WORST_CASE(Omega(n^1),O(n^2)) proof of input_wi57you2GL.trs # AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 263 ms] (2) CpxRelTRS (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms] (4) CdtProblem (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CdtProblem (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] (8) CdtProblem (9) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms] (10) CdtProblem (11) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms] (12) CdtProblem (13) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (14) CpxRelTRS (15) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (16) CpxWeightedTrs (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (18) CpxTypedWeightedTrs (19) CompletionProof [UPPER BOUND(ID), 0 ms] (20) CpxTypedWeightedCompleteTrs (21) NarrowingProof [BOTH BOUNDS(ID, ID), 47 ms] (22) CpxTypedWeightedCompleteTrs (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (24) CpxRNTS (25) InliningProof [UPPER BOUND(ID), 1080 ms] (26) CpxRNTS (27) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms] (28) CpxRNTS (29) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 787 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 390 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 4 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 350 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 197 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 309 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 154 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 381 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 932 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 225 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 256 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 148 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 759 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 157 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 44 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 259 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 24 ms] (102) CpxRNTS (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 517 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (108) CpxRNTS (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 198 ms] (112) CpxRNTS (113) IntTrsBoundProof [UPPER BOUND(ID), 63 ms] (114) CpxRNTS (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (116) CpxRNTS (117) IntTrsBoundProof [UPPER BOUND(ID), 271 ms] (118) CpxRNTS (119) IntTrsBoundProof [UPPER BOUND(ID), 23 ms] (120) CpxRNTS (121) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (122) CpxRNTS (123) IntTrsBoundProof [UPPER BOUND(ID), 380 ms] (124) CpxRNTS (125) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (126) CpxRNTS (127) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (128) CpxRNTS (129) IntTrsBoundProof [UPPER BOUND(ID), 509 ms] (130) CpxRNTS (131) IntTrsBoundProof [UPPER BOUND(ID), 74 ms] (132) CpxRNTS (133) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (134) CpxRNTS (135) IntTrsBoundProof [UPPER BOUND(ID), 108 ms] (136) CpxRNTS (137) IntTrsBoundProof [UPPER BOUND(ID), 53 ms] (138) CpxRNTS (139) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (140) CpxRNTS (141) IntTrsBoundProof [UPPER BOUND(ID), 158 ms] (142) CpxRNTS (143) IntTrsBoundProof [UPPER BOUND(ID), 23 ms] (144) CpxRNTS (145) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (146) CpxRNTS (147) IntTrsBoundProof [UPPER BOUND(ID), 128 ms] (148) CpxRNTS (149) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (150) CpxRNTS (151) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (152) CpxRNTS (153) IntTrsBoundProof [UPPER BOUND(ID), 390 ms] (154) CpxRNTS (155) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (156) CpxRNTS (157) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (158) CpxRNTS (159) IntTrsBoundProof [UPPER BOUND(ID), 1128 ms] (160) CpxRNTS (161) IntTrsBoundProof [UPPER BOUND(ID), 259 ms] (162) CpxRNTS (163) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (164) CpxRNTS (165) IntTrsBoundProof [UPPER BOUND(ID), 1777 ms] (166) CpxRNTS (167) IntTrsBoundProof [UPPER BOUND(ID), 362 ms] (168) CpxRNTS (169) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (170) CpxRNTS (171) IntTrsBoundProof [UPPER BOUND(ID), 228 ms] (172) CpxRNTS (173) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (174) CpxRNTS (175) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (176) CpxRNTS (177) IntTrsBoundProof [UPPER BOUND(ID), 409 ms] (178) CpxRNTS (179) IntTrsBoundProof [UPPER BOUND(ID), 209 ms] (180) CpxRNTS (181) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (182) CpxRNTS (183) IntTrsBoundProof [UPPER BOUND(ID), 3614 ms] (184) CpxRNTS (185) IntTrsBoundProof [UPPER BOUND(ID), 1324 ms] (186) CpxRNTS (187) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (188) CpxRNTS (189) IntTrsBoundProof [UPPER BOUND(ID), 86 ms] (190) CpxRNTS (191) IntTrsBoundProof [UPPER BOUND(ID), 13 ms] (192) CpxRNTS (193) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (194) CpxRNTS (195) IntTrsBoundProof [UPPER BOUND(ID), 149 ms] (196) CpxRNTS (197) IntTrsBoundProof [UPPER BOUND(ID), 42 ms] (198) CpxRNTS (199) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (200) CpxRNTS (201) IntTrsBoundProof [UPPER BOUND(ID), 2017 ms] (202) CpxRNTS (203) IntTrsBoundProof [UPPER BOUND(ID), 1006 ms] (204) CpxRNTS (205) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (206) CpxRNTS (207) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (208) CpxRNTS (209) IntTrsBoundProof [UPPER BOUND(ID), 64 ms] (210) CpxRNTS (211) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (212) CpxRNTS (213) IntTrsBoundProof [UPPER BOUND(ID), 201 ms] (214) CpxRNTS (215) IntTrsBoundProof [UPPER BOUND(ID), 67 ms] (216) CpxRNTS (217) FinalProof [FINISHED, 0 ms] (218) BOUNDS(1, n^2) (219) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 50 ms] (220) CdtProblem (221) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (222) CpxRelTRS (223) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms] (224) CpxRelTRS (225) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (226) typed CpxTrs (227) OrderProof [LOWER BOUND(ID), 35 ms] (228) typed CpxTrs (229) RewriteLemmaProof [LOWER BOUND(ID), 1851 ms] (230) typed CpxTrs (231) RewriteLemmaProof [LOWER BOUND(ID), 1593 ms] (232) typed CpxTrs (233) RewriteLemmaProof [LOWER BOUND(ID), 114 ms] (234) typed CpxTrs (235) RewriteLemmaProof [LOWER BOUND(ID), 93 ms] (236) typed CpxTrs (237) RewriteLemmaProof [LOWER BOUND(ID), 278 ms] (238) typed CpxTrs (239) RewriteLemmaProof [LOWER BOUND(ID), 152 ms] (240) typed CpxTrs (241) RewriteLemmaProof [LOWER BOUND(ID), 14.2 s] (242) BEST (243) proven lower bound (244) LowerBoundPropagationProof [FINISHED, 0 ms] (245) BOUNDS(n^1, INF) (246) typed CpxTrs (247) RewriteLemmaProof [LOWER BOUND(ID), 485 ms] (248) typed CpxTrs (249) RewriteLemmaProof [LOWER BOUND(ID), 14.4 s] (250) typed CpxTrs (251) RewriteLemmaProof [LOWER BOUND(ID), 111 ms] (252) typed CpxTrs (253) RewriteLemmaProof [LOWER BOUND(ID), 104 ms] (254) typed CpxTrs (255) RewriteLemmaProof [LOWER BOUND(ID), 14.6 s] (256) typed CpxTrs (257) RewriteLemmaProof [LOWER BOUND(ID), 500 ms] (258) typed CpxTrs (259) RewriteLemmaProof [LOWER BOUND(ID), 14.9 s] (260) typed CpxTrs (261) RewriteLemmaProof [LOWER BOUND(ID), 468 ms] (262) typed CpxTrs (263) RewriteLemmaProof [LOWER BOUND(ID), 523 ms] (264) typed CpxTrs (265) RewriteLemmaProof [LOWER BOUND(ID), 456 ms] (266) typed CpxTrs (267) RewriteLemmaProof [LOWER BOUND(ID), 3180 ms] (268) typed CpxTrs (269) RewriteLemmaProof [LOWER BOUND(ID), 470 ms] (270) typed CpxTrs (271) RewriteLemmaProof [LOWER BOUND(ID), 498 ms] (272) typed CpxTrs ---------------------------------------- (0) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) append(@l, @ys) -> append#1(@l, @ys) append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) append#1(nil, @ys) -> @ys appendD(@l, @ys) -> appendD#1(@l, @ys) appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) appendD#1(nil, @ys) -> @ys quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) quicksortD(@l) -> quicksortD#1(@l) quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys))) split(@pivot, @l) -> split#1(@l, @pivot) split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x) split#1(nil, @pivot) -> tuple#2(nil, nil) split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x) split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) splitD(@pivot, @l) -> splitD#1(@l, @pivot) splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x) splitD#1(nil, @pivot) -> tuple#2(nil, nil) splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x) splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(@x) -> quicksort(testList(#unit)) testQuicksort2(@x) -> quicksort(testList(#unit)) The (relative) TRS S consists of the following rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2). The TRS R consists of the following rules: #abs(#0) -> #0 #abs(#neg(@x)) -> #pos(@x) #abs(#pos(@x)) -> #pos(@x) #abs(#s(@x)) -> #pos(#s(@x)) #greater(@x, @y) -> #ckgt(#compare(@x, @y)) append(@l, @ys) -> append#1(@l, @ys) append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) append#1(nil, @ys) -> @ys appendD(@l, @ys) -> appendD#1(@l, @ys) appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) appendD#1(nil, @ys) -> @ys quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) quicksortD(@l) -> quicksortD#1(@l) quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys))) split(@pivot, @l) -> split#1(@l, @pivot) split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x) split#1(nil, @pivot) -> tuple#2(nil, nil) split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x) split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) splitD(@pivot, @l) -> splitD#1(@l, @pivot) splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x) splitD#1(nil, @pivot) -> tuple#2(nil, nil) splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x) splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(@x) -> quicksort(testList(#unit)) testQuicksort2(@x) -> quicksort(testList(#unit)) The (relative) TRS S consists of the following rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(@y)) -> #GT #compare(#0, #pos(@y)) -> #LT #compare(#0, #s(@y)) -> #LT #compare(#neg(@x), #0) -> #LT #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) #compare(#neg(@x), #pos(@y)) -> #LT #compare(#pos(@x), #0) -> #GT #compare(#pos(@x), #neg(@y)) -> #GT #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) #compare(#s(@x), #0) -> #GT #compare(#s(@x), #s(@y)) -> #compare(@x, @y) Rewrite Strategy: PARALLEL_INNERMOST ---------------------------------------- (3) CpxTrsToCdtProof (UPPER BOUND(ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (4) Obligation: Complexity Dependency Tuples Problem Rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Tuples: #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) S tuples: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) K tuples:none Defined Rule Symbols: #abs_1, #greater_2, append_2, append#1_2, appendD_2, appendD#1_2, quicksort_1, quicksort#1_1, quicksort#2_2, quicksortD_1, quicksortD#1_1, quicksortD#2_2, split_2, split#1_2, split#2_3, split#3_4, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, testList_1, testQuicksort_1, testQuicksort2_1, #ckgt_1, #compare_2 Defined Pair Symbols: #CKGT_1, #COMPARE_2, #ABS_1, #GREATER_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLIT#2_3, SPLIT#3_4, SPLITD_2, SPLITD#1_2, SPLITD#2_3, SPLITD#3_4, TESTLIST_1, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8_1, c9, c10, c11, c12_1, c13, c14_1, c15, c16, c17, c18, c19_2, c20_1, c21_1, c22, c23_1, c24_1, c25, c26_1, c27_2, c28, c29_2, c30_2, c31_1, c32_2, c33, c34_2, c35_2, c36_1, c37_2, c38, c39_2, c40, c41, c42_1, c43_2, c44, c45_2, c46, c47, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_2, c59_2 ---------------------------------------- (5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) Removed 36 trailing nodes: TESTLIST(z0) -> c48(#ABS(#0)) #ABS(#0) -> c15 TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) #ABS(#pos(z0)) -> c17 #COMPARE(#pos(z0), #0) -> c10 SPLIT#3(#true, z0, z1, z2) -> c41 APPENDD#1(nil, z0) -> c25 SPLIT#1(nil, z0) -> c38 TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) QUICKSORT#1(nil) -> c28 #COMPARE(#0, #pos(z0)) -> c5 TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) #COMPARE(#neg(z0), #0) -> c7 SPLIT#3(#false, z0, z1, z2) -> c40 TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) #ABS(#neg(z0)) -> c16 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#0, #neg(z0)) -> c4 APPEND#1(nil, z0) -> c22 SPLITD#3(#true, z0, z1, z2) -> c47 #COMPARE(#s(z0), #0) -> c13 TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) #COMPARE(#0, #0) -> c3 SPLITD#3(#false, z0, z1, z2) -> c46 #CKGT(#GT) -> c1 TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) #CKGT(#EQ) -> c #CKGT(#LT) -> c2 QUICKSORTD#1(nil) -> c33 #COMPARE(#neg(z0), #pos(z1)) -> c9 #ABS(#s(z0)) -> c18 #COMPARE(#0, #s(z0)) -> c6 SPLITD#1(nil, z0) -> c44 ---------------------------------------- (6) Obligation: Complexity Dependency Tuples Problem Rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) S tuples: #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) K tuples:none Defined Rule Symbols: #abs_1, #greater_2, append_2, append#1_2, appendD_2, appendD#1_2, quicksort_1, quicksort#1_1, quicksort#2_2, quicksortD_1, quicksortD#1_1, quicksortD#2_2, split_2, split#1_2, split#2_3, split#3_4, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, testList_1, testQuicksort_1, testQuicksort2_1, #ckgt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, #GREATER_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLIT#2_3, SPLITD_2, SPLITD#1_2, SPLITD#2_3, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c8_1, c12_1, c14_1, c19_2, c20_1, c21_1, c23_1, c24_1, c26_1, c27_2, c29_2, c30_2, c31_1, c32_2, c34_2, c35_2, c36_1, c37_2, c39_2, c42_1, c43_2, c45_2, c58_2, c59_2 ---------------------------------------- (7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) Removed 5 trailing tuple parts ---------------------------------------- (8) Obligation: Complexity Dependency Tuples Problem Rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) S tuples: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) K tuples:none Defined Rule Symbols: #abs_1, #greater_2, append_2, append#1_2, appendD_2, appendD#1_2, quicksort_1, quicksort#1_1, quicksort#2_2, quicksortD_1, quicksortD#1_1, quicksortD#2_2, split_2, split#1_2, split#2_3, split#3_4, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, testList_1, testQuicksort_1, testQuicksort2_1, #ckgt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLITD_2, SPLITD#1_2, #GREATER_2, SPLIT#2_3, SPLITD#2_3, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c8_1, c12_1, c14_1, c20_1, c21_1, c23_1, c24_1, c26_1, c27_2, c29_2, c30_2, c31_1, c32_2, c34_2, c35_2, c36_1, c37_2, c42_1, c43_2, c19_1, c39_1, c45_1, c58_1, c59_1 ---------------------------------------- (9) CdtKnowledgeProof (BOTH BOUNDS(ID, ID)) The following tuples could be moved from S to K by knowledge propagation: TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) ---------------------------------------- (10) Obligation: Complexity Dependency Tuples Problem Rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) S tuples: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) K tuples: TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) Defined Rule Symbols: #abs_1, #greater_2, append_2, append#1_2, appendD_2, appendD#1_2, quicksort_1, quicksort#1_1, quicksort#2_2, quicksortD_1, quicksortD#1_1, quicksortD#2_2, split_2, split#1_2, split#2_3, split#3_4, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, testList_1, testQuicksort_1, testQuicksort2_1, #ckgt_1, #compare_2 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLITD_2, SPLITD#1_2, #GREATER_2, SPLIT#2_3, SPLITD#2_3, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c8_1, c12_1, c14_1, c20_1, c21_1, c23_1, c24_1, c26_1, c27_2, c29_2, c30_2, c31_1, c32_2, c34_2, c35_2, c36_1, c37_2, c42_1, c43_2, c19_1, c39_1, c45_1, c58_1, c59_1 ---------------------------------------- (11) CdtUsableRulesProof (BOTH BOUNDS(ID, ID)) The following rules are not usable and were removed: #abs(#neg(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) ---------------------------------------- (12) Obligation: Complexity Dependency Tuples Problem Rules: split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #abs(#0) -> #0 #abs(#pos(z0)) -> #pos(z0) Tuples: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) S tuples: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) K tuples: TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) Defined Rule Symbols: split_2, split#1_2, split#2_3, split#3_4, #greater_2, #ckgt_1, #compare_2, quicksort_1, quicksort#1_1, quicksort#2_2, append_2, append#1_2, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, quicksortD_1, quicksortD#1_1, quicksortD#2_2, appendD_2, appendD#1_2, testList_1, #abs_1 Defined Pair Symbols: #COMPARE_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLITD_2, SPLITD#1_2, #GREATER_2, SPLIT#2_3, SPLITD#2_3, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c8_1, c12_1, c14_1, c20_1, c21_1, c23_1, c24_1, c26_1, c27_2, c29_2, c30_2, c31_1, c32_2, c34_2, c35_2, c36_1, c37_2, c42_1, c43_2, c19_1, c39_1, c45_1, c58_1, c59_1 ---------------------------------------- (13) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (14) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) The (relative) TRS S consists of the following rules: #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) #abs(#0) -> #0 #abs(#pos(z0)) -> #pos(z0) Rewrite Strategy: INNERMOST ---------------------------------------- (15) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (16) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2). The TRS R consists of the following rules: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) [1] APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) [1] APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) [1] APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) [1] QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) [1] QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) [1] QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) [1] QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) [1] SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) [1] SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) [1] SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) [1] SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) [1] #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) [1] SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) [1] SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) [1] #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) [0] #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) [0] #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) [0] TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) [0] TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) [0] split(z0, z1) -> split#1(z1, z0) [0] split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) [0] split#1(nil, z0) -> tuple#2(nil, nil) [0] split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) [0] split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] #greater(z0, z1) -> #ckgt(#compare(z0, z1)) [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(z0)) -> #GT [0] #compare(#0, #pos(z0)) -> #LT [0] #compare(#0, #s(z0)) -> #LT [0] #compare(#neg(z0), #0) -> #LT [0] #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) [0] #compare(#neg(z0), #pos(z1)) -> #LT [0] #compare(#pos(z0), #0) -> #GT [0] #compare(#pos(z0), #neg(z1)) -> #GT [0] #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) [0] #compare(#s(z0), #0) -> #GT [0] #compare(#s(z0), #s(z1)) -> #compare(z0, z1) [0] quicksort(z0) -> quicksort#1(z0) [0] quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) [0] quicksort#1(nil) -> nil [0] quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) [0] append(z0, z1) -> append#1(z0, z1) [0] append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) [0] append#1(nil, z0) -> z0 [0] splitD(z0, z1) -> splitD#1(z1, z0) [0] splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) [0] splitD#1(nil, z0) -> tuple#2(nil, nil) [0] splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) [0] splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] quicksortD(z0) -> quicksortD#1(z0) [0] quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) [0] quicksortD#1(nil) -> nil [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) [0] appendD(z0, z1) -> appendD#1(z0, z1) [0] appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) [0] appendD#1(nil, z0) -> z0 [0] testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [0] #abs(#0) -> #0 [0] #abs(#pos(z0)) -> #pos(z0) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (17) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (18) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) [1] APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) [1] APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) [1] APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) [1] QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) [1] QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) [1] QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) [1] QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) [1] SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) [1] SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) [1] SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) [1] SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) [1] #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) [1] SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) [1] SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) [1] #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) [0] #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) [0] #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) [0] TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) [0] TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) [0] split(z0, z1) -> split#1(z1, z0) [0] split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) [0] split#1(nil, z0) -> tuple#2(nil, nil) [0] split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) [0] split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] #greater(z0, z1) -> #ckgt(#compare(z0, z1)) [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(z0)) -> #GT [0] #compare(#0, #pos(z0)) -> #LT [0] #compare(#0, #s(z0)) -> #LT [0] #compare(#neg(z0), #0) -> #LT [0] #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) [0] #compare(#neg(z0), #pos(z1)) -> #LT [0] #compare(#pos(z0), #0) -> #GT [0] #compare(#pos(z0), #neg(z1)) -> #GT [0] #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) [0] #compare(#s(z0), #0) -> #GT [0] #compare(#s(z0), #s(z1)) -> #compare(z0, z1) [0] quicksort(z0) -> quicksort#1(z0) [0] quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) [0] quicksort#1(nil) -> nil [0] quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) [0] append(z0, z1) -> append#1(z0, z1) [0] append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) [0] append#1(nil, z0) -> z0 [0] splitD(z0, z1) -> splitD#1(z1, z0) [0] splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) [0] splitD#1(nil, z0) -> tuple#2(nil, nil) [0] splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) [0] splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] quicksortD(z0) -> quicksortD#1(z0) [0] quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) [0] quicksortD#1(nil) -> nil [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) [0] appendD(z0, z1) -> appendD#1(z0, z1) [0] appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) [0] appendD#1(nil, z0) -> z0 [0] testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [0] #abs(#0) -> #0 [0] #abs(#pos(z0)) -> #pos(z0) [0] The TRS has the following type information: APPEND :: :::nil -> :::nil -> c20 c20 :: c21 -> c20 APPEND#1 :: :::nil -> :::nil -> c21 :: :: #neg:#pos:#s:#0 -> :::nil -> :::nil c21 :: c20 -> c21 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24 c24 :: c23 -> c24 QUICKSORT :: :::nil -> c26 c26 :: c27 -> c26 QUICKSORT#1 :: :::nil -> c27 c27 :: c29:c30 -> c36 -> c27 QUICKSORT#2 :: tuple#2 -> #neg:#pos:#s:#0 -> c29:c30 split :: #neg:#pos:#s:#0 -> :::nil -> tuple#2 SPLIT :: #neg:#pos:#s:#0 -> :::nil -> c36 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32 -> c31 QUICKSORTD#1 :: :::nil -> c32 c32 :: c34:c35 -> c42 -> c32 QUICKSORTD#2 :: tuple#2 -> #neg:#pos:#s:#0 -> c34:c35 splitD :: #neg:#pos:#s:#0 -> :::nil -> tuple#2 SPLITD :: #neg:#pos:#s:#0 -> :::nil -> c42 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37 -> c36 SPLIT#1 :: :::nil -> #neg:#pos:#s:#0 -> c37 c37 :: c39 -> c36 -> c37 SPLIT#2 :: tuple#2 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c39 c42 :: c43 -> c42 SPLITD#1 :: :::nil -> #neg:#pos:#s:#0 -> c43 c43 :: c45 -> c42 -> c43 SPLITD#2 :: tuple#2 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c45 #GREATER :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c19 c19 :: c8:c12:c14 -> c19 #COMPARE :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c8:c12:c14 c39 :: c19 -> c39 c45 :: c19 -> c45 #neg :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c8 :: c8:c12:c14 -> c8:c12:c14 #pos :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c12 :: c8:c12:c14 -> c8:c12:c14 #s :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c14 :: c8:c12:c14 -> c8:c12:c14 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c59 split#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2 split#2 :: tuple#2 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2 nil :: :::nil split#3 :: #false:#true -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2 #greater :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #false:#true #false :: #false:#true #true :: #false:#true #ckgt :: #EQ:#GT:#LT -> #false:#true #compare :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #EQ:#GT:#LT #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: #neg:#pos:#s:#0 quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #neg:#pos:#s:#0 -> :::nil append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil splitD#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2 splitD#2 :: tuple#2 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2 quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #neg:#pos:#s:#0 -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil #abs :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 Rewrite Strategy: INNERMOST ---------------------------------------- (19) 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: APPEND_2 APPEND#1_2 APPENDD_2 APPENDD#1_2 QUICKSORT_1 QUICKSORT#1_1 QUICKSORT#2_2 QUICKSORTD_1 QUICKSORTD#1_1 QUICKSORTD#2_2 SPLIT_2 SPLIT#1_2 SPLITD_2 SPLITD#1_2 #GREATER_2 SPLIT#2_3 SPLITD#2_3 (c) The following functions are completely defined: #COMPARE_2 TESTQUICKSORT_1 TESTQUICKSORT2_1 split_2 split#1_2 split#2_3 split#3_4 #greater_2 #ckgt_1 #compare_2 quicksort_1 quicksort#1_1 quicksort#2_2 append_2 append#1_2 splitD_2 splitD#1_2 splitD#2_3 splitD#3_4 quicksortD_1 quicksortD#1_1 quicksortD#2_2 appendD_2 appendD#1_2 testList_1 #abs_1 Due to the following rules being added: #COMPARE(v0, v1) -> const18 [0] TESTQUICKSORT(v0) -> const19 [0] TESTQUICKSORT2(v0) -> const21 [0] split(v0, v1) -> const8 [0] split#1(v0, v1) -> const8 [0] split#2(v0, v1, v2) -> const8 [0] split#3(v0, v1, v2, v3) -> const8 [0] #greater(v0, v1) -> null_#greater [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] quicksort(v0) -> nil [0] quicksort#1(v0) -> nil [0] quicksort#2(v0, v1) -> nil [0] append(v0, v1) -> nil [0] append#1(v0, v1) -> nil [0] splitD(v0, v1) -> const8 [0] splitD#1(v0, v1) -> const8 [0] splitD#2(v0, v1, v2) -> const8 [0] splitD#3(v0, v1, v2, v3) -> const8 [0] quicksortD(v0) -> nil [0] quicksortD#1(v0) -> nil [0] quicksortD#2(v0, v1) -> nil [0] appendD(v0, v1) -> nil [0] appendD#1(v0, v1) -> nil [0] testList(v0) -> nil [0] #abs(v0) -> #0 [0] And the following fresh constants: const18, const19, const21, const8, null_#greater, null_#ckgt, null_#compare, const, const1, const2, const3, const4, const5, const6, const7, const9, const10, const11, const12, const13, const14, const15, const16, const17, const20, const22 ---------------------------------------- (20) 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: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) [1] APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) [1] APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) [1] APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) [1] QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) [1] QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) [1] QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) [1] QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) [1] SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) [1] SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) [1] SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) [1] SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) [1] #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) [1] SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) [1] SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) [1] #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) [0] #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) [0] #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) [0] TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit))) [0] TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit))) [0] split(z0, z1) -> split#1(z1, z0) [0] split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) [0] split#1(nil, z0) -> tuple#2(nil, nil) [0] split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) [0] split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] #greater(z0, z1) -> #ckgt(#compare(z0, z1)) [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(z0)) -> #GT [0] #compare(#0, #pos(z0)) -> #LT [0] #compare(#0, #s(z0)) -> #LT [0] #compare(#neg(z0), #0) -> #LT [0] #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) [0] #compare(#neg(z0), #pos(z1)) -> #LT [0] #compare(#pos(z0), #0) -> #GT [0] #compare(#pos(z0), #neg(z1)) -> #GT [0] #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) [0] #compare(#s(z0), #0) -> #GT [0] #compare(#s(z0), #s(z1)) -> #compare(z0, z1) [0] quicksort(z0) -> quicksort#1(z0) [0] quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) [0] quicksort#1(nil) -> nil [0] quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) [0] append(z0, z1) -> append#1(z0, z1) [0] append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) [0] append#1(nil, z0) -> z0 [0] splitD(z0, z1) -> splitD#1(z1, z0) [0] splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) [0] splitD#1(nil, z0) -> tuple#2(nil, nil) [0] splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) [0] splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] quicksortD(z0) -> quicksortD#1(z0) [0] quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) [0] quicksortD#1(nil) -> nil [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) [0] appendD(z0, z1) -> appendD#1(z0, z1) [0] appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) [0] appendD#1(nil, z0) -> z0 [0] testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [0] #abs(#0) -> #0 [0] #abs(#pos(z0)) -> #pos(z0) [0] #COMPARE(v0, v1) -> const18 [0] TESTQUICKSORT(v0) -> const19 [0] TESTQUICKSORT2(v0) -> const21 [0] split(v0, v1) -> const8 [0] split#1(v0, v1) -> const8 [0] split#2(v0, v1, v2) -> const8 [0] split#3(v0, v1, v2, v3) -> const8 [0] #greater(v0, v1) -> null_#greater [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] quicksort(v0) -> nil [0] quicksort#1(v0) -> nil [0] quicksort#2(v0, v1) -> nil [0] append(v0, v1) -> nil [0] append#1(v0, v1) -> nil [0] splitD(v0, v1) -> const8 [0] splitD#1(v0, v1) -> const8 [0] splitD#2(v0, v1, v2) -> const8 [0] splitD#3(v0, v1, v2, v3) -> const8 [0] quicksortD(v0) -> nil [0] quicksortD#1(v0) -> nil [0] quicksortD#2(v0, v1) -> nil [0] appendD(v0, v1) -> nil [0] appendD#1(v0, v1) -> nil [0] testList(v0) -> nil [0] #abs(v0) -> #0 [0] The TRS has the following type information: APPEND :: :::nil -> :::nil -> c20 c20 :: c21 -> c20 APPEND#1 :: :::nil -> :::nil -> c21 :: :: #neg:#pos:#s:#0 -> :::nil -> :::nil c21 :: c20 -> c21 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24 c24 :: c23 -> c24 QUICKSORT :: :::nil -> c26 c26 :: c27 -> c26 QUICKSORT#1 :: :::nil -> c27 c27 :: c29:c30 -> c36 -> c27 QUICKSORT#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> c29:c30 split :: #neg:#pos:#s:#0 -> :::nil -> tuple#2:const8 SPLIT :: #neg:#pos:#s:#0 -> :::nil -> c36 tuple#2 :: :::nil -> :::nil -> tuple#2:const8 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32 -> c31 QUICKSORTD#1 :: :::nil -> c32 c32 :: c34:c35 -> c42 -> c32 QUICKSORTD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> c34:c35 splitD :: #neg:#pos:#s:#0 -> :::nil -> tuple#2:const8 SPLITD :: #neg:#pos:#s:#0 -> :::nil -> c42 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37 -> c36 SPLIT#1 :: :::nil -> #neg:#pos:#s:#0 -> c37 c37 :: c39 -> c36 -> c37 SPLIT#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c39 c42 :: c43 -> c42 SPLITD#1 :: :::nil -> #neg:#pos:#s:#0 -> c43 c43 :: c45 -> c42 -> c43 SPLITD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c45 #GREATER :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c19 c19 :: c8:c12:c14:const18 -> c19 #COMPARE :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c8:c12:c14:const18 c39 :: c19 -> c39 c45 :: c19 -> c45 #neg :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c8 :: c8:c12:c14:const18 -> c8:c12:c14:const18 #pos :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c12 :: c8:c12:c14:const18 -> c8:c12:c14:const18 #s :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c14 :: c8:c12:c14:const18 -> c8:c12:c14:const18 TESTQUICKSORT :: a -> c58:const19 c58 :: c26 -> c58:const19 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59:const21 c59 :: c26 -> c59:const21 split#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 split#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2:const8 nil :: :::nil split#3 :: #false:#true:null_#greater:null_#ckgt -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 #greater :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #false:#true:null_#greater:null_#ckgt #false :: #false:#true:null_#greater:null_#ckgt #true :: #false:#true:null_#greater:null_#ckgt #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#greater:null_#ckgt #compare :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #EQ:#GT:#LT:null_#compare #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #neg:#pos:#s:#0 quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> :::nil append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil splitD#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 splitD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2:const8 splitD#3 :: #false:#true:null_#greater:null_#ckgt -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil #abs :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 const18 :: c8:c12:c14:const18 const19 :: c58:const19 const21 :: c59:const21 const8 :: tuple#2:const8 null_#greater :: #false:#true:null_#greater:null_#ckgt null_#ckgt :: #false:#true:null_#greater:null_#ckgt null_#compare :: #EQ:#GT:#LT:null_#compare const :: c20 const1 :: c21 const2 :: c23 const3 :: c24 const4 :: c26 const5 :: c27 const6 :: c29:c30 const7 :: c36 const9 :: c31 const10 :: c32 const11 :: c34:c35 const12 :: c42 const13 :: c37 const14 :: c39 const15 :: c43 const16 :: c45 const17 :: c19 const20 :: a const22 :: b Rewrite Strategy: INNERMOST ---------------------------------------- (21) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (22) 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: APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) [1] APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) [1] APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) [1] APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) [1] QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) [1] QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split#1(z1, z0), z0), SPLIT(z0, z1)) [1] QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(const8, z0), SPLIT(z0, z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort#1(z0), ::(z2, quicksort#1(z1))), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort#1(z0), ::(z2, nil)), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(nil, ::(z2, quicksort#1(z1))), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(nil, ::(z2, nil)), QUICKSORT(z0)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort#1(z0), ::(z2, quicksort#1(z1))), QUICKSORT(z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort#1(z0), ::(z2, nil)), QUICKSORT(z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(nil, ::(z2, quicksort#1(z1))), QUICKSORT(z1)) [1] QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(nil, ::(z2, nil)), QUICKSORT(z1)) [1] QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) [1] QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD#1(z1, z0), z0), SPLITD(z0, z1)) [1] QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(const8, z0), SPLITD(z0, z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD#1(z0), ::(z2, quicksortD#1(z1))), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD#1(z0), ::(z2, nil)), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(nil, ::(z2, quicksortD#1(z1))), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(nil, ::(z2, nil)), QUICKSORTD(z0)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD#1(z0), ::(z2, quicksortD#1(z1))), QUICKSORTD(z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD#1(z0), ::(z2, nil)), QUICKSORTD(z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(nil, ::(z2, quicksortD#1(z1))), QUICKSORTD(z1)) [1] QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(nil, ::(z2, nil)), QUICKSORTD(z1)) [1] SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) [1] SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split#1(z1, z2), z2, z0), SPLIT(z2, z1)) [1] SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(const8, z2, z0), SPLIT(z2, z1)) [1] SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) [1] SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD#1(z1, z2), z2, z0), SPLITD(z2, z1)) [1] SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(const8, z2, z0), SPLITD(z2, z1)) [1] #GREATER(z0, z1) -> c19(#COMPARE(z0, z1)) [1] SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(#GREATER(z3, z2)) [1] SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(#GREATER(z3, z2)) [1] #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) [0] #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) [0] #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) [0] TESTQUICKSORT(z0) -> c58(QUICKSORT(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))))) [0] TESTQUICKSORT(z0) -> c58(QUICKSORT(nil)) [0] TESTQUICKSORT2(z0) -> c59(QUICKSORT(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))))) [0] TESTQUICKSORT2(z0) -> c59(QUICKSORT(nil)) [0] split(z0, z1) -> split#1(z1, z0) [0] split#1(::(z0, z1), z2) -> split#2(split#1(z1, z2), z2, z0) [0] split#1(::(z0, z1), z2) -> split#2(const8, z2, z0) [0] split#1(nil, z0) -> tuple#2(nil, nil) [0] split#2(tuple#2(z0, z1), z2, z3) -> split#3(#ckgt(#compare(z3, z2)), z0, z1, z3) [0] split#2(tuple#2(z0, z1), z2, z3) -> split#3(null_#greater, z0, z1, z3) [0] split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] #greater(#0, #0) -> #ckgt(#EQ) [0] #greater(#0, #neg(z0')) -> #ckgt(#GT) [0] #greater(#0, #pos(z0'')) -> #ckgt(#LT) [0] #greater(#0, #s(z01)) -> #ckgt(#LT) [0] #greater(#neg(z02), #0) -> #ckgt(#LT) [0] #greater(#neg(z03), #neg(z1')) -> #ckgt(#compare(z1', z03)) [0] #greater(#neg(z04), #pos(z1'')) -> #ckgt(#LT) [0] #greater(#pos(z05), #0) -> #ckgt(#GT) [0] #greater(#pos(z06), #neg(z11)) -> #ckgt(#GT) [0] #greater(#pos(z07), #pos(z12)) -> #ckgt(#compare(z07, z12)) [0] #greater(#s(z08), #0) -> #ckgt(#GT) [0] #greater(#s(z09), #s(z13)) -> #ckgt(#compare(z09, z13)) [0] #greater(z0, z1) -> #ckgt(null_#compare) [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(z0)) -> #GT [0] #compare(#0, #pos(z0)) -> #LT [0] #compare(#0, #s(z0)) -> #LT [0] #compare(#neg(z0), #0) -> #LT [0] #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) [0] #compare(#neg(z0), #pos(z1)) -> #LT [0] #compare(#pos(z0), #0) -> #GT [0] #compare(#pos(z0), #neg(z1)) -> #GT [0] #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) [0] #compare(#s(z0), #0) -> #GT [0] #compare(#s(z0), #s(z1)) -> #compare(z0, z1) [0] quicksort(z0) -> quicksort#1(z0) [0] quicksort#1(::(z0, z1)) -> quicksort#2(split#1(z1, z0), z0) [0] quicksort#1(::(z0, z1)) -> quicksort#2(const8, z0) [0] quicksort#1(nil) -> nil [0] quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort#1(z0), ::(z2, quicksort#1(z1))) [0] quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort#1(z0), ::(z2, nil)) [0] quicksort#2(tuple#2(z0, z1), z2) -> append(nil, ::(z2, quicksort#1(z1))) [0] quicksort#2(tuple#2(z0, z1), z2) -> append(nil, ::(z2, nil)) [0] append(z0, z1) -> append#1(z0, z1) [0] append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) [0] append#1(nil, z0) -> z0 [0] splitD(z0, z1) -> splitD#1(z1, z0) [0] splitD#1(::(z0, z1), z2) -> splitD#2(splitD#1(z1, z2), z2, z0) [0] splitD#1(::(z0, z1), z2) -> splitD#2(const8, z2, z0) [0] splitD#1(nil, z0) -> tuple#2(nil, nil) [0] splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#ckgt(#compare(z3, z2)), z0, z1, z3) [0] splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(null_#greater, z0, z1, z3) [0] splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) [0] splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) [0] quicksortD(z0) -> quicksortD#1(z0) [0] quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD#1(z1, z0), z0) [0] quicksortD#1(::(z0, z1)) -> quicksortD#2(const8, z0) [0] quicksortD#1(nil) -> nil [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD#1(z0), ::(z2, quicksortD#1(z1))) [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD#1(z0), ::(z2, nil)) [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(nil, ::(z2, quicksortD#1(z1))) [0] quicksortD#2(tuple#2(z0, z1), z2) -> appendD(nil, ::(z2, nil)) [0] appendD(z0, z1) -> appendD#1(z0, z1) [0] appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) [0] appendD#1(nil, z0) -> z0 [0] testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [0] #abs(#0) -> #0 [0] #abs(#pos(z0)) -> #pos(z0) [0] #COMPARE(v0, v1) -> const18 [0] TESTQUICKSORT(v0) -> const19 [0] TESTQUICKSORT2(v0) -> const21 [0] split(v0, v1) -> const8 [0] split#1(v0, v1) -> const8 [0] split#2(v0, v1, v2) -> const8 [0] split#3(v0, v1, v2, v3) -> const8 [0] #greater(v0, v1) -> null_#greater [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] quicksort(v0) -> nil [0] quicksort#1(v0) -> nil [0] quicksort#2(v0, v1) -> nil [0] append(v0, v1) -> nil [0] append#1(v0, v1) -> nil [0] splitD(v0, v1) -> const8 [0] splitD#1(v0, v1) -> const8 [0] splitD#2(v0, v1, v2) -> const8 [0] splitD#3(v0, v1, v2, v3) -> const8 [0] quicksortD(v0) -> nil [0] quicksortD#1(v0) -> nil [0] quicksortD#2(v0, v1) -> nil [0] appendD(v0, v1) -> nil [0] appendD#1(v0, v1) -> nil [0] testList(v0) -> nil [0] #abs(v0) -> #0 [0] The TRS has the following type information: APPEND :: :::nil -> :::nil -> c20 c20 :: c21 -> c20 APPEND#1 :: :::nil -> :::nil -> c21 :: :: #neg:#pos:#s:#0 -> :::nil -> :::nil c21 :: c20 -> c21 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24 c24 :: c23 -> c24 QUICKSORT :: :::nil -> c26 c26 :: c27 -> c26 QUICKSORT#1 :: :::nil -> c27 c27 :: c29:c30 -> c36 -> c27 QUICKSORT#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> c29:c30 split :: #neg:#pos:#s:#0 -> :::nil -> tuple#2:const8 SPLIT :: #neg:#pos:#s:#0 -> :::nil -> c36 tuple#2 :: :::nil -> :::nil -> tuple#2:const8 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32 -> c31 QUICKSORTD#1 :: :::nil -> c32 c32 :: c34:c35 -> c42 -> c32 QUICKSORTD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> c34:c35 splitD :: #neg:#pos:#s:#0 -> :::nil -> tuple#2:const8 SPLITD :: #neg:#pos:#s:#0 -> :::nil -> c42 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37 -> c36 SPLIT#1 :: :::nil -> #neg:#pos:#s:#0 -> c37 c37 :: c39 -> c36 -> c37 SPLIT#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c39 c42 :: c43 -> c42 SPLITD#1 :: :::nil -> #neg:#pos:#s:#0 -> c43 c43 :: c45 -> c42 -> c43 SPLITD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c45 #GREATER :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c19 c19 :: c8:c12:c14:const18 -> c19 #COMPARE :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> c8:c12:c14:const18 c39 :: c19 -> c39 c45 :: c19 -> c45 #neg :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c8 :: c8:c12:c14:const18 -> c8:c12:c14:const18 #pos :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c12 :: c8:c12:c14:const18 -> c8:c12:c14:const18 #s :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 c14 :: c8:c12:c14:const18 -> c8:c12:c14:const18 TESTQUICKSORT :: a -> c58:const19 c58 :: c26 -> c58:const19 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59:const21 c59 :: c26 -> c59:const21 split#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 split#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2:const8 nil :: :::nil split#3 :: #false:#true:null_#greater:null_#ckgt -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 #greater :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #false:#true:null_#greater:null_#ckgt #false :: #false:#true:null_#greater:null_#ckgt #true :: #false:#true:null_#greater:null_#ckgt #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#greater:null_#ckgt #compare :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> #EQ:#GT:#LT:null_#compare #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: #neg:#pos:#s:#0 quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> :::nil append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil splitD#1 :: :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 splitD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 -> tuple#2:const8 splitD#3 :: #false:#true:null_#greater:null_#ckgt -> :::nil -> :::nil -> #neg:#pos:#s:#0 -> tuple#2:const8 quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2:const8 -> #neg:#pos:#s:#0 -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil #abs :: #neg:#pos:#s:#0 -> #neg:#pos:#s:#0 const18 :: c8:c12:c14:const18 const19 :: c58:const19 const21 :: c59:const21 const8 :: tuple#2:const8 null_#greater :: #false:#true:null_#greater:null_#ckgt null_#ckgt :: #false:#true:null_#greater:null_#ckgt null_#compare :: #EQ:#GT:#LT:null_#compare const :: c20 const1 :: c21 const2 :: c23 const3 :: c24 const4 :: c26 const5 :: c27 const6 :: c29:c30 const7 :: c36 const9 :: c31 const10 :: c32 const11 :: c34:c35 const12 :: c42 const13 :: c37 const14 :: c39 const15 :: c43 const16 :: c45 const17 :: c19 const20 :: a const22 :: b Rewrite Strategy: INNERMOST ---------------------------------------- (23) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: #unit => 0 nil => 0 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 #0 => 0 const18 => 0 const19 => 0 const21 => 0 const8 => 0 null_#greater => 0 null_#ckgt => 0 null_#compare => 0 const => 0 const1 => 0 const2 => 0 const3 => 0 const4 => 0 const5 => 0 const6 => 0 const7 => 0 const9 => 0 const10 => 0 const11 => 0 const12 => 0 const13 => 0 const14 => 0 const15 => 0 const16 => 0 const17 => 0 const20 => 0 const22 => 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #abs(z) -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z0 >= 0, z' = 1 + z0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z = 1 + z0, z0 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> #compare(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #greater(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #greater(z, z') -{ 0 }-> #ckgt(3) :|: z' = 1 + z0'', z0'' >= 0, z = 0 #greater(z, z') -{ 0 }-> #ckgt(3) :|: z01 >= 0, z' = 1 + z01, z = 0 #greater(z, z') -{ 0 }-> #ckgt(3) :|: z = 1 + z02, z02 >= 0, z' = 0 #greater(z, z') -{ 0 }-> #ckgt(3) :|: z04 >= 0, z' = 1 + z1'', z = 1 + z04, z1'' >= 0 #greater(z, z') -{ 0 }-> #ckgt(2) :|: z0' >= 0, z = 0, z' = 1 + z0' #greater(z, z') -{ 0 }-> #ckgt(2) :|: z05 >= 0, z = 1 + z05, z' = 0 #greater(z, z') -{ 0 }-> #ckgt(2) :|: z' = 1 + z11, z11 >= 0, z06 >= 0, z = 1 + z06 #greater(z, z') -{ 0 }-> #ckgt(2) :|: z08 >= 0, z = 1 + z08, z' = 0 #greater(z, z') -{ 0 }-> #ckgt(1) :|: z = 0, z' = 0 #greater(z, z') -{ 0 }-> #ckgt(0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z07, z12)) :|: z' = 1 + z12, z07 >= 0, z = 1 + z07, z12 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z09, z13)) :|: z' = 1 + z13, z = 1 + z09, z09 >= 0, z13 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z1', z03)) :|: z' = 1 + z1', z1' >= 0, z = 1 + z03, z03 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z0) :|: z = z0, z0 >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + 0) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + 0) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + 0) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + 0) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z0) :|: z = z0, z0 >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + 0) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + 0) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + 0) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + 0) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z2), z2, z0) + SPLIT(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z2, z0) + SPLIT(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z2), z2, z0) + SPLITD(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z2, z0) + SPLITD(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z = z0, z0 >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = z0, z0 >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z = z0, z0 >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = z0, z0 >= 0 append(z, z') -{ 0 }-> append#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 appendD(z, z') -{ 0 }-> appendD#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 appendD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 appendD#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 appendD#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort(z) -{ 0 }-> quicksort#1(z0) :|: z = z0, z0 >= 0 quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z2 + quicksort#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quicksortD(z) -{ 0 }-> quicksortD#1(z0) :|: z = z0, z0 >= 0 quicksortD(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z2 + quicksortD#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split(z, z') -{ 0 }-> split#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 split(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z2), z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 split#1(z, z') -{ 0 }-> split#2(0, z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 split#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z0 >= 0, z = 0, z' = z0 split#2(z, z', z'') -{ 0 }-> split#3(0, z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z3, z2)), z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 splitD(z, z') -{ 0 }-> splitD#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 splitD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z2), z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z0 >= 0, z = 0, z' = z0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(0, z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z3, z2)), z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 testList(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z = z0, z0 >= 0 ---------------------------------------- (25) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #abs(z) -{ 0 }-> 1 + z0 :|: z = 1 + z0, z0 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 #compare(z, z') -{ 0 }-> 3 :|: z0 >= 0, z' = 1 + z0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z = 1 + z0, z0 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> 2 :|: z0 >= 0, z' = 1 + z0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z = 1 + z0, z0 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #compare(z, z') -{ 0 }-> #compare(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #compare(z, z') -{ 0 }-> #compare(z1, z0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1 #greater(z, z') -{ 0 }-> 2 :|: z0' >= 0, z = 0, z' = 1 + z0', 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z05 >= 0, z = 1 + z05, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' = 1 + z11, z11 >= 0, z06 >= 0, z = 1 + z06, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z08 >= 0, z = 1 + z08, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' = 1 + z0'', z0'' >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z01 >= 0, z' = 1 + z01, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z = 1 + z02, z02 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z04 >= 0, z' = 1 + z1'', z = 1 + z04, z1'' >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z0' >= 0, z = 0, z' = 1 + z0', v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' = 1 + z0'', z0'' >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z01 >= 0, z' = 1 + z01, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z = 1 + z02, z02 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z04 >= 0, z' = 1 + z1'', z = 1 + z04, z1'' >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z05 >= 0, z = 1 + z05, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' = 1 + z11, z11 >= 0, z06 >= 0, z = 1 + z06, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z08 >= 0, z = 1 + z08, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z07, z12)) :|: z' = 1 + z12, z07 >= 0, z = 1 + z07, z12 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z09, z13)) :|: z' = 1 + z13, z = 1 + z09, z09 >= 0, z13 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z1', z03)) :|: z' = 1 + z1', z1' >= 0, z = 1 + z03, z03 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z0) :|: z = z0, z0 >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + 0) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z2 + 0) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + 0) + QUICKSORT(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z2 + 0) + QUICKSORT(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z0) :|: z = z0, z0 >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + 0) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z2 + 0) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + 0) + QUICKSORTD(z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z2 + 0) + QUICKSORTD(z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z2), z2, z0) + SPLIT(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z2, z0) + SPLIT(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z2), z2, z0) + SPLITD(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z2, z0) + SPLITD(z2, z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z3, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z = z0, z0 >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = z0, z0 >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z = z0, z0 >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = z0, z0 >= 0 append(z, z') -{ 0 }-> append#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 appendD(z, z') -{ 0 }-> appendD#1(z0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 appendD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 appendD#1(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0 appendD#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort(z) -{ 0 }-> quicksort#1(z0) :|: z = z0, z0 >= 0 quicksort(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z2 + quicksort#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z2 + quicksort#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 quicksortD(z) -{ 0 }-> quicksortD#1(z0) :|: z = z0, z0 >= 0 quicksortD(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z2 + quicksortD#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z2 + quicksortD#1(z1)) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z2 + 0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split(z, z') -{ 0 }-> split#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 split(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z2), z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 split#1(z, z') -{ 0 }-> split#2(0, z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 split#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z0 >= 0, z = 0, z' = z0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z3, z2)), z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, z3 = v3, v2 >= 0, v3 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 splitD(z, z') -{ 0 }-> splitD#1(z1, z0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0 splitD(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z2), z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z2, z0) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z0 >= 0, z = 0, z' = z0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z3, z2)), z0, z1, z3) :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z' = z2, z0 >= 0, z'' = z3, z = 1 + z0 + z1, z2 >= 0, z3 >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, z3 = v3, v2 >= 0, v3 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z4 = v3, v2 >= 0, v3 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z0 + (1 + z2 + z1) :|: z = 2, z1 >= 0, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z2 + z0) + z1 :|: z1 >= 0, z = 1, z0 >= 0, z' = z0, z2 >= 0, z'' = z1, z4 = z2 testList(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z = z0, z0 >= 0 ---------------------------------------- (27) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> appendD#1(z, z') :|: z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 ---------------------------------------- (29) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { appendD, appendD#1 } { split#3 } { splitD#3 } { #ckgt } { #COMPARE } { APPEND#1, APPEND } { #compare } { APPENDD, APPENDD#1 } { append#1, append } { #abs } { split#2 } { splitD#2 } { #greater } { #GREATER } { testList } { split#1 } { splitD#1 } { SPLITD#2 } { SPLIT#2 } { split } { SPLIT#1, SPLIT } { quicksort#1, quicksort#2 } { quicksortD#2, quicksortD#1 } { splitD } { SPLITD, SPLITD#1 } { QUICKSORT#2, QUICKSORT, QUICKSORT#1 } { quicksort } { quicksortD } { QUICKSORTD, QUICKSORTD#1, QUICKSORTD#2 } { TESTQUICKSORT2 } { TESTQUICKSORT } ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> appendD#1(z, z') :|: z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} ---------------------------------------- (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: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> appendD#1(z, z') :|: z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} ---------------------------------------- (33) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: appendD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using CoFloCo for: appendD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> appendD#1(z, z') :|: z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: ?, size: O(n^1) [z + z'] appendD#1: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (35) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: appendD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: appendD#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> appendD#1(z, z') :|: z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + appendD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (37) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (38) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: split#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z4 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {split#3}, {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: split#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (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: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitD#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z4 ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {splitD#3}, {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitD#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (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: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #ckgt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (52) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#ckgt}, {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: ?, size: O(1) [2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #ckgt after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (54) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (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: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #COMPARE after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#COMPARE}, {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #COMPARE after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + #COMPARE(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + #COMPARE(z, z') :|: z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (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: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (63) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: APPEND#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: APPEND after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPEND#1,APPEND}, {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: ?, size: O(1) [0] APPEND: runtime: ?, size: O(1) [1] ---------------------------------------- (65) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: APPEND#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4*z Computed RUNTIME bound using CoFloCo for: APPEND after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 4*z ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 }-> 1 + APPEND#1(z, z') :|: z' >= 0, z >= 0 APPEND#1(z, z') -{ 1 }-> 1 + APPEND(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] ---------------------------------------- (67) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (68) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#compare}, {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #compare after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> split#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> splitD#3(#ckgt(#compare(z'', z')), z0, z1, z'') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (73) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (74) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: APPENDD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: APPENDD#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {APPENDD,APPENDD#1}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: ?, size: O(1) [0] APPENDD#1: runtime: ?, size: O(1) [1] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: APPENDD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z Computed RUNTIME bound using CoFloCo for: APPENDD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 }-> 1 + APPENDD#1(z, z') :|: z' >= 0, z >= 0 APPENDD#1(z, z') -{ 1 }-> 1 + APPENDD(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] ---------------------------------------- (79) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (80) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' Computed SIZE bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: ?, size: O(n^1) [z + z'] append: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> append#1(z, z') :|: z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + append(z1, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (85) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (86) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: ?, size: O(n^1) [z] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #abs after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0 Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (91) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (92) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: split#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z'' ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: ?, size: O(n^1) [1 + z + z''] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: split#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] ---------------------------------------- (97) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (98) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] ---------------------------------------- (99) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z'' ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#2}, {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: ?, size: O(n^1) [1 + z + z''] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitD#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(0, z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] ---------------------------------------- (103) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (104) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] ---------------------------------------- (105) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #greater after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#greater}, {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: ?, size: O(1) [2] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #greater after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (109) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (110) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (111) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #GREATER after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z' ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {#GREATER}, {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: ?, size: O(n^1) [1 + z + z'] ---------------------------------------- (113) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #GREATER after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 1 }-> 1 + #GREATER(z'', z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (115) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (116) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] ---------------------------------------- (117) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: testList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 64 ---------------------------------------- (118) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: ?, size: O(1) [64] ---------------------------------------- (119) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: testList after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (120) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] ---------------------------------------- (121) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (122) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] ---------------------------------------- (123) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: split#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (124) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split#1}, {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (125) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: split#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (126) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(split#1(z1, z0), z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(split#1(z1, z'), z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(split#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> split#1(z', z) :|: z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> split#2(split#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (127) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (128) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (129) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z ---------------------------------------- (130) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD#1}, {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: ?, size: O(n^1) [1 + z] ---------------------------------------- (131) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitD#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (132) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(splitD#1(z1, z0), z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(splitD#1(z1, z'), z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(splitD#1(z1, z0), z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> splitD#1(z', z) :|: z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> splitD#2(splitD#1(z1, z'), z', z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (133) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (134) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(s68, z', z0) + SPLITD(z', z1) :|: s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] ---------------------------------------- (135) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: SPLITD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' ---------------------------------------- (136) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(s68, z', z0) + SPLITD(z', z1) :|: s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD#2}, {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: ?, size: O(n^1) [2 + z' + z''] ---------------------------------------- (137) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: SPLITD#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (138) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(s68, z', z0) + SPLITD(z', z1) :|: s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 1 }-> 1 + SPLITD#2(0, z', z0) + SPLITD(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] ---------------------------------------- (139) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (140) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] ---------------------------------------- (141) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: SPLIT#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' ---------------------------------------- (142) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#2}, {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: ?, size: O(n^1) [2 + z' + z''] ---------------------------------------- (143) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: SPLIT#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (144) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(s62, z', z0) + SPLIT(z', z1) :|: s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 1 }-> 1 + SPLIT#2(0, z', z0) + SPLIT(z', z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] ---------------------------------------- (145) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (146) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] ---------------------------------------- (147) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: split after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (148) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {split}, {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (149) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: split after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (150) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] ---------------------------------------- (151) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (152) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] ---------------------------------------- (153) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: SPLIT#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: SPLIT after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (154) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLIT#1,SPLIT}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: ?, size: O(1) [0] SPLIT: runtime: ?, size: O(1) [1] ---------------------------------------- (155) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: SPLIT#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 14*z Computed RUNTIME bound using CoFloCo for: SPLIT after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 14*z' ---------------------------------------- (156) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(s61, z0) + SPLIT(z0, z1) :|: s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 1 }-> 1 + QUICKSORT#2(0, z0) + SPLIT(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 }-> 1 + SPLIT#1(z', z) :|: z' >= 0, z >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s75 + SPLIT(z', z1) :|: s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 3 }-> 1 + s76 + SPLIT(z', z1) :|: s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] ---------------------------------------- (157) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (158) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] ---------------------------------------- (159) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: quicksort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z Computed SIZE bound using KoAT for: quicksort#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z + z' ---------------------------------------- (160) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: ?, size: O(n^1) [z] quicksort#2: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (161) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksort#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: quicksort#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (162) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(quicksort#1(z0), 1 + z' + 0) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 1 }-> 1 + APPEND(0, 1 + z' + quicksort#1(z1)) + QUICKSORT(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> quicksort#1(z) :|: z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> quicksort#2(s66, z0) :|: s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> quicksort#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(quicksort#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> append(0, 1 + z' + quicksort#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (163) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (164) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] ---------------------------------------- (165) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: quicksortD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 3*z + 3*z' Computed SIZE bound using CoFloCo for: quicksortD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 3*z ---------------------------------------- (166) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: ?, size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: ?, size: O(n^1) [1 + 3*z] ---------------------------------------- (167) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksortD#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed RUNTIME bound using CoFloCo for: quicksortD#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (168) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(quicksortD#1(z0), 1 + z' + 0) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 1 }-> 1 + APPENDD(0, 1 + z' + quicksortD#1(z1)) + QUICKSORTD(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> quicksortD#1(z) :|: z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> quicksortD#2(s72, z0) :|: s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> quicksortD#2(0, z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(quicksortD#1(z0), 1 + z' + 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> appendD(0, 1 + z' + quicksortD#1(z1)) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] ---------------------------------------- (169) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (170) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] ---------------------------------------- (171) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z' ---------------------------------------- (172) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {splitD}, {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: ?, size: O(n^1) [1 + z'] ---------------------------------------- (173) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (174) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] ---------------------------------------- (175) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (176) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] ---------------------------------------- (177) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: SPLITD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: SPLITD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z + z' ---------------------------------------- (178) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {SPLITD,SPLITD#1}, {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: ?, size: O(1) [0] SPLITD#1: runtime: ?, size: O(n^1) [2 + z + z'] ---------------------------------------- (179) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: SPLITD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 4*z' Computed RUNTIME bound using CoFloCo for: SPLITD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4*z ---------------------------------------- (180) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(s67, z0) + SPLITD(z0, z1) :|: s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 1 }-> 1 + QUICKSORTD#2(0, z0) + SPLITD(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 }-> 1 + SPLITD#1(z', z) :|: z' >= 0, z >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s73 + SPLITD(z', z1) :|: s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 3 }-> 1 + s74 + SPLITD(z', z1) :|: s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] ---------------------------------------- (181) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (182) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] ---------------------------------------- (183) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: QUICKSORT#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: QUICKSORT after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 Computed SIZE bound using CoFloCo for: QUICKSORT#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (184) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORT#2,QUICKSORT,QUICKSORT#1}, {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: ?, size: O(1) [0] QUICKSORT: runtime: ?, size: O(1) [3] QUICKSORT#1: runtime: ?, size: O(1) [2] ---------------------------------------- (185) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: QUICKSORT#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 63*z + 132*z^2 Computed RUNTIME bound using KoAT for: QUICKSORT after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 200 + 355*z + 132*z^2 Computed RUNTIME bound using KoAT for: QUICKSORT#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 199 + 355*z + 132*z^2 ---------------------------------------- (186) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 1 }-> 1 + QUICKSORT#1(z) :|: z >= 0 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(s61, z0) + s77 :|: s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + QUICKSORT#2(0, z0) + s78 :|: s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 2 }-> 1 + s6 + QUICKSORT(z0) :|: s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s7 + QUICKSORT(z1) :|: s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s82 }-> 1 + s84 + QUICKSORT(z0) :|: s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s85 }-> 1 + s86 + QUICKSORT(z0) :|: s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s88 + QUICKSORT(z0) :|: s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s89 }-> 1 + s91 + QUICKSORT(z1) :|: s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 + 4*s92 }-> 1 + s93 + QUICKSORT(z1) :|: s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 2 }-> 1 + s95 + QUICKSORT(z1) :|: s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT(z) -{ 0 }-> 1 + QUICKSORT(1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + 0)))))))))) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(0) :|: z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 1 + QUICKSORT(1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + (1 + s44 + (1 + s45 + (1 + s46 + 0)))))))))) :|: s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] ---------------------------------------- (187) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (188) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] ---------------------------------------- (189) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: quicksort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: z ---------------------------------------- (190) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksort}, {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: ?, size: O(n^1) [z] ---------------------------------------- (191) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksort after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (192) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (193) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (194) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] ---------------------------------------- (195) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: quicksortD after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 3*z ---------------------------------------- (196) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {quicksortD}, {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: ?, size: O(n^1) [1 + 3*z] ---------------------------------------- (197) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksortD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (198) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] ---------------------------------------- (199) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (200) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] ---------------------------------------- (201) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: QUICKSORTD after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 Computed SIZE bound using CoFloCo for: QUICKSORTD#1 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 Computed SIZE bound using CoFloCo for: QUICKSORTD#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (202) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {QUICKSORTD,QUICKSORTD#1,QUICKSORTD#2}, {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: ?, size: O(1) [0] QUICKSORTD#1: runtime: ?, size: O(1) [2] QUICKSORTD#2: runtime: ?, size: O(1) [1] ---------------------------------------- (203) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: QUICKSORTD after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 25 + 84*z + 64*z^2 Computed RUNTIME bound using KoAT for: QUICKSORTD#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 228 + 704*z + 512*z^2 Computed RUNTIME bound using KoAT for: QUICKSORTD#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 224 + 696*z + 512*z^2 ---------------------------------------- (204) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 1 }-> 1 + QUICKSORTD#1(z) :|: z >= 0 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(s67, z0) + s130 :|: s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 2 + 4*z1 }-> 1 + QUICKSORTD#2(0, z0) + s131 :|: s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 2 + 2*s106 }-> 1 + s108 + QUICKSORTD(z0) :|: s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s109 }-> 1 + s110 + QUICKSORTD(z0) :|: s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s112 + QUICKSORTD(z0) :|: s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s113 }-> 1 + s115 + QUICKSORTD(z1) :|: s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 + 2*s116 }-> 1 + s117 + QUICKSORTD(z1) :|: s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s119 + QUICKSORTD(z1) :|: s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s22 + QUICKSORTD(z0) :|: s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 2 }-> 1 + s23 + QUICKSORTD(z1) :|: s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] ---------------------------------------- (205) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (206) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] ---------------------------------------- (207) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: TESTQUICKSORT2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (208) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT2}, {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] TESTQUICKSORT2: runtime: ?, size: O(1) [4] ---------------------------------------- (209) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: TESTQUICKSORT2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 563592 ---------------------------------------- (210) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] TESTQUICKSORT2: runtime: O(1) [563592], size: O(1) [4] ---------------------------------------- (211) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (212) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] TESTQUICKSORT2: runtime: O(1) [563592], size: O(1) [4] ---------------------------------------- (213) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: TESTQUICKSORT after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 4 ---------------------------------------- (214) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: {TESTQUICKSORT} Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] TESTQUICKSORT2: runtime: O(1) [563592], size: O(1) [4] TESTQUICKSORT: runtime: ?, size: O(1) [4] ---------------------------------------- (215) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: TESTQUICKSORT after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 563592 ---------------------------------------- (216) Obligation: Complexity RNTS consisting of the following rules: #COMPARE(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z' - 1 + (z - 1), z' - 1 >= 0, z - 1 >= 0 #COMPARE(z, z') -{ 0 }-> 1 + s3 :|: s3 >= 0, s3 <= z - 1 + (z' - 1), z' - 1 >= 0, z - 1 >= 0 #GREATER(z, z') -{ 1 }-> 1 + s1 :|: s1 >= 0, s1 <= z + z', z' >= 0, z >= 0 #abs(z) -{ 0 }-> 0 :|: z = 0 #abs(z) -{ 0 }-> 0 :|: z >= 0 #abs(z) -{ 0 }-> 1 + (z - 1) :|: z - 1 >= 0 #ckgt(z) -{ 0 }-> 2 :|: z = 2 #ckgt(z) -{ 0 }-> 1 :|: z = 1 #ckgt(z) -{ 0 }-> 1 :|: z = 3 #ckgt(z) -{ 0 }-> 0 :|: z >= 0 #compare(z, z') -{ 0 }-> s8 :|: s8 >= 0, s8 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> s9 :|: s9 >= 0, s9 <= 3, z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0 #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0 #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0 #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> s11 :|: s10 >= 0, s10 <= 3, s11 >= 0, s11 <= 2, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 0 }-> s13 :|: s12 >= 0, s12 <= 3, s13 >= 0, s13 <= 2, z - 1 >= 0, z' - 1 >= 0 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 0 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 2 = 2 #greater(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 0 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 3 = 3 #greater(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #greater(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 0 }-> 0 :|: z' >= 0, z >= 0, v0 >= 0, 0 = v0 APPEND(z, z') -{ 1 + 4*z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z >= 0 APPEND#1(z, z') -{ 2 + 4*z1 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 APPENDD(z, z') -{ 1 + 2*z }-> 1 + s20 :|: s20 >= 0, s20 <= 1, z' >= 0, z >= 0 APPENDD#1(z, z') -{ 2 + 2*z1 }-> 1 + s21 :|: s21 >= 0, s21 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT(z) -{ 200 + 355*z + 132*z^2 }-> 1 + s135 :|: s135 >= 0, s135 <= 2, z >= 0 QUICKSORT#1(z) -{ 2 + 63*s61 + 132*s61^2 + 14*z1 }-> 1 + s136 + s77 :|: s136 >= 0, s136 <= 0, s77 >= 0, s77 <= 1, s61 >= 0, s61 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#1(z) -{ 2 + 14*z1 }-> 1 + s137 + s78 :|: s137 >= 0, s137 <= 0, s78 >= 0, s78 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s6 + s141 :|: s141 >= 0, s141 <= 3, s6 >= 0, s6 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s7 + s145 :|: s145 >= 0, s145 <= 3, s7 >= 0, s7 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s82 + 355*z0 + 132*z0^2 }-> 1 + s84 + s138 :|: s138 >= 0, s138 <= 3, s82 >= 0, s82 <= z0, s83 >= 0, s83 <= z1, s84 >= 0, s84 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s85 + 355*z0 + 132*z0^2 }-> 1 + s86 + s139 :|: s139 >= 0, s139 <= 3, s85 >= 0, s85 <= z0, s86 >= 0, s86 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z0 + 132*z0^2 }-> 1 + s88 + s140 :|: s140 >= 0, s140 <= 3, s87 >= 0, s87 <= z1, s88 >= 0, s88 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s89 + 355*z1 + 132*z1^2 }-> 1 + s91 + s142 :|: s142 >= 0, s142 <= 3, s89 >= 0, s89 <= z0, s90 >= 0, s90 <= z1, s91 >= 0, s91 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 4*s92 + 355*z1 + 132*z1^2 }-> 1 + s93 + s143 :|: s143 >= 0, s143 <= 3, s92 >= 0, s92 <= z0, s93 >= 0, s93 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORT#2(z, z') -{ 202 + 355*z1 + 132*z1^2 }-> 1 + s95 + s144 :|: s144 >= 0, s144 <= 3, s94 >= 0, s94 <= z1, s95 >= 0, s95 <= 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD(z) -{ 229 + 704*z + 512*z^2 }-> 1 + s150 :|: s150 >= 0, s150 <= 2, z >= 0 QUICKSORTD#1(z) -{ 226 + 696*s67 + 512*s67^2 + 4*z1 }-> 1 + s151 + s130 :|: s151 >= 0, s151 <= 1, s130 >= 0, s130 <= 0, s67 >= 0, s67 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#1(z) -{ 226 + 4*z1 }-> 1 + s152 + s131 :|: s152 >= 0, s152 <= 1, s131 >= 0, s131 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 QUICKSORTD#2(z, z') -{ 27 + 2*s106 + 84*z0 + 64*z0^2 }-> 1 + s108 + s153 :|: s153 >= 0, s153 <= 0, s106 >= 0, s106 <= 3 * z0 + 1, s107 >= 0, s107 <= 3 * z1 + 1, s108 >= 0, s108 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s109 + 84*z0 + 64*z0^2 }-> 1 + s110 + s154 :|: s154 >= 0, s154 <= 0, s109 >= 0, s109 <= 3 * z0 + 1, s110 >= 0, s110 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s112 + s155 :|: s155 >= 0, s155 <= 0, s111 >= 0, s111 <= 3 * z1 + 1, s112 >= 0, s112 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s113 + 84*z1 + 64*z1^2 }-> 1 + s115 + s157 :|: s157 >= 0, s157 <= 0, s113 >= 0, s113 <= 3 * z0 + 1, s114 >= 0, s114 <= 3 * z1 + 1, s115 >= 0, s115 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 2*s116 + 84*z1 + 64*z1^2 }-> 1 + s117 + s158 :|: s158 >= 0, s158 <= 0, s116 >= 0, s116 <= 3 * z0 + 1, s117 >= 0, s117 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s119 + s159 :|: s159 >= 0, s159 <= 0, s118 >= 0, s118 <= 3 * z1 + 1, s119 >= 0, s119 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z0 + 64*z0^2 }-> 1 + s22 + s156 :|: s156 >= 0, s156 <= 0, s22 >= 0, s22 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 QUICKSORTD#2(z, z') -{ 27 + 84*z1 + 64*z1^2 }-> 1 + s23 + s160 :|: s160 >= 0, s160 <= 0, s23 >= 0, s23 <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT(z, z') -{ 1 + 14*z' }-> 1 + s79 :|: s79 >= 0, s79 <= 0, z' >= 0, z >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s75 + s80 :|: s80 >= 0, s80 <= 1, s75 >= 0, s75 <= z' + z0 + 2, s62 >= 0, s62 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#1(z, z') -{ 4 + 14*z1 }-> 1 + s76 + s81 :|: s81 >= 0, s81 <= 1, s76 >= 0, s76 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLIT#2(z, z', z'') -{ 2 }-> 1 + s59 :|: s59 >= 0, s59 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 SPLITD(z, z') -{ 1 + 4*z' }-> 1 + s132 :|: s132 >= 0, s132 <= z' + z + 2, z' >= 0, z >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s73 + s133 :|: s133 >= 0, s133 <= 0, s73 >= 0, s73 <= z' + z0 + 2, s68 >= 0, s68 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#1(z, z') -{ 4 + 4*z1 }-> 1 + s74 + s134 :|: s134 >= 0, s134 <= 0, s74 >= 0, s74 <= z' + z0 + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 SPLITD#2(z, z', z'') -{ 2 }-> 1 + s60 :|: s60 >= 0, s60 <= z'' + z' + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 TESTQUICKSORT(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT(z) -{ 16950 + 2995*s27 + 264*s27*s28 + 264*s27*s29 + 264*s27*s30 + 264*s27*s31 + 264*s27*s32 + 264*s27*s33 + 264*s27*s34 + 264*s27*s35 + 264*s27*s36 + 132*s27^2 + 2995*s28 + 264*s28*s29 + 264*s28*s30 + 264*s28*s31 + 264*s28*s32 + 264*s28*s33 + 264*s28*s34 + 264*s28*s35 + 264*s28*s36 + 132*s28^2 + 2995*s29 + 264*s29*s30 + 264*s29*s31 + 264*s29*s32 + 264*s29*s33 + 264*s29*s34 + 264*s29*s35 + 264*s29*s36 + 132*s29^2 + 2995*s30 + 264*s30*s31 + 264*s30*s32 + 264*s30*s33 + 264*s30*s34 + 264*s30*s35 + 264*s30*s36 + 132*s30^2 + 2995*s31 + 264*s31*s32 + 264*s31*s33 + 264*s31*s34 + 264*s31*s35 + 264*s31*s36 + 132*s31^2 + 2995*s32 + 264*s32*s33 + 264*s32*s34 + 264*s32*s35 + 264*s32*s36 + 132*s32^2 + 2995*s33 + 264*s33*s34 + 264*s33*s35 + 264*s33*s36 + 132*s33^2 + 2995*s34 + 264*s34*s35 + 264*s34*s36 + 132*s34^2 + 2995*s35 + 264*s35*s36 + 132*s35^2 + 2995*s36 + 132*s36^2 }-> 1 + s146 :|: s146 >= 0, s146 <= 3, s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + 0)))), s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s32 >= 0, s32 <= 1 + (1 + 0), s33 >= 0, s33 <= 1 + (1 + (1 + 0)), s34 >= 0, s34 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s36 >= 0, s36 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT(z) -{ 200 }-> 1 + s147 :|: s147 >= 0, s147 <= 3, z >= 0 TESTQUICKSORT2(z) -{ 0 }-> 0 :|: z >= 0 TESTQUICKSORT2(z) -{ 16950 + 2995*s37 + 264*s37*s38 + 264*s37*s39 + 264*s37*s40 + 264*s37*s41 + 264*s37*s42 + 264*s37*s43 + 264*s37*s44 + 264*s37*s45 + 264*s37*s46 + 132*s37^2 + 2995*s38 + 264*s38*s39 + 264*s38*s40 + 264*s38*s41 + 264*s38*s42 + 264*s38*s43 + 264*s38*s44 + 264*s38*s45 + 264*s38*s46 + 132*s38^2 + 2995*s39 + 264*s39*s40 + 264*s39*s41 + 264*s39*s42 + 264*s39*s43 + 264*s39*s44 + 264*s39*s45 + 264*s39*s46 + 132*s39^2 + 2995*s40 + 264*s40*s41 + 264*s40*s42 + 264*s40*s43 + 264*s40*s44 + 264*s40*s45 + 264*s40*s46 + 132*s40^2 + 2995*s41 + 264*s41*s42 + 264*s41*s43 + 264*s41*s44 + 264*s41*s45 + 264*s41*s46 + 132*s41^2 + 2995*s42 + 264*s42*s43 + 264*s42*s44 + 264*s42*s45 + 264*s42*s46 + 132*s42^2 + 2995*s43 + 264*s43*s44 + 264*s43*s45 + 264*s43*s46 + 132*s43^2 + 2995*s44 + 264*s44*s45 + 264*s44*s46 + 132*s44^2 + 2995*s45 + 264*s45*s46 + 132*s45^2 + 2995*s46 + 132*s46^2 }-> 1 + s148 :|: s148 >= 0, s148 <= 3, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + 0)))), s39 >= 0, s39 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s40 >= 0, s40 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s42 >= 0, s42 <= 1 + (1 + 0), s43 >= 0, s43 <= 1 + (1 + (1 + 0)), s44 >= 0, s44 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s45 >= 0, s45 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s46 >= 0, s46 <= 1 + (1 + (1 + (1 + 0))), z >= 0 TESTQUICKSORT2(z) -{ 200 }-> 1 + s149 :|: s149 >= 0, s149 <= 3, z >= 0 append(z, z') -{ 0 }-> s25 :|: s25 >= 0, s25 <= z + z', z' >= 0, z >= 0 append(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 0 }-> 1 + z0 + s26 :|: s26 >= 0, s26 <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 appendD(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0 appendD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> z' :|: z' >= 0, z = 0 appendD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 appendD#1(z, z') -{ 0 }-> 1 + z0 + s'' :|: s'' >= 0, s'' <= z1 + z', z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort(z) -{ 0 }-> s96 :|: s96 >= 0, s96 <= z, z >= 0 quicksort(z) -{ 0 }-> 0 :|: z >= 0 quicksort#1(z) -{ 0 }-> s97 :|: s97 >= 0, s97 <= s66 + z0, s66 >= 0, s66 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> s98 :|: s98 >= 0, s98 <= 0 + z0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksort#1(z) -{ 0 }-> 0 :|: z = 0 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 0 }-> s101 :|: s99 >= 0, s99 <= z0, s100 >= 0, s100 <= z1, s101 >= 0, s101 <= s99 + (1 + z' + s100), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s103 :|: s102 >= 0, s102 <= z0, s103 >= 0, s103 <= s102 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s105 :|: s104 >= 0, s104 <= z1, s105 >= 0, s105 <= 0 + (1 + z' + s104), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> s24 :|: s24 >= 0, s24 <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 quicksortD(z) -{ 0 }-> s120 :|: s120 >= 0, s120 <= 3 * z + 1, z >= 0 quicksortD(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#1(z) -{ 0 }-> s121 :|: s121 >= 0, s121 <= 3 * z0 + 1 + 3 * s72, s72 >= 0, s72 <= z1 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> s122 :|: s122 >= 0, s122 <= 3 * z0 + 1 + 3 * 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1 quicksortD#1(z) -{ 0 }-> 0 :|: z = 0 quicksortD#1(z) -{ 0 }-> 0 :|: z >= 0 quicksortD#2(z, z') -{ 0 }-> s :|: s >= 0, s <= 0 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s125 :|: s123 >= 0, s123 <= 3 * z0 + 1, s124 >= 0, s124 <= 3 * z1 + 1, s125 >= 0, s125 <= s123 + (1 + z' + s124), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s127 :|: s126 >= 0, s126 <= 3 * z0 + 1, s127 >= 0, s127 <= s126 + (1 + z' + 0), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> s129 :|: s128 >= 0, s128 <= 3 * z1 + 1, s129 >= 0, s129 <= 0 + (1 + z' + s128), z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split(z, z') -{ 0 }-> s63 :|: s63 >= 0, s63 <= z' + 1, z' >= 0, z >= 0 split(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> s57 :|: s57 >= 0, s57 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> s65 :|: s64 >= 0, s64 <= z1 + 1, s65 >= 0, s65 <= s64 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 split#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 split#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 split#2(z, z', z'') -{ 0 }-> s16 :|: s14 >= 0, s14 <= 3, s15 >= 0, s15 <= 2, s16 >= 0, s16 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 split#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 split#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 splitD(z, z') -{ 0 }-> s69 :|: s69 >= 0, s69 <= z' + 1, z' >= 0, z >= 0 splitD(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> s58 :|: s58 >= 0, s58 <= 0 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> s71 :|: s70 >= 0, s70 <= z1 + 1, s71 >= 0, s71 <= s70 + z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0 splitD#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitD#1(z, z') -{ 0 }-> 1 + 0 + 0 :|: z' >= 0, z = 0 splitD#2(z, z', z'') -{ 0 }-> s19 :|: s17 >= 0, s17 <= 3, s18 >= 0, s18 <= 2, s19 >= 0, s19 <= z0 + z1 + z'' + 2, z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitD#2(z, z', z'') -{ 0 }-> 0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0, z'' >= 0, v0 >= 0, z1 = v2, v1 >= 0, 0 = v0, z0 = v1, v2 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + z' + (1 + z4 + z'') :|: z = 2, z'' >= 0, z' >= 0, z4 >= 0 splitD#3(z, z', z'', z4) -{ 0 }-> 1 + (1 + z4 + z') + z'' :|: z'' >= 0, z = 1, z' >= 0, z4 >= 0 testList(z) -{ 0 }-> 0 :|: z >= 0 testList(z) -{ 0 }-> 1 + s47 + (1 + s48 + (1 + s49 + (1 + s50 + (1 + s51 + (1 + s52 + (1 + s53 + (1 + s54 + (1 + s55 + (1 + s56 + 0))))))))) :|: s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 1 + (1 + (1 + (1 + (1 + 0)))), s49 >= 0, s49 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))), s50 >= 0, s50 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))), s51 >= 0, s51 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))), s52 >= 0, s52 <= 1 + (1 + 0), s53 >= 0, s53 <= 1 + (1 + (1 + 0)), s54 >= 0, s54 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))), s55 >= 0, s55 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))), s56 >= 0, s56 <= 1 + (1 + (1 + (1 + 0))), z >= 0 Function symbols to be analyzed: Previous analysis results are: appendD: runtime: O(1) [0], size: O(n^1) [z + z'] appendD#1: runtime: O(1) [0], size: O(n^1) [z + z'] split#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] splitD#3: runtime: O(1) [0], size: O(n^1) [2 + z' + z'' + z4] #ckgt: runtime: O(1) [0], size: O(1) [2] #COMPARE: runtime: O(1) [0], size: O(n^1) [z + z'] APPEND#1: runtime: O(n^1) [4*z], size: O(1) [0] APPEND: runtime: O(n^1) [1 + 4*z], size: O(1) [1] #compare: runtime: O(1) [0], size: O(1) [3] APPENDD: runtime: O(n^1) [1 + 2*z], size: O(1) [0] APPENDD#1: runtime: O(n^1) [2*z], size: O(1) [1] append#1: runtime: O(1) [0], size: O(n^1) [z + z'] append: runtime: O(1) [0], size: O(n^1) [z + z'] #abs: runtime: O(1) [0], size: O(n^1) [z] split#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] splitD#2: runtime: O(1) [0], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [0], size: O(1) [2] #GREATER: runtime: O(1) [1], size: O(n^1) [1 + z + z'] testList: runtime: O(1) [0], size: O(1) [64] split#1: runtime: O(1) [0], size: O(n^1) [1 + z] splitD#1: runtime: O(1) [0], size: O(n^1) [1 + z] SPLITD#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] SPLIT#2: runtime: O(1) [2], size: O(n^1) [2 + z' + z''] split: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLIT#1: runtime: O(n^1) [14*z], size: O(1) [0] SPLIT: runtime: O(n^1) [1 + 14*z'], size: O(1) [1] quicksort#1: runtime: O(1) [0], size: O(n^1) [z] quicksort#2: runtime: O(1) [0], size: O(n^1) [z + z'] quicksortD#2: runtime: O(1) [0], size: O(n^1) [1 + 3*z + 3*z'] quicksortD#1: runtime: O(1) [0], size: O(n^1) [1 + 3*z] splitD: runtime: O(1) [0], size: O(n^1) [1 + z'] SPLITD: runtime: O(n^1) [1 + 4*z'], size: O(1) [0] SPLITD#1: runtime: O(n^1) [4*z], size: O(n^1) [2 + z + z'] QUICKSORT#2: runtime: O(n^2) [63*z + 132*z^2], size: O(1) [0] QUICKSORT: runtime: O(n^2) [200 + 355*z + 132*z^2], size: O(1) [3] QUICKSORT#1: runtime: O(n^2) [199 + 355*z + 132*z^2], size: O(1) [2] quicksort: runtime: O(1) [0], size: O(n^1) [z] quicksortD: runtime: O(1) [0], size: O(n^1) [1 + 3*z] QUICKSORTD: runtime: O(n^2) [25 + 84*z + 64*z^2], size: O(1) [0] QUICKSORTD#1: runtime: O(n^2) [228 + 704*z + 512*z^2], size: O(1) [2] QUICKSORTD#2: runtime: O(n^2) [224 + 696*z + 512*z^2], size: O(1) [1] TESTQUICKSORT2: runtime: O(1) [563592], size: O(1) [4] TESTQUICKSORT: runtime: O(1) [563592], size: O(1) [4] ---------------------------------------- (217) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (218) BOUNDS(1, n^2) ---------------------------------------- (219) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID)) Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT ---------------------------------------- (220) Obligation: Complexity Dependency Tuples Problem Rules: #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Tuples: #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) S tuples: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) K tuples:none Defined Rule Symbols: #abs_1, #greater_2, append_2, append#1_2, appendD_2, appendD#1_2, quicksort_1, quicksort#1_1, quicksort#2_2, quicksortD_1, quicksortD#1_1, quicksortD#2_2, split_2, split#1_2, split#2_3, split#3_4, splitD_2, splitD#1_2, splitD#2_3, splitD#3_4, testList_1, testQuicksort_1, testQuicksort2_1, #ckgt_1, #compare_2 Defined Pair Symbols: #CKGT_1, #COMPARE_2, #ABS_1, #GREATER_2, APPEND_2, APPEND#1_2, APPENDD_2, APPENDD#1_2, QUICKSORT_1, QUICKSORT#1_1, QUICKSORT#2_2, QUICKSORTD_1, QUICKSORTD#1_1, QUICKSORTD#2_2, SPLIT_2, SPLIT#1_2, SPLIT#2_3, SPLIT#3_4, SPLITD_2, SPLITD#1_2, SPLITD#2_3, SPLITD#3_4, TESTLIST_1, TESTQUICKSORT_1, TESTQUICKSORT2_1 Compound Symbols: c, c1, c2, c3, c4, c5, c6, c7, c8_1, c9, c10, c11, c12_1, c13, c14_1, c15, c16, c17, c18, c19_2, c20_1, c21_1, c22, c23_1, c24_1, c25, c26_1, c27_2, c28, c29_2, c30_2, c31_1, c32_2, c33, c34_2, c35_2, c36_1, c37_2, c38, c39_2, c40, c41, c42_1, c43_2, c44, c45_2, c46, c47, c48_1, c49_1, c50_1, c51_1, c52_1, c53_1, c54_1, c55_1, c56_1, c57_1, c58_2, c59_2 ---------------------------------------- (221) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID)) Converted S to standard rules, and D \ S as well as R to relative rules. ---------------------------------------- (222) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) The (relative) TRS S consists of the following rules: #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Rewrite Strategy: INNERMOST ---------------------------------------- (223) RenamingProof (BOTH BOUNDS(ID, ID)) Renamed function symbols to avoid clashes with predefined symbol. ---------------------------------------- (224) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF). The TRS R consists of the following rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) The (relative) TRS S consists of the following rules: #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Rewrite Strategy: INNERMOST ---------------------------------------- (225) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Inferred types. ---------------------------------------- (226) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil ---------------------------------------- (227) OrderProof (LOWER BOUND(ID)) Heuristically decided to analyse the following defined symbols: #compare, #COMPARE, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, append, append#1, appendD, appendD#1, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 appendD = appendD#1 appendD < quicksortD#1 ---------------------------------------- (228) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: #compare, #COMPARE, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, append, append#1, appendD, appendD#1, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 appendD = appendD#1 appendD < quicksortD#1 ---------------------------------------- (229) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) Induction Base: #compare(gen_#0:#neg:#pos:#s36_60(0), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) #EQ Induction Step: #compare(gen_#0:#neg:#pos:#s36_60(+(n40_60, 1)), gen_#0:#neg:#pos:#s36_60(+(n40_60, 1))) ->_R^Omega(0) #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) ->_IH #EQ We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (230) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: #COMPARE, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, append, append#1, appendD, appendD#1, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 appendD = appendD#1 appendD < quicksortD#1 ---------------------------------------- (231) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) Induction Base: #COMPARE(gen_#0:#neg:#pos:#s36_60(0), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) c3 Induction Step: #COMPARE(gen_#0:#neg:#pos:#s36_60(+(n319207_60, 1)), gen_#0:#neg:#pos:#s36_60(+(n319207_60, 1))) ->_R^Omega(0) c8(#COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60))) ->_IH c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(c319208_60)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (232) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: appendD#1, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, append, append#1, appendD, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 appendD = appendD#1 appendD < quicksortD#1 ---------------------------------------- (233) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) Induction Base: appendD#1(gen_:::nil38_60(0), gen_:::nil38_60(b)) ->_R^Omega(0) gen_:::nil38_60(b) Induction Step: appendD#1(gen_:::nil38_60(+(n638822_60, 1)), gen_:::nil38_60(b)) ->_R^Omega(0) ::(#0, appendD(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b))) ->_R^Omega(0) ::(#0, appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b))) ->_IH ::(#0, gen_:::nil38_60(+(b, c638823_60))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (234) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: appendD, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, append, append#1, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 appendD = appendD#1 appendD < quicksortD#1 ---------------------------------------- (235) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) Induction Base: append#1(gen_:::nil38_60(0), gen_:::nil38_60(b)) ->_R^Omega(0) gen_:::nil38_60(b) Induction Step: append#1(gen_:::nil38_60(+(n642011_60, 1)), gen_:::nil38_60(b)) ->_R^Omega(0) ::(#0, append(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b))) ->_R^Omega(0) ::(#0, append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b))) ->_IH ::(#0, gen_:::nil38_60(+(b, c642012_60))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (236) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: append, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, splitD, SPLITD, quicksortD, SPLIT#1, SPLITD#1, quicksort#1, quicksortD#1, split#1, splitD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 append = append#1 append < quicksort#1 ---------------------------------------- (237) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) Induction Base: splitD#1(gen_:::nil38_60(0), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) tuple#2(nil, nil) Induction Step: splitD#1(gen_:::nil38_60(+(n645222_60, 1)), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) splitD#2(splitD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n645222_60)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_R^Omega(0) splitD#2(splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_IH splitD#2(tuple#2(gen_:::nil38_60(c645223_60), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_R^Omega(0) splitD#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(n645222_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) splitD#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(n645222_60), gen_:::nil38_60(0), #0) ->_L^Omega(0) splitD#3(#ckgt(#EQ), gen_:::nil38_60(n645222_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) splitD#3(#false, gen_:::nil38_60(n645222_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) tuple#2(::(#0, gen_:::nil38_60(n645222_60)), gen_:::nil38_60(0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (238) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: splitD, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLITD, quicksortD, SPLIT#1, SPLITD#1, quicksort#1, quicksortD#1, split#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 splitD < QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 splitD < SPLITD#1 splitD < quicksortD#1 splitD = splitD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 ---------------------------------------- (239) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) Induction Base: quicksortD#1(gen_:::nil38_60(0)) ->_R^Omega(0) nil Induction Step: quicksortD#1(gen_:::nil38_60(+(n648994_60, 1))) ->_R^Omega(0) quicksortD#2(splitD(#0, gen_:::nil38_60(n648994_60)), #0) ->_R^Omega(0) quicksortD#2(splitD#1(gen_:::nil38_60(n648994_60), #0), #0) ->_L^Omega(0) quicksortD#2(tuple#2(gen_:::nil38_60(n648994_60), gen_:::nil38_60(0)), #0) ->_R^Omega(0) appendD(quicksortD(gen_:::nil38_60(n648994_60)), ::(#0, quicksortD(gen_:::nil38_60(0)))) ->_R^Omega(0) appendD(quicksortD#1(gen_:::nil38_60(n648994_60)), ::(#0, quicksortD(gen_:::nil38_60(0)))) ->_IH appendD(gen_:::nil38_60(c648995_60), ::(#0, quicksortD(gen_:::nil38_60(0)))) ->_R^Omega(0) appendD(gen_:::nil38_60(n648994_60), ::(#0, quicksortD#1(gen_:::nil38_60(0)))) ->_R^Omega(0) appendD(gen_:::nil38_60(n648994_60), ::(#0, nil)) ->_R^Omega(0) appendD#1(gen_:::nil38_60(n648994_60), ::(#0, nil)) ->_L^Omega(0) gen_:::nil38_60(+(n648994_60, +(0, 1))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (240) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: quicksortD, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLITD, SPLIT#1, SPLITD#1, quicksort#1, split#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 SPLITD < QUICKSORTD#1 quicksortD < QUICKSORTD#1 SPLITD = SPLITD#1 quicksortD = quicksortD#1 ---------------------------------------- (241) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLITD#1(gen_:::nil38_60(+(1, n651314_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n651314_60) Induction Base: SPLITD#1(gen_:::nil38_60(+(1, 0)), gen_#0:#neg:#pos:#s36_60(0)) Induction Step: SPLITD#1(gen_:::nil38_60(+(1, +(n651314_60, 1))), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(1) c43(SPLITD#2(splitD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60))), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(0) c43(SPLITD#2(splitD#1(gen_:::nil38_60(+(1, n651314_60)), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_L^Omega(0) c43(SPLITD#2(tuple#2(gen_:::nil38_60(+(1, n651314_60)), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(1) c43(c45(SPLITD#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(+(1, n651314_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(0) c43(c45(SPLITD#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(+(1, n651314_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_L^Omega(0) c43(c45(SPLITD#3(#ckgt(#EQ), gen_:::nil38_60(+(1, n651314_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(0) c43(c45(SPLITD#3(#false, gen_:::nil38_60(+(1, n651314_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(1) c43(c45(c46, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(1) c43(c45(c46, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_L^Omega(0) c43(c45(c46, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(0) c43(c45(c46, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_L^Omega(0) c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n651314_60)))) ->_R^Omega(1) c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c42(SPLITD#1(gen_:::nil38_60(+(1, n651314_60)), gen_#0:#neg:#pos:#s36_60(0)))) ->_IH c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c42(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (242) Complex Obligation (BEST) ---------------------------------------- (243) Obligation: Proved the lower bound n^1 for the following obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: SPLITD#1, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLITD, SPLIT#1, quicksort#1, split#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 SPLITD < QUICKSORTD#1 SPLITD = SPLITD#1 ---------------------------------------- (244) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (245) BOUNDS(n^1, INF) ---------------------------------------- (246) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n651314_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n651314_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: SPLITD, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLIT#1, quicksort#1, split#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 SPLITD < QUICKSORTD#1 SPLITD = SPLITD#1 ---------------------------------------- (247) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) Induction Base: SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(0)) Induction Step: SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(n920889_60, 1))) ->_R^Omega(1) c42(SPLITD#1(gen_:::nil38_60(+(n920889_60, 1)), gen_#0:#neg:#pos:#s36_60(0))) ->_R^Omega(1) c42(c43(SPLITD#2(splitD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(0) c42(c43(SPLITD#2(splitD#1(gen_:::nil38_60(n920889_60), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_L^Omega(0) c42(c43(SPLITD#2(tuple#2(gen_:::nil38_60(n920889_60), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(1) c42(c43(c45(SPLITD#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(n920889_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(0) c42(c43(c45(SPLITD#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(n920889_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_L^Omega(0) c42(c43(c45(SPLITD#3(#ckgt(#EQ), gen_:::nil38_60(n920889_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(0) c42(c43(c45(SPLITD#3(#false, gen_:::nil38_60(n920889_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(1) c42(c43(c45(c46, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(1) c42(c43(c45(c46, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_L^Omega(0) c42(c43(c45(c46, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_R^Omega(0) c42(c43(c45(c46, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_L^Omega(0) c42(c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)))) ->_IH c42(c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), *39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (248) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n651314_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n651314_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: SPLITD#1, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLIT#1, quicksort#1, split#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 SPLITD < QUICKSORTD#1 SPLITD = SPLITD#1 ---------------------------------------- (249) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) Induction Base: SPLITD#1(gen_:::nil38_60(+(1, 0)), gen_#0:#neg:#pos:#s36_60(0)) Induction Step: SPLITD#1(gen_:::nil38_60(+(1, +(n930953_60, 1))), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(1) c43(SPLITD#2(splitD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60))), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(0) c43(SPLITD#2(splitD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_L^Omega(0) c43(SPLITD#2(tuple#2(gen_:::nil38_60(+(1, n930953_60)), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(1) c43(c45(SPLITD#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(+(1, n930953_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(0) c43(c45(SPLITD#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(+(1, n930953_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_L^Omega(0) c43(c45(SPLITD#3(#ckgt(#EQ), gen_:::nil38_60(+(1, n930953_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(0) c43(c45(SPLITD#3(#false, gen_:::nil38_60(+(1, n930953_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(1) c43(c45(c46, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(1) c43(c45(c46, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_L^Omega(0) c43(c45(c46, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(0) c43(c45(c46, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_L^Omega(0) c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n930953_60)))) ->_R^Omega(1) c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c42(SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)))) ->_IH c43(c45(c46, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c42(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (250) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: split#1, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, split, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLIT#1, quicksort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (251) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) Induction Base: split#1(gen_:::nil38_60(0), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) tuple#2(nil, nil) Induction Step: split#1(gen_:::nil38_60(+(n1206352_60, 1)), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(0) split#2(split(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1206352_60)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_R^Omega(0) split#2(split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_IH split#2(tuple#2(gen_:::nil38_60(c1206353_60), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0) ->_R^Omega(0) split#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) split#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0), #0) ->_L^Omega(0) split#3(#ckgt(#EQ), gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) split#3(#false, gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0), #0) ->_R^Omega(0) tuple#2(::(#0, gen_:::nil38_60(n1206352_60)), gen_:::nil38_60(0)) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (252) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: split, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, SPLIT, quicksort, QUICKSORTD, QUICKSORTD#1, SPLIT#1, quicksort#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 split < QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 split < SPLIT#1 split < quicksort#1 split = split#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (253) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) Induction Base: quicksort#1(gen_:::nil38_60(0)) ->_R^Omega(0) nil Induction Step: quicksort#1(gen_:::nil38_60(+(n1211300_60, 1))) ->_R^Omega(0) quicksort#2(split(#0, gen_:::nil38_60(n1211300_60)), #0) ->_R^Omega(0) quicksort#2(split#1(gen_:::nil38_60(n1211300_60), #0), #0) ->_L^Omega(0) quicksort#2(tuple#2(gen_:::nil38_60(n1211300_60), gen_:::nil38_60(0)), #0) ->_R^Omega(0) append(quicksort(gen_:::nil38_60(n1211300_60)), ::(#0, quicksort(gen_:::nil38_60(0)))) ->_R^Omega(0) append(quicksort#1(gen_:::nil38_60(n1211300_60)), ::(#0, quicksort(gen_:::nil38_60(0)))) ->_IH append(gen_:::nil38_60(c1211301_60), ::(#0, quicksort(gen_:::nil38_60(0)))) ->_R^Omega(0) append(gen_:::nil38_60(n1211300_60), ::(#0, quicksort#1(gen_:::nil38_60(0)))) ->_R^Omega(0) append(gen_:::nil38_60(n1211300_60), ::(#0, nil)) ->_R^Omega(0) append#1(gen_:::nil38_60(n1211300_60), ::(#0, nil)) ->_L^Omega(0) gen_:::nil38_60(+(n1211300_60, +(0, 1))) We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0). ---------------------------------------- (254) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: quicksort, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, SPLIT, QUICKSORTD, QUICKSORTD#1, SPLIT#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 SPLIT < QUICKSORT#1 quicksort < QUICKSORT#1 SPLIT = SPLIT#1 quicksort = quicksort#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (255) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLIT#1(gen_:::nil38_60(+(1, n1214292_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1214292_60) Induction Base: SPLIT#1(gen_:::nil38_60(+(1, 0)), gen_#0:#neg:#pos:#s36_60(0)) Induction Step: SPLIT#1(gen_:::nil38_60(+(1, +(n1214292_60, 1))), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(1) c37(SPLIT#2(split(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60))), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(0) c37(SPLIT#2(split#1(gen_:::nil38_60(+(1, n1214292_60)), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_L^Omega(0) c37(SPLIT#2(tuple#2(gen_:::nil38_60(+(1, n1214292_60)), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(1) c37(c39(SPLIT#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(+(1, n1214292_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(0) c37(c39(SPLIT#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(+(1, n1214292_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_L^Omega(0) c37(c39(SPLIT#3(#ckgt(#EQ), gen_:::nil38_60(+(1, n1214292_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(0) c37(c39(SPLIT#3(#false, gen_:::nil38_60(+(1, n1214292_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(1) c37(c39(c40, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(1) c37(c39(c40, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_L^Omega(0) c37(c39(c40, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(0) c37(c39(c40, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_L^Omega(0) c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1214292_60)))) ->_R^Omega(1) c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c36(SPLIT#1(gen_:::nil38_60(+(1, n1214292_60)), gen_#0:#neg:#pos:#s36_60(0)))) ->_IH c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c36(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (256) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1214292_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1214292_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: SPLIT, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, QUICKSORTD, QUICKSORTD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 SPLIT < QUICKSORT#1 SPLIT = SPLIT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (257) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) Induction Base: SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(0)) Induction Step: SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(n1493911_60, 1))) ->_R^Omega(1) c36(SPLIT#1(gen_:::nil38_60(+(n1493911_60, 1)), gen_#0:#neg:#pos:#s36_60(0))) ->_R^Omega(1) c36(c37(SPLIT#2(split(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(0) c36(c37(SPLIT#2(split#1(gen_:::nil38_60(n1493911_60), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_L^Omega(0) c36(c37(SPLIT#2(tuple#2(gen_:::nil38_60(n1493911_60), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(1) c36(c37(c39(SPLIT#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(n1493911_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(0) c36(c37(c39(SPLIT#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(n1493911_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_L^Omega(0) c36(c37(c39(SPLIT#3(#ckgt(#EQ), gen_:::nil38_60(n1493911_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(0) c36(c37(c39(SPLIT#3(#false, gen_:::nil38_60(n1493911_60), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(1) c36(c37(c39(c40, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(1) c36(c37(c39(c40, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_L^Omega(0) c36(c37(c39(c40, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_R^Omega(0) c36(c37(c39(c40, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_L^Omega(0) c36(c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)))) ->_IH c36(c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), *39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (258) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1214292_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1214292_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: SPLIT#1, APPEND, APPEND#1, APPENDD, APPENDD#1, QUICKSORT, QUICKSORT#1, QUICKSORTD, QUICKSORTD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 SPLIT < QUICKSORT#1 SPLIT = SPLIT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (259) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) Induction Base: SPLIT#1(gen_:::nil38_60(+(1, 0)), gen_#0:#neg:#pos:#s36_60(0)) Induction Step: SPLIT#1(gen_:::nil38_60(+(1, +(n1505819_60, 1))), gen_#0:#neg:#pos:#s36_60(0)) ->_R^Omega(1) c37(SPLIT#2(split(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60))), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(0) c37(SPLIT#2(split#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_L^Omega(0) c37(SPLIT#2(tuple#2(gen_:::nil38_60(+(1, n1505819_60)), gen_:::nil38_60(0)), gen_#0:#neg:#pos:#s36_60(0), #0), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(1) c37(c39(SPLIT#3(#greater(#0, gen_#0:#neg:#pos:#s36_60(0)), gen_:::nil38_60(+(1, n1505819_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(0) c37(c39(SPLIT#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), gen_:::nil38_60(+(1, n1505819_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_L^Omega(0) c37(c39(SPLIT#3(#ckgt(#EQ), gen_:::nil38_60(+(1, n1505819_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(0) c37(c39(SPLIT#3(#false, gen_:::nil38_60(+(1, n1505819_60)), gen_:::nil38_60(0), #0), #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(1) c37(c39(c40, #GREATER(#0, gen_#0:#neg:#pos:#s36_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(1) c37(c39(c40, c19(#CKGT(#compare(#0, gen_#0:#neg:#pos:#s36_60(0))), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_L^Omega(0) c37(c39(c40, c19(#CKGT(#EQ), #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(0) c37(c39(c40, c19(c, #COMPARE(#0, gen_#0:#neg:#pos:#s36_60(0)))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_L^Omega(0) c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(+(1, n1505819_60)))) ->_R^Omega(1) c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c36(SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)))) ->_IH c37(c39(c40, c19(c, gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0))), c36(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (260) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: APPENDD#1, APPEND, APPEND#1, APPENDD, QUICKSORT, QUICKSORT#1, QUICKSORTD, QUICKSORTD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (261) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPENDD#1(gen_:::nil38_60(+(1, n1791262_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1791262_60) Induction Base: APPENDD#1(gen_:::nil38_60(+(1, 0)), gen_:::nil38_60(b)) Induction Step: APPENDD#1(gen_:::nil38_60(+(1, +(n1791262_60, 1))), gen_:::nil38_60(b)) ->_R^Omega(1) c24(APPENDD(gen_:::nil38_60(+(1, n1791262_60)), gen_:::nil38_60(b))) ->_R^Omega(1) c24(c23(APPENDD#1(gen_:::nil38_60(+(1, n1791262_60)), gen_:::nil38_60(b)))) ->_IH c24(c23(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (262) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1791262_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1791262_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: APPENDD, APPEND, APPEND#1, QUICKSORT, QUICKSORT#1, QUICKSORTD, QUICKSORTD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (263) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) Induction Base: APPENDD(gen_:::nil38_60(0), gen_:::nil38_60(b)) Induction Step: APPENDD(gen_:::nil38_60(+(n1797809_60, 1)), gen_:::nil38_60(b)) ->_R^Omega(1) c23(APPENDD#1(gen_:::nil38_60(+(n1797809_60, 1)), gen_:::nil38_60(b))) ->_R^Omega(1) c23(c24(APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)))) ->_IH c23(c24(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (264) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1791262_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1791262_60) APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: APPENDD#1, APPEND, APPEND#1, QUICKSORT, QUICKSORT#1, QUICKSORTD, QUICKSORTD#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 APPENDD = APPENDD#1 APPENDD < QUICKSORTD#1 QUICKSORT = QUICKSORT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (265) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1805315_60) Induction Base: APPENDD#1(gen_:::nil38_60(+(1, 0)), gen_:::nil38_60(b)) Induction Step: APPENDD#1(gen_:::nil38_60(+(1, +(n1805315_60, 1))), gen_:::nil38_60(b)) ->_R^Omega(1) c24(APPENDD(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b))) ->_R^Omega(1) c24(c23(APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)))) ->_IH c24(c23(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (266) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1805315_60) APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: QUICKSORTD#1, APPEND, APPEND#1, QUICKSORT, QUICKSORT#1, QUICKSORTD They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 QUICKSORT = QUICKSORT#1 QUICKSORTD = QUICKSORTD#1 ---------------------------------------- (267) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND#1(gen_:::nil38_60(+(1, n1951812_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1951812_60) Induction Base: APPEND#1(gen_:::nil38_60(+(1, 0)), gen_:::nil38_60(b)) Induction Step: APPEND#1(gen_:::nil38_60(+(1, +(n1951812_60, 1))), gen_:::nil38_60(b)) ->_R^Omega(1) c21(APPEND(gen_:::nil38_60(+(1, n1951812_60)), gen_:::nil38_60(b))) ->_R^Omega(1) c21(c20(APPEND#1(gen_:::nil38_60(+(1, n1951812_60)), gen_:::nil38_60(b)))) ->_IH c21(c20(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (268) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1805315_60) APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) APPEND#1(gen_:::nil38_60(+(1, n1951812_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1951812_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: APPEND, QUICKSORT, QUICKSORT#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 QUICKSORT = QUICKSORT#1 ---------------------------------------- (269) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND(gen_:::nil38_60(n1961291_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1961291_60) Induction Base: APPEND(gen_:::nil38_60(0), gen_:::nil38_60(b)) Induction Step: APPEND(gen_:::nil38_60(+(n1961291_60, 1)), gen_:::nil38_60(b)) ->_R^Omega(1) c20(APPEND#1(gen_:::nil38_60(+(n1961291_60, 1)), gen_:::nil38_60(b))) ->_R^Omega(1) c20(c21(APPEND(gen_:::nil38_60(n1961291_60), gen_:::nil38_60(b)))) ->_IH c20(c21(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (270) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1805315_60) APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) APPEND#1(gen_:::nil38_60(+(1, n1951812_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1951812_60) APPEND(gen_:::nil38_60(n1961291_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1961291_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: APPEND#1, QUICKSORT, QUICKSORT#1 They will be analysed ascendingly in the following order: APPEND = APPEND#1 APPEND < QUICKSORT#1 QUICKSORT = QUICKSORT#1 ---------------------------------------- (271) RewriteLemmaProof (LOWER BOUND(ID)) Proved the following rewrite lemma: APPEND#1(gen_:::nil38_60(+(1, n1971817_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1971817_60) Induction Base: APPEND#1(gen_:::nil38_60(+(1, 0)), gen_:::nil38_60(b)) Induction Step: APPEND#1(gen_:::nil38_60(+(1, +(n1971817_60, 1))), gen_:::nil38_60(b)) ->_R^Omega(1) c21(APPEND(gen_:::nil38_60(+(1, n1971817_60)), gen_:::nil38_60(b))) ->_R^Omega(1) c21(c20(APPEND#1(gen_:::nil38_60(+(1, n1971817_60)), gen_:::nil38_60(b)))) ->_IH c21(c20(*39_60)) We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n). ---------------------------------------- (272) Obligation: Innermost TRS: Rules: #ABS(#0) -> c15 #ABS(#neg(z0)) -> c16 #ABS(#pos(z0)) -> c17 #ABS(#s(z0)) -> c18 #GREATER(z0, z1) -> c19(#CKGT(#compare(z0, z1)), #COMPARE(z0, z1)) APPEND(z0, z1) -> c20(APPEND#1(z0, z1)) APPEND#1(::(z0, z1), z2) -> c21(APPEND(z1, z2)) APPEND#1(nil, z0) -> c22 APPENDD(z0, z1) -> c23(APPENDD#1(z0, z1)) APPENDD#1(::(z0, z1), z2) -> c24(APPENDD(z1, z2)) APPENDD#1(nil, z0) -> c25 QUICKSORT(z0) -> c26(QUICKSORT#1(z0)) QUICKSORT#1(::(z0, z1)) -> c27(QUICKSORT#2(split(z0, z1), z0), SPLIT(z0, z1)) QUICKSORT#1(nil) -> c28 QUICKSORT#2(tuple#2(z0, z1), z2) -> c29(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z0)) QUICKSORT#2(tuple#2(z0, z1), z2) -> c30(APPEND(quicksort(z0), ::(z2, quicksort(z1))), QUICKSORT(z1)) QUICKSORTD(z0) -> c31(QUICKSORTD#1(z0)) QUICKSORTD#1(::(z0, z1)) -> c32(QUICKSORTD#2(splitD(z0, z1), z0), SPLITD(z0, z1)) QUICKSORTD#1(nil) -> c33 QUICKSORTD#2(tuple#2(z0, z1), z2) -> c34(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z0)) QUICKSORTD#2(tuple#2(z0, z1), z2) -> c35(APPENDD(quicksortD(z0), ::(z2, quicksortD(z1))), QUICKSORTD(z1)) SPLIT(z0, z1) -> c36(SPLIT#1(z1, z0)) SPLIT#1(::(z0, z1), z2) -> c37(SPLIT#2(split(z2, z1), z2, z0), SPLIT(z2, z1)) SPLIT#1(nil, z0) -> c38 SPLIT#2(tuple#2(z0, z1), z2, z3) -> c39(SPLIT#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLIT#3(#false, z0, z1, z2) -> c40 SPLIT#3(#true, z0, z1, z2) -> c41 SPLITD(z0, z1) -> c42(SPLITD#1(z1, z0)) SPLITD#1(::(z0, z1), z2) -> c43(SPLITD#2(splitD(z2, z1), z2, z0), SPLITD(z2, z1)) SPLITD#1(nil, z0) -> c44 SPLITD#2(tuple#2(z0, z1), z2, z3) -> c45(SPLITD#3(#greater(z3, z2), z0, z1, z3), #GREATER(z3, z2)) SPLITD#3(#false, z0, z1, z2) -> c46 SPLITD#3(#true, z0, z1, z2) -> c47 TESTLIST(z0) -> c48(#ABS(#0)) TESTLIST(z0) -> c49(#ABS(#pos(#s(#s(#s(#s(#0))))))) TESTLIST(z0) -> c50(#ABS(#pos(#s(#s(#s(#s(#s(#0)))))))) TESTLIST(z0) -> c51(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))))) TESTLIST(z0) -> c52(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#0)))))))))) TESTLIST(z0) -> c53(#ABS(#pos(#s(#0)))) TESTLIST(z0) -> c54(#ABS(#pos(#s(#s(#0))))) TESTLIST(z0) -> c55(#ABS(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))) TESTLIST(z0) -> c56(#ABS(#pos(#s(#s(#s(#s(#s(#s(#0))))))))) TESTLIST(z0) -> c57(#ABS(#pos(#s(#s(#s(#0)))))) TESTQUICKSORT(z0) -> c58(QUICKSORT(testList(#unit)), TESTLIST(#unit)) TESTQUICKSORT2(z0) -> c59(QUICKSORT(testList(#unit)), TESTLIST(#unit)) #CKGT(#EQ) -> c #CKGT(#GT) -> c1 #CKGT(#LT) -> c2 #COMPARE(#0, #0) -> c3 #COMPARE(#0, #neg(z0)) -> c4 #COMPARE(#0, #pos(z0)) -> c5 #COMPARE(#0, #s(z0)) -> c6 #COMPARE(#neg(z0), #0) -> c7 #COMPARE(#neg(z0), #neg(z1)) -> c8(#COMPARE(z1, z0)) #COMPARE(#neg(z0), #pos(z1)) -> c9 #COMPARE(#pos(z0), #0) -> c10 #COMPARE(#pos(z0), #neg(z1)) -> c11 #COMPARE(#pos(z0), #pos(z1)) -> c12(#COMPARE(z0, z1)) #COMPARE(#s(z0), #0) -> c13 #COMPARE(#s(z0), #s(z1)) -> c14(#COMPARE(z0, z1)) #ckgt(#EQ) -> #false #ckgt(#GT) -> #true #ckgt(#LT) -> #false #compare(#0, #0) -> #EQ #compare(#0, #neg(z0)) -> #GT #compare(#0, #pos(z0)) -> #LT #compare(#0, #s(z0)) -> #LT #compare(#neg(z0), #0) -> #LT #compare(#neg(z0), #neg(z1)) -> #compare(z1, z0) #compare(#neg(z0), #pos(z1)) -> #LT #compare(#pos(z0), #0) -> #GT #compare(#pos(z0), #neg(z1)) -> #GT #compare(#pos(z0), #pos(z1)) -> #compare(z0, z1) #compare(#s(z0), #0) -> #GT #compare(#s(z0), #s(z1)) -> #compare(z0, z1) #abs(#0) -> #0 #abs(#neg(z0)) -> #pos(z0) #abs(#pos(z0)) -> #pos(z0) #abs(#s(z0)) -> #pos(#s(z0)) #greater(z0, z1) -> #ckgt(#compare(z0, z1)) append(z0, z1) -> append#1(z0, z1) append#1(::(z0, z1), z2) -> ::(z0, append(z1, z2)) append#1(nil, z0) -> z0 appendD(z0, z1) -> appendD#1(z0, z1) appendD#1(::(z0, z1), z2) -> ::(z0, appendD(z1, z2)) appendD#1(nil, z0) -> z0 quicksort(z0) -> quicksort#1(z0) quicksort#1(::(z0, z1)) -> quicksort#2(split(z0, z1), z0) quicksort#1(nil) -> nil quicksort#2(tuple#2(z0, z1), z2) -> append(quicksort(z0), ::(z2, quicksort(z1))) quicksortD(z0) -> quicksortD#1(z0) quicksortD#1(::(z0, z1)) -> quicksortD#2(splitD(z0, z1), z0) quicksortD#1(nil) -> nil quicksortD#2(tuple#2(z0, z1), z2) -> appendD(quicksortD(z0), ::(z2, quicksortD(z1))) split(z0, z1) -> split#1(z1, z0) split#1(::(z0, z1), z2) -> split#2(split(z2, z1), z2, z0) split#1(nil, z0) -> tuple#2(nil, nil) split#2(tuple#2(z0, z1), z2, z3) -> split#3(#greater(z3, z2), z0, z1, z3) split#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) split#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) splitD(z0, z1) -> splitD#1(z1, z0) splitD#1(::(z0, z1), z2) -> splitD#2(splitD(z2, z1), z2, z0) splitD#1(nil, z0) -> tuple#2(nil, nil) splitD#2(tuple#2(z0, z1), z2, z3) -> splitD#3(#greater(z3, z2), z0, z1, z3) splitD#3(#false, z0, z1, z2) -> tuple#2(::(z2, z0), z1) splitD#3(#true, z0, z1, z2) -> tuple#2(z0, ::(z2, z1)) testList(z0) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) testQuicksort(z0) -> quicksort(testList(#unit)) testQuicksort2(z0) -> quicksort(testList(#unit)) Types: #ABS :: #0:#neg:#pos:#s -> c15:c16:c17:c18 #0 :: #0:#neg:#pos:#s c15 :: c15:c16:c17:c18 #neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c16 :: c15:c16:c17:c18 #pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c17 :: c15:c16:c17:c18 #s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s c18 :: c15:c16:c17:c18 #GREATER :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c19 c19 :: c:c1:c2 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c19 #CKGT :: #EQ:#GT:#LT -> c:c1:c2 #compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT #COMPARE :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 APPEND :: :::nil -> :::nil -> c20 c20 :: c21:c22 -> c20 APPEND#1 :: :::nil -> :::nil -> c21:c22 :: :: #0:#neg:#pos:#s -> :::nil -> :::nil c21 :: c20 -> c21:c22 nil :: :::nil c22 :: c21:c22 APPENDD :: :::nil -> :::nil -> c23 c23 :: c24:c25 -> c23 APPENDD#1 :: :::nil -> :::nil -> c24:c25 c24 :: c23 -> c24:c25 c25 :: c24:c25 QUICKSORT :: :::nil -> c26 c26 :: c27:c28 -> c26 QUICKSORT#1 :: :::nil -> c27:c28 c27 :: c29:c30 -> c36 -> c27:c28 QUICKSORT#2 :: tuple#2 -> #0:#neg:#pos:#s -> c29:c30 split :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLIT :: #0:#neg:#pos:#s -> :::nil -> c36 c28 :: c27:c28 tuple#2 :: :::nil -> :::nil -> tuple#2 c29 :: c20 -> c26 -> c29:c30 quicksort :: :::nil -> :::nil c30 :: c20 -> c26 -> c29:c30 QUICKSORTD :: :::nil -> c31 c31 :: c32:c33 -> c31 QUICKSORTD#1 :: :::nil -> c32:c33 c32 :: c34:c35 -> c42 -> c32:c33 QUICKSORTD#2 :: tuple#2 -> #0:#neg:#pos:#s -> c34:c35 splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2 SPLITD :: #0:#neg:#pos:#s -> :::nil -> c42 c33 :: c32:c33 c34 :: c23 -> c31 -> c34:c35 quicksortD :: :::nil -> :::nil c35 :: c23 -> c31 -> c34:c35 c36 :: c37:c38 -> c36 SPLIT#1 :: :::nil -> #0:#neg:#pos:#s -> c37:c38 c37 :: c39 -> c36 -> c37:c38 SPLIT#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c39 c38 :: c37:c38 c39 :: c40:c41 -> c19 -> c39 SPLIT#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c40:c41 #greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true #false :: #false:#true c40 :: c40:c41 #true :: #false:#true c41 :: c40:c41 c42 :: c43:c44 -> c42 SPLITD#1 :: :::nil -> #0:#neg:#pos:#s -> c43:c44 c43 :: c45 -> c42 -> c43:c44 SPLITD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> c45 c44 :: c43:c44 c45 :: c46:c47 -> c19 -> c45 SPLITD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> c46:c47 c46 :: c46:c47 c47 :: c46:c47 TESTLIST :: #unit -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c48 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c49 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c50 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c51 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c52 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c53 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c54 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c55 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c56 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 c57 :: c15:c16:c17:c18 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 TESTQUICKSORT :: a -> c58 c58 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c58 testList :: #unit -> :::nil #unit :: #unit TESTQUICKSORT2 :: b -> c59 c59 :: c26 -> c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 -> c59 #EQ :: #EQ:#GT:#LT c :: c:c1:c2 #GT :: #EQ:#GT:#LT c1 :: c:c1:c2 #LT :: #EQ:#GT:#LT c2 :: c:c1:c2 c3 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c4 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c5 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c6 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c7 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c8 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c9 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c10 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c11 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c12 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c13 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 c14 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 #ckgt :: #EQ:#GT:#LT -> #false:#true #abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s append :: :::nil -> :::nil -> :::nil append#1 :: :::nil -> :::nil -> :::nil appendD :: :::nil -> :::nil -> :::nil appendD#1 :: :::nil -> :::nil -> :::nil quicksort#1 :: :::nil -> :::nil quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil quicksortD#1 :: :::nil -> :::nil quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2 splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2 splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2 testQuicksort :: c -> :::nil testQuicksort2 :: d -> :::nil hole_c15:c16:c17:c181_60 :: c15:c16:c17:c18 hole_#0:#neg:#pos:#s2_60 :: #0:#neg:#pos:#s hole_c193_60 :: c19 hole_c:c1:c24_60 :: c:c1:c2 hole_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c145_60 :: c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 hole_#EQ:#GT:#LT6_60 :: #EQ:#GT:#LT hole_c207_60 :: c20 hole_:::nil8_60 :: :::nil hole_c21:c229_60 :: c21:c22 hole_c2310_60 :: c23 hole_c24:c2511_60 :: c24:c25 hole_c2612_60 :: c26 hole_c27:c2813_60 :: c27:c28 hole_c29:c3014_60 :: c29:c30 hole_c3615_60 :: c36 hole_tuple#216_60 :: tuple#2 hole_c3117_60 :: c31 hole_c32:c3318_60 :: c32:c33 hole_c34:c3519_60 :: c34:c35 hole_c4220_60 :: c42 hole_c37:c3821_60 :: c37:c38 hole_c3922_60 :: c39 hole_c40:c4123_60 :: c40:c41 hole_#false:#true24_60 :: #false:#true hole_c43:c4425_60 :: c43:c44 hole_c4526_60 :: c45 hole_c46:c4727_60 :: c46:c47 hole_c48:c49:c50:c51:c52:c53:c54:c55:c56:c5728_60 :: c48:c49:c50:c51:c52:c53:c54:c55:c56:c57 hole_#unit29_60 :: #unit hole_c5830_60 :: c58 hole_a31_60 :: a hole_c5932_60 :: c59 hole_b33_60 :: b hole_c34_60 :: c hole_d35_60 :: d gen_#0:#neg:#pos:#s36_60 :: Nat -> #0:#neg:#pos:#s gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60 :: Nat -> c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c14 gen_:::nil38_60 :: Nat -> :::nil Lemmas: #compare(gen_#0:#neg:#pos:#s36_60(n40_60), gen_#0:#neg:#pos:#s36_60(n40_60)) -> #EQ, rt in Omega(0) #COMPARE(gen_#0:#neg:#pos:#s36_60(n319207_60), gen_#0:#neg:#pos:#s36_60(n319207_60)) -> gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(n319207_60), rt in Omega(0) appendD#1(gen_:::nil38_60(n638822_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n638822_60, b)), rt in Omega(0) append#1(gen_:::nil38_60(n642011_60), gen_:::nil38_60(b)) -> gen_:::nil38_60(+(n642011_60, b)), rt in Omega(0) splitD#1(gen_:::nil38_60(n645222_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n645222_60), gen_:::nil38_60(0)), rt in Omega(0) quicksortD#1(gen_:::nil38_60(n648994_60)) -> gen_:::nil38_60(n648994_60), rt in Omega(0) SPLITD#1(gen_:::nil38_60(+(1, n930953_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n930953_60) SPLITD(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n920889_60)) -> *39_60, rt in Omega(n920889_60) split#1(gen_:::nil38_60(n1206352_60), gen_#0:#neg:#pos:#s36_60(0)) -> tuple#2(gen_:::nil38_60(n1206352_60), gen_:::nil38_60(0)), rt in Omega(0) quicksort#1(gen_:::nil38_60(n1211300_60)) -> gen_:::nil38_60(n1211300_60), rt in Omega(0) SPLIT#1(gen_:::nil38_60(+(1, n1505819_60)), gen_#0:#neg:#pos:#s36_60(0)) -> *39_60, rt in Omega(n1505819_60) SPLIT(gen_#0:#neg:#pos:#s36_60(0), gen_:::nil38_60(n1493911_60)) -> *39_60, rt in Omega(n1493911_60) APPENDD#1(gen_:::nil38_60(+(1, n1805315_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1805315_60) APPENDD(gen_:::nil38_60(n1797809_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1797809_60) APPEND#1(gen_:::nil38_60(+(1, n1971817_60)), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1971817_60) APPEND(gen_:::nil38_60(n1961291_60), gen_:::nil38_60(b)) -> *39_60, rt in Omega(n1961291_60) Generator Equations: gen_#0:#neg:#pos:#s36_60(0) <=> #0 gen_#0:#neg:#pos:#s36_60(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s36_60(x)) gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(0) <=> c3 gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(+(x, 1)) <=> c8(gen_c3:c4:c5:c6:c7:c8:c9:c10:c11:c12:c13:c1437_60(x)) gen_:::nil38_60(0) <=> nil gen_:::nil38_60(+(x, 1)) <=> ::(#0, gen_:::nil38_60(x)) The following defined symbols remain to be analysed: QUICKSORT#1, QUICKSORT They will be analysed ascendingly in the following order: QUICKSORT = QUICKSORT#1