS(s(0), cons(x1, nil)) -> c5(QS(0, filterhigh(x1, cons(x1, nil)))) QS(s(s(z0)), cons(x1, nil)) -> c5(QS(s(half(z0)), filterhigh(x1, cons(x1, nil))), HALF(s(s(z0)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c5(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), HALF(s(s(z0)))) QS(s(0), cons(x1, cons(x2, x3))) -> c5(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3))))) QS(s(0), cons(x1, x2)) -> c6(QS(0, if2(ge(x1, get(s(0), cons(x1, x2))), get(s(0), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(0), cons(x1, x2)), cons(x1, x2)), GET(s(0), cons(x1, x2))) QS(s(s(z0)), cons(x1, x2)) -> c6(QS(s(half(z0)), if2(ge(x1, get(s(s(z0)), cons(x1, x2))), get(s(s(z0)), cons(x1, x2)), x1, x2)), FILTERHIGH(get(s(s(z0)), cons(x1, x2)), cons(x1, x2)), GET(s(s(z0)), cons(x1, x2))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(FILTERHIGH(get(s(x0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3)))) QS(s(x0), cons(x1, cons(x2, x3))) -> c7(GET(s(x0), cons(x1, cons(x2, x3)))) QS(s(0), cons(x1, cons(x2, x3))) -> c6(QS(0, filterhigh(get(0, cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(0), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(0), cons(x1, cons(x2, x3)))) QS(s(s(z0)), cons(x1, cons(x2, x3))) -> c6(QS(s(half(z0)), filterhigh(get(s(z0), cons(x2, x3)), cons(x1, cons(x2, x3)))), FILTERHIGH(get(s(s(z0)), cons(x1, cons(x2, x3))), cons(x1, cons(x2, x3))), GET(s(s(z0)), cons(x1, cons(x2, x3)))) QS(x0, cons(x1, nil)) -> c6(FILTERHIGH(get(x0, cons(x1, nil)), cons(x1, nil))) QS(s(0), cons(z1, nil)) -> c6(QS(0, filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(0), cons(z1, nil)), cons(z1, nil))) QS(s(s(x0)), cons(z1, nil)) -> c6(QS(s(half(x0)), filterhigh(z1, cons(z1, nil))), FILTERHIGH(get(s(s(x0)), cons(z1, nil)), cons(z1, nil))) Defined Rule Symbols: ge_2, half_1, filterlow_2, get_2, if1_4, filterhigh_2, if2_4, length_1 Defined Pair Symbols: IF1_4, IF2_4, GE_2, APPEND_2, LENGTH_1, HALF_1, GET_2, QS_2, FILTERLOW_2, FILTERHIGH_2, QSORT_1 Compound Symbols: c9_1, c10_1, c13_1, c14_1, c17_1, c19_1, c21_1, c24_1, c28_1, c4_1, c8_1, c12_1, c1_1, c8_2, c12_2, c2_2, c2_1, c3_3, c3_2, c_1, c3_1, c5_2, c5_1, c6_3, c6_2, c7_1, c6_1