/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1), O(n^3)) proof of /export/starexec/sandbox/benchmark/theBenchmark.xml # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). (0) CpxRelTRS (1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 188 ms] (2) CpxRelTRS (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms] (4) CpxWeightedTrs (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms] (6) CpxTypedWeightedTrs (7) CompletionProof [UPPER BOUND(ID), 43 ms] (8) CpxTypedWeightedCompleteTrs (9) NarrowingProof [BOTH BOUNDS(ID, ID), 714 ms] (10) CpxTypedWeightedCompleteTrs (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms] (12) CpxRNTS (13) InliningProof [UPPER BOUND(ID), 1083 ms] (14) CpxRNTS (15) SimplificationProof [BOTH BOUNDS(ID, ID), 7 ms] (16) CpxRNTS (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms] (18) CpxRNTS (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (20) CpxRNTS (21) IntTrsBoundProof [UPPER BOUND(ID), 321 ms] (22) CpxRNTS (23) IntTrsBoundProof [UPPER BOUND(ID), 76 ms] (24) CpxRNTS (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (26) CpxRNTS (27) IntTrsBoundProof [UPPER BOUND(ID), 925 ms] (28) CpxRNTS (29) IntTrsBoundProof [UPPER BOUND(ID), 186 ms] (30) CpxRNTS (31) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (32) CpxRNTS (33) IntTrsBoundProof [UPPER BOUND(ID), 220 ms] (34) CpxRNTS (35) IntTrsBoundProof [UPPER BOUND(ID), 32 ms] (36) CpxRNTS (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (38) CpxRNTS (39) IntTrsBoundProof [UPPER BOUND(ID), 312 ms] (40) CpxRNTS (41) IntTrsBoundProof [UPPER BOUND(ID), 94 ms] (42) CpxRNTS (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (44) CpxRNTS (45) IntTrsBoundProof [UPPER BOUND(ID), 642 ms] (46) CpxRNTS (47) IntTrsBoundProof [UPPER BOUND(ID), 198 ms] (48) CpxRNTS (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (50) CpxRNTS (51) IntTrsBoundProof [UPPER BOUND(ID), 706 ms] (52) CpxRNTS (53) IntTrsBoundProof [UPPER BOUND(ID), 298 ms] (54) CpxRNTS (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (56) CpxRNTS (57) IntTrsBoundProof [UPPER BOUND(ID), 199 ms] (58) CpxRNTS (59) IntTrsBoundProof [UPPER BOUND(ID), 72 ms] (60) CpxRNTS (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (62) CpxRNTS (63) IntTrsBoundProof [UPPER BOUND(ID), 528 ms] (64) CpxRNTS (65) IntTrsBoundProof [UPPER BOUND(ID), 118 ms] (66) CpxRNTS (67) ResultPropagationProof [UPPER BOUND(ID), 1 ms] (68) CpxRNTS (69) IntTrsBoundProof [UPPER BOUND(ID), 3334 ms] (70) CpxRNTS (71) IntTrsBoundProof [UPPER BOUND(ID), 744 ms] (72) CpxRNTS (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (74) CpxRNTS (75) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (76) CpxRNTS (77) IntTrsBoundProof [UPPER BOUND(ID), 51 ms] (78) CpxRNTS (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (80) CpxRNTS (81) IntTrsBoundProof [UPPER BOUND(ID), 348 ms] (82) CpxRNTS (83) IntTrsBoundProof [UPPER BOUND(ID), 115 ms] (84) CpxRNTS (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (86) CpxRNTS (87) IntTrsBoundProof [UPPER BOUND(ID), 329 ms] (88) CpxRNTS (89) IntTrsBoundProof [UPPER BOUND(ID), 114 ms] (90) CpxRNTS (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (92) CpxRNTS (93) IntTrsBoundProof [UPPER BOUND(ID), 127 ms] (94) CpxRNTS (95) IntTrsBoundProof [UPPER BOUND(ID), 3 ms] (96) CpxRNTS (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (98) CpxRNTS (99) IntTrsBoundProof [UPPER BOUND(ID), 2179 ms] (100) CpxRNTS (101) IntTrsBoundProof [UPPER BOUND(ID), 368 ms] (102) CpxRNTS (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (104) CpxRNTS (105) IntTrsBoundProof [UPPER BOUND(ID), 237 ms] (106) CpxRNTS (107) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (108) CpxRNTS (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (110) CpxRNTS (111) IntTrsBoundProof [UPPER BOUND(ID), 187 ms] (112) CpxRNTS (113) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (114) CpxRNTS (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (116) CpxRNTS (117) IntTrsBoundProof [UPPER BOUND(ID), 891 ms] (118) CpxRNTS (119) IntTrsBoundProof [UPPER BOUND(ID), 107 ms] (120) CpxRNTS (121) ResultPropagationProof [UPPER BOUND(ID), 0 ms] (122) CpxRNTS (123) IntTrsBoundProof [UPPER BOUND(ID), 166 ms] (124) CpxRNTS (125) IntTrsBoundProof [UPPER BOUND(ID), 0 ms] (126) CpxRNTS (127) FinalProof [FINISHED, 0 ms] (128) BOUNDS(1, n^3) (129) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] (130) TRS for Loop Detection (131) DecreasingLoopProof [LOWER BOUND(ID), 97 ms] (132) BEST (133) proven lower bound (134) LowerBoundPropagationProof [FINISHED, 0 ms] (135) BOUNDS(n^1, INF) (136) TRS for Loop Detection ---------------------------------------- (0) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #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 insert(@x, @l) -> insert#1(@x, @l, @x) insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) sortAll(@l) -> sortAll#1(@l) sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) sortAll#1(nil) -> nil sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) split(@l) -> split#1(@l) split#1(::(@x, @xs)) -> insert(@x, split(@xs)) split#1(nil) -> nil splitAndSort(@l) -> sortAll(split(@l)) splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) splitqs#1(nil, @pivot) -> tuple#2(nil, nil) splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #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) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #eq(nil, tuple#2(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), nil) -> #false #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID)) proved innermost termination of relative rules ---------------------------------------- (2) Obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #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 insert(@x, @l) -> insert#1(@x, @l, @x) insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) sortAll(@l) -> sortAll#1(@l) sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) sortAll#1(nil) -> nil sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) split(@l) -> split#1(@l) split#1(::(@x, @xs)) -> insert(@x, split(@xs)) split#1(nil) -> nil splitAndSort(@l) -> sortAll(split(@l)) splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) splitqs#1(nil, @pivot) -> tuple#2(nil, nil) splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #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) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #eq(nil, tuple#2(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), nil) -> #false #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID)) Transformed relative TRS to weighted TRS ---------------------------------------- (4) Obligation: The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] append(@l, @ys) -> append#1(@l, @ys) [1] append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1] append#1(nil, @ys) -> @ys [1] insert(@x, @l) -> insert#1(@x, @l, @x) [1] insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1] insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1] insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1] insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1] insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1] insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1] quicksort(@l) -> quicksort#1(@l) [1] quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1] quicksort#1(nil) -> nil [1] quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1] sortAll(@l) -> sortAll#1(@l) [1] sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1] sortAll#1(nil) -> nil [1] sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1] split(@l) -> split#1(@l) [1] split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1] split#1(nil) -> nil [1] splitAndSort(@l) -> sortAll(split(@l)) [1] splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1] splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1] splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1] splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1] splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1] splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), nil) -> #false [0] #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] Rewrite Strategy: INNERMOST ---------------------------------------- (5) TypeInferenceProof (BOTH BOUNDS(ID, ID)) Infered types. ---------------------------------------- (6) Obligation: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] append(@l, @ys) -> append#1(@l, @ys) [1] append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1] append#1(nil, @ys) -> @ys [1] insert(@x, @l) -> insert#1(@x, @l, @x) [1] insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1] insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1] insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1] insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1] insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1] insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1] quicksort(@l) -> quicksort#1(@l) [1] quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1] quicksort#1(nil) -> nil [1] quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1] sortAll(@l) -> sortAll#1(@l) [1] sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1] sortAll#1(nil) -> nil [1] sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1] split(@l) -> split#1(@l) [1] split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1] split#1(nil) -> nil [1] splitAndSort(@l) -> sortAll(split(@l)) [1] splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1] splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1] splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1] splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1] splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1] splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), nil) -> #false [0] #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] The TRS has the following type information: #equal :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true #eq :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true #greater :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true #ckgt :: #EQ:#GT:#LT -> #false:#true #compare :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #EQ:#GT:#LT append :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s :: :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s nil :: :::nil:tuple#2:#0:#neg:#pos:#s insert :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s insert#4 :: #false:#true -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s #false :: #false:#true #true :: #false:#true quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s split :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s splitqs#3 :: #false:#true -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s #and :: #false:#true -> #false:#true -> #false:#true #EQ :: #EQ:#GT:#LT #GT :: #EQ:#GT:#LT #LT :: #EQ:#GT:#LT #0 :: :::nil:tuple#2:#0:#neg:#pos:#s #neg :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s #pos :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s #s :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s Rewrite Strategy: INNERMOST ---------------------------------------- (7) 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: sortAll_1 sortAll#1_1 sortAll#2_2 splitAndSort_1 (c) The following functions are completely defined: quicksort_1 split_1 #greater_2 #equal_2 splitqs_2 split#1_1 insert_2 splitqs#1_2 splitqs#2_3 insert#1_3 quicksort#1_1 insert#2_4 insert#3_5 splitqs#3_4 insert#4_6 quicksort#2_2 append_2 append#1_2 #and_2 #ckgt_1 #compare_2 #eq_2 Due to the following rules being added: #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] split#1(v0) -> null_split#1 [0] splitqs#1(v0, v1) -> null_splitqs#1 [0] splitqs#2(v0, v1, v2) -> null_splitqs#2 [0] insert#1(v0, v1, v2) -> null_insert#1 [0] quicksort#1(v0) -> null_quicksort#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0] splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0] insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0] quicksort#2(v0, v1) -> null_quicksort#2 [0] append#1(v0, v1) -> null_append#1 [0] And the following fresh constants: null_#and, null_#ckgt, null_#compare, null_#eq, null_split#1, null_splitqs#1, null_splitqs#2, null_insert#1, null_quicksort#1, null_insert#2, null_insert#3, null_splitqs#3, null_insert#4, null_quicksort#2, null_append#1 ---------------------------------------- (8) 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: #equal(@x, @y) -> #eq(@x, @y) [1] #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1] append(@l, @ys) -> append#1(@l, @ys) [1] append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1] append#1(nil, @ys) -> @ys [1] insert(@x, @l) -> insert#1(@x, @l, @x) [1] insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1] insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1] insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1] insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1] insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1] insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1] quicksort(@l) -> quicksort#1(@l) [1] quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1] quicksort#1(nil) -> nil [1] quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1] sortAll(@l) -> sortAll#1(@l) [1] sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1] sortAll#1(nil) -> nil [1] sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1] split(@l) -> split#1(@l) [1] split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1] split#1(nil) -> nil [1] splitAndSort(@l) -> sortAll(split(@l)) [1] splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1] splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1] splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1] splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1] splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1] splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), nil) -> #false [0] #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] split#1(v0) -> null_split#1 [0] splitqs#1(v0, v1) -> null_splitqs#1 [0] splitqs#2(v0, v1, v2) -> null_splitqs#2 [0] insert#1(v0, v1, v2) -> null_insert#1 [0] quicksort#1(v0) -> null_quicksort#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0] splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0] insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0] quicksort#2(v0, v1) -> null_quicksort#2 [0] append#1(v0, v1) -> null_append#1 [0] The TRS has the following type information: #equal :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #eq :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #greater :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq #compare :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #EQ:#GT:#LT:null_#compare append :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 :: :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 nil :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#4 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #false :: #false:#true:null_#and:null_#ckgt:null_#eq #true :: #false:#true:null_#and:null_#ckgt:null_#eq quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 split :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#3 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #neg :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #pos :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #s :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq null_split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#4 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 Rewrite Strategy: INNERMOST ---------------------------------------- (9) NarrowingProof (BOTH BOUNDS(ID, ID)) Narrowed the inner basic terms of all right-hand sides by a single narrowing step. ---------------------------------------- (10) Obligation: Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is: Runtime Complexity Weighted TRS with Types. The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) [1] #greater(#0, #0) -> #ckgt(#EQ) [1] #greater(#0, #neg(@y')) -> #ckgt(#GT) [1] #greater(#0, #pos(@y'')) -> #ckgt(#LT) [1] #greater(#0, #s(@y1)) -> #ckgt(#LT) [1] #greater(#neg(@x'), #0) -> #ckgt(#LT) [1] #greater(#neg(@x''), #neg(@y2)) -> #ckgt(#compare(@y2, @x'')) [1] #greater(#neg(@x1), #pos(@y3)) -> #ckgt(#LT) [1] #greater(#pos(@x2), #0) -> #ckgt(#GT) [1] #greater(#pos(@x3), #neg(@y4)) -> #ckgt(#GT) [1] #greater(#pos(@x4), #pos(@y5)) -> #ckgt(#compare(@x4, @y5)) [1] #greater(#s(@x5), #0) -> #ckgt(#GT) [1] #greater(#s(@x6), #s(@y6)) -> #ckgt(#compare(@x6, @y6)) [1] #greater(@x, @y) -> #ckgt(null_#compare) [1] append(@l, @ys) -> append#1(@l, @ys) [1] append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1] append#1(nil, @ys) -> @ys [1] insert(@x, @l) -> insert#1(@x, @l, @x) [1] insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1] insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1] insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1] insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [2] insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1] insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1] quicksort(@l) -> quicksort#1(@l) [1] quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs#1(@zs, @z), @z) [2] quicksort#1(nil) -> nil [1] quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort#1(@xs), ::(@z, quicksort#1(@ys))) [3] sortAll(@l) -> sortAll#1(@l) [1] sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1] sortAll#1(nil) -> nil [1] sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1] split(@l) -> split#1(@l) [1] split#1(::(@x, @xs)) -> insert(@x, split#1(@xs)) [2] split#1(nil) -> nil [1] splitAndSort(@l) -> sortAll(split#1(@l)) [2] splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1] splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) [2] splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1] splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) [2] splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1] splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1] #and(#false, #false) -> #false [0] #and(#false, #true) -> #false [0] #and(#true, #false) -> #false [0] #and(#true, #true) -> #true [0] #ckgt(#EQ) -> #false [0] #ckgt(#GT) -> #true [0] #ckgt(#LT) -> #false [0] #compare(#0, #0) -> #EQ [0] #compare(#0, #neg(@y)) -> #GT [0] #compare(#0, #pos(@y)) -> #LT [0] #compare(#0, #s(@y)) -> #LT [0] #compare(#neg(@x), #0) -> #LT [0] #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0] #compare(#neg(@x), #pos(@y)) -> #LT [0] #compare(#pos(@x), #0) -> #GT [0] #compare(#pos(@x), #neg(@y)) -> #GT [0] #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0] #compare(#s(@x), #0) -> #GT [0] #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0] #eq(#0, #0) -> #true [0] #eq(#0, #neg(@y)) -> #false [0] #eq(#0, #pos(@y)) -> #false [0] #eq(#0, #s(@y)) -> #false [0] #eq(#neg(@x), #0) -> #false [0] #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0] #eq(#neg(@x), #pos(@y)) -> #false [0] #eq(#pos(@x), #0) -> #false [0] #eq(#pos(@x), #neg(@y)) -> #false [0] #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0] #eq(#s(@x), #0) -> #false [0] #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0] #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #eq(::(@x_1, @x_2), nil) -> #false [0] #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0] #eq(nil, ::(@y_1, @y_2)) -> #false [0] #eq(nil, nil) -> #true [0] #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0] #eq(tuple#2(@x_1, @x_2), nil) -> #false [0] #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0] #and(v0, v1) -> null_#and [0] #ckgt(v0) -> null_#ckgt [0] #compare(v0, v1) -> null_#compare [0] #eq(v0, v1) -> null_#eq [0] split#1(v0) -> null_split#1 [0] splitqs#1(v0, v1) -> null_splitqs#1 [0] splitqs#2(v0, v1, v2) -> null_splitqs#2 [0] insert#1(v0, v1, v2) -> null_insert#1 [0] quicksort#1(v0) -> null_quicksort#1 [0] insert#2(v0, v1, v2, v3) -> null_insert#2 [0] insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0] splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0] insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0] quicksort#2(v0, v1) -> null_quicksort#2 [0] append#1(v0, v1) -> null_append#1 [0] The TRS has the following type information: #equal :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #eq :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #greater :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq #ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq #compare :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #EQ:#GT:#LT:null_#compare append :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 :: :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 nil :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 insert#4 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #false :: #false:#true:null_#and:null_#ckgt:null_#eq #true :: #false:#true:null_#and:null_#ckgt:null_#eq quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 split :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 splitqs#3 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq #EQ :: #EQ:#GT:#LT:null_#compare #GT :: #EQ:#GT:#LT:null_#compare #LT :: #EQ:#GT:#LT:null_#compare #0 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #neg :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #pos :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 #s :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq null_#compare :: #EQ:#GT:#LT:null_#compare null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq null_split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_splitqs#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_insert#4 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 null_append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 Rewrite Strategy: INNERMOST ---------------------------------------- (11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID)) Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction. The constant constructors are abstracted as follows: nil => 1 #false => 1 #true => 2 #EQ => 1 #GT => 2 #LT => 3 #0 => 0 null_#and => 0 null_#ckgt => 0 null_#compare => 0 null_#eq => 0 null_split#1 => 0 null_splitqs#1 => 0 null_splitqs#2 => 0 null_insert#1 => 0 null_quicksort#1 => 0 null_insert#2 => 0 null_insert#3 => 0 null_splitqs#3 => 0 null_insert#4 => 0 null_quicksort#2 => 0 null_append#1 => 0 ---------------------------------------- (12) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #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 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #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(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: z = 1 + @x', @x' >= 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @y' >= 0, z' = 1 + @y', z = 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(1) :|: z = 0, z' = 0 #greater(z, z') -{ 1 }-> #ckgt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0 append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 1, @ys >= 0 append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0 insert(z, z') -{ 1 }-> insert#1(@x, @l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z', z'') -{ 1 }-> insert#2(@l, @keyX, @valX, @x) :|: @valX >= 0, @keyX >= 0, @l >= 0, z = 1 + @valX + @keyX, @x >= 0, z' = @l, z'' = @x insert#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, @keyX, @ls, @valX, @x) :|: z1 = @x, @ls >= 0, @keyX >= 0, @valX >= 0, @l1 >= 0, @x >= 0, z = 1 + @l1 + @ls, z'' = @valX, z' = @keyX insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + @valX + 1) + @keyX) + 1 :|: z1 = @x, @keyX >= 0, @valX >= 0, @x >= 0, z = 1, z'' = @valX, z' = @keyX insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) :|: @key1 >= 0, @keyX >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @x, z = 1 + @vals1 + @key1, z' = @keyX, z'' = @ls insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z1 = v3, z3 = v5, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, v5 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + @vals1 + @key1) + insert(@x, @ls) :|: @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z = 1, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + @valX + @vals1) + @key1) + @ls :|: z = 2, @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + @z + quicksort#1(@ys)) :|: z' = @z, z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, @z >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 sortAll(z) -{ 1 }-> sortAll#1(@l) :|: z = @l, @l >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(@xs) :|: z' = @xs, @vals >= 0, z = 1 + @vals + @key, @xs >= 0, @key >= 0 split(z) -{ 1 }-> split#1(@l) :|: z = @l, @l >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 splitAndSort(z) -{ 2 }-> sortAll(split#1(@l)) :|: z = @l, @l >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' = @pivot, @pivot >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs ---------------------------------------- (13) InliningProof (UPPER BOUND(ID)) Inlined the following terminating rules on right-hand sides where appropriate: splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #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 ---------------------------------------- (14) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #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 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0 #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #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(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0 #greater(z, z') -{ 1 }-> 2 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0 append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0 append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 1, @ys >= 0 append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0 insert(z, z') -{ 1 }-> insert#1(@x, @l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l insert#1(z, z', z'') -{ 1 }-> insert#2(@l, @keyX, @valX, @x) :|: @valX >= 0, @keyX >= 0, @l >= 0, z = 1 + @valX + @keyX, @x >= 0, z' = @l, z'' = @x insert#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, @keyX, @ls, @valX, @x) :|: z1 = @x, @ls >= 0, @keyX >= 0, @valX >= 0, @l1 >= 0, @x >= 0, z = 1 + @l1 + @ls, z'' = @valX, z' = @keyX insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + @valX + 1) + @keyX) + 1 :|: z1 = @x, @keyX >= 0, @valX >= 0, @x >= 0, z = 1, z'' = @valX, z' = @keyX insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) :|: @key1 >= 0, @keyX >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @x, z = 1 + @vals1 + @key1, z' = @keyX, z'' = @ls insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z1 = v3, z3 = v5, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, v5 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + @vals1 + @key1) + insert(@x, @ls) :|: @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z = 1, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + @valX + @vals1) + @key1) + @ls :|: z = 2, @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + @z + quicksort#1(@ys)) :|: z' = @z, z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, @z >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 sortAll(z) -{ 1 }-> sortAll#1(@l) :|: z = @l, @l >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(@xs) :|: z' = @xs, @vals >= 0, z = 1 + @vals + @key, @xs >= 0, @key >= 0 split(z) -{ 1 }-> split#1(@l) :|: z = @l, @l >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0 splitAndSort(z) -{ 2 }-> sortAll(split#1(@l)) :|: z = @l, @l >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' = @pivot, @pivot >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs ---------------------------------------- (15) SimplificationProof (BOTH BOUNDS(ID, ID)) Simplified the RNTS by moving equalities from the constraints into the right-hand sides. ---------------------------------------- (16) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 ---------------------------------------- (17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID)) Found the following analysis order by SCC decomposition: { #and } { #compare } { #ckgt } { splitqs#3 } { append#1, append } { #eq } { splitqs#2 } { #greater } { insert#4, insert#2, insert, insert#3, insert#1 } { #equal } { splitqs#1 } { split#1 } { splitqs } { quicksort#1, quicksort#2 } { split } { quicksort } { sortAll#2, sortAll, sortAll#1 } { splitAndSort } ---------------------------------------- (18) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} ---------------------------------------- (19) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (20) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} ---------------------------------------- (21) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (22) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: ?, size: O(1) [2] ---------------------------------------- (23) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #and after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 0 ---------------------------------------- (24) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (25) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (26) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (27) 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 ---------------------------------------- (28) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: ?, size: O(1) [3] ---------------------------------------- (29) 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 ---------------------------------------- (30) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] ---------------------------------------- (33) 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 ---------------------------------------- (34) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: ?, size: O(1) [2] ---------------------------------------- (35) 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 ---------------------------------------- (36) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (39) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitqs#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1 ---------------------------------------- (40) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (41) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitqs#3 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (42) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (43) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (44) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] ---------------------------------------- (45) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT 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' ---------------------------------------- (46) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: ?, size: O(n^1) [z + z'] append: runtime: ?, size: O(n^1) [z + z'] ---------------------------------------- (47) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: append#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 2*z Computed RUNTIME bound using CoFloCo for: append after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z ---------------------------------------- (48) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] ---------------------------------------- (49) ResultPropagationProof (UPPER BOUND(ID)) Applied inner abstraction using the recently inferred runtime/size bounds where possible. ---------------------------------------- (50) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] ---------------------------------------- (51) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #eq 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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: ?, size: O(1) [2] ---------------------------------------- (53) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #eq 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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: 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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] ---------------------------------------- (57) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitqs#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + z + z'' ---------------------------------------- (58) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: ?, size: O(n^1) [1 + z + z''] ---------------------------------------- (59) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitqs#2 after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 3 ---------------------------------------- (60) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] ---------------------------------------- (63) 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 ---------------------------------------- (64) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: ?, size: O(1) [2] ---------------------------------------- (65) 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 ---------------------------------------- (66) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (69) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: insert#4 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 6 + z' + z'' + z1 + z2 + z3 Computed SIZE bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + z + z' + z'' + z1 Computed SIZE bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + 2*z + z' Computed SIZE bound using CoFloCo for: insert#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 5 + z + z'' + z1 + z2 Computed SIZE bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4 + z + z' + z'' ---------------------------------------- (70) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: ?, size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: ?, size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: ?, size: O(n^1) [4 + 2*z + z'] insert#3: runtime: ?, size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: ?, size: O(n^1) [4 + z + z' + z''] ---------------------------------------- (71) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: insert#4 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 10 + 6*z'' Computed RUNTIME bound using CoFloCo for: insert#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 6*z Computed RUNTIME bound using CoFloCo for: insert after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 3 + 6*z' Computed RUNTIME bound using CoFloCo for: insert#3 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 12 + 6*z'' Computed RUNTIME bound using CoFloCo for: insert#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 6*z' ---------------------------------------- (72) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0 insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] ---------------------------------------- (75) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 2 ---------------------------------------- (76) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: ?, size: O(1) [2] ---------------------------------------- (77) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: #equal after applying outer abstraction to obtain an ITS, resulting in: O(1) with polynomial bound: 1 ---------------------------------------- (78) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] ---------------------------------------- (81) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: splitqs#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z ---------------------------------------- (82) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: ?, size: O(n^1) [2 + z] ---------------------------------------- (83) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: splitqs#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 5*z ---------------------------------------- (84) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0 splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + 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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] ---------------------------------------- (87) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: split#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 4*z ---------------------------------------- (88) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: ?, size: O(n^1) [4*z] ---------------------------------------- (89) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: split#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 1 + 5*z + 24*z^2 ---------------------------------------- (90) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 1 }-> split#1(z) :|: z >= 0 split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] ---------------------------------------- (93) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitqs after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + z' ---------------------------------------- (94) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: ?, size: O(n^1) [2 + z'] ---------------------------------------- (95) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: splitqs after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2 + 5*z' ---------------------------------------- (96) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + 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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] ---------------------------------------- (99) 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: 1 + 2*z Computed SIZE bound using CoFloCo for: quicksort#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z + z' ---------------------------------------- (100) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: ?, size: O(n^1) [1 + 2*z] quicksort#2: runtime: ?, size: O(n^1) [1 + 2*z + z'] ---------------------------------------- (101) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using CoFloCo for: quicksort#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 69 + 184*z + 90*z^2 Computed RUNTIME bound using KoAT for: quicksort#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 147 + 372*z + 180*z^2 ---------------------------------------- (102) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0 quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] ---------------------------------------- (105) 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: 4*z ---------------------------------------- (106) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: ?, size: O(n^1) [4*z] ---------------------------------------- (107) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: split after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 2 + 5*z + 24*z^2 ---------------------------------------- (108) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] ---------------------------------------- (111) 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: 1 + 2*z ---------------------------------------- (112) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: ?, size: O(n^1) [1 + 2*z] ---------------------------------------- (113) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: quicksort after applying outer abstraction to obtain an ITS, resulting in: O(n^2) with polynomial bound: 70 + 184*z + 90*z^2 ---------------------------------------- (114) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] ---------------------------------------- (117) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using KoAT for: sortAll#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 1 + 2*z + 2*z' Computed SIZE bound using KoAT for: sortAll after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z Computed SIZE bound using KoAT for: sortAll#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 2*z ---------------------------------------- (118) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] sortAll#2: runtime: ?, size: O(n^1) [1 + 2*z + 2*z'] sortAll: runtime: ?, size: O(n^1) [2*z] sortAll#1: runtime: ?, size: O(n^1) [2*z] ---------------------------------------- (119) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: sortAll#2 after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3 Computed RUNTIME bound using KoAT for: sortAll after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 150 + 955*z + 1464*z^2 + 540*z^3 Computed RUNTIME bound using KoAT for: sortAll#1 after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 149 + 955*z + 1464*z^2 + 540*z^3 ---------------------------------------- (120) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0 sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z'] sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] ---------------------------------------- (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: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0 sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z'] sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] ---------------------------------------- (123) IntTrsBoundProof (UPPER BOUND(ID)) Computed SIZE bound using CoFloCo for: splitAndSort after applying outer abstraction to obtain an ITS, resulting in: O(n^1) with polynomial bound: 8*z ---------------------------------------- (124) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0 sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: {splitAndSort} Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z'] sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] splitAndSort: runtime: ?, size: O(n^1) [8*z] ---------------------------------------- (125) IntTrsBoundProof (UPPER BOUND(ID)) Computed RUNTIME bound using KoAT for: splitAndSort after applying outer abstraction to obtain an ITS, resulting in: O(n^3) with polynomial bound: 153 + 3825*z + 23448*z^2 + 34560*z^3 ---------------------------------------- (126) Obligation: Complexity RNTS consisting of the following rules: #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2 #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1 #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1 #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1 #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 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 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0 #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 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 = 0, z' - 1 >= 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 #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0 #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0 #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1 #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2 #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0 #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0 #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2 #greater(z, z') -{ 1 }-> 2 :|: z >= 0, z' >= 0, 0 = 2 #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3 #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3 #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0 #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0 append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0 append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0 append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0 insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0 insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1 insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1 insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0 insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1 insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0 quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0 quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0 quicksort#1(z) -{ 1 }-> 1 :|: z = 1 quicksort#1(z) -{ 0 }-> 0 :|: z >= 0 quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0 quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0 sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0 sortAll#1(z) -{ 1 }-> 1 :|: z = 1 sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0 split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0 split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0 split#1(z) -{ 1 }-> 1 :|: z = 1 split#1(z) -{ 0 }-> 0 :|: z >= 0 splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0 splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0 splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0 splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0 splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0 splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0 splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0 splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0 Function symbols to be analyzed: Previous analysis results are: #and: runtime: O(1) [0], size: O(1) [2] #compare: runtime: O(1) [0], size: O(1) [3] #ckgt: runtime: O(1) [0], size: O(1) [2] splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1] append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z'] append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z'] #eq: runtime: O(1) [0], size: O(1) [2] splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z''] #greater: runtime: O(1) [1], size: O(1) [2] insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3] insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1] insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z'] insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2] insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z''] #equal: runtime: O(1) [1], size: O(1) [2] splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z] split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z] splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z'] quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z'] split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z] quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z] sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z'] sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z] splitAndSort: runtime: O(n^3) [153 + 3825*z + 23448*z^2 + 34560*z^3], size: O(n^1) [8*z] ---------------------------------------- (127) FinalProof (FINISHED) Computed overall runtime complexity ---------------------------------------- (128) BOUNDS(1, n^3) ---------------------------------------- (129) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) Transformed a relative TRS into a decreasing-loop problem. ---------------------------------------- (130) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #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 insert(@x, @l) -> insert#1(@x, @l, @x) insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) sortAll(@l) -> sortAll#1(@l) sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) sortAll#1(nil) -> nil sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) split(@l) -> split#1(@l) split#1(::(@x, @xs)) -> insert(@x, split(@xs)) split#1(nil) -> nil splitAndSort(@l) -> sortAll(split(@l)) splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) splitqs#1(nil, @pivot) -> tuple#2(nil, nil) splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #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) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #eq(nil, tuple#2(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), nil) -> #false #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (131) DecreasingLoopProof (LOWER BOUND(ID)) The following loop(s) give(s) rise to the lower bound Omega(n^1): The rewrite sequence splitqs(@pivot, ::(@x1_0, @xs2_0)) ->^+ splitqs#2(splitqs(@pivot, @xs2_0), @pivot, @x1_0) gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. The pumping substitution is [@xs2_0 / ::(@x1_0, @xs2_0)]. The result substitution is [ ]. ---------------------------------------- (132) Complex Obligation (BEST) ---------------------------------------- (133) Obligation: Proved the lower bound n^1 for the following obligation: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #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 insert(@x, @l) -> insert#1(@x, @l, @x) insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) sortAll(@l) -> sortAll#1(@l) sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) sortAll#1(nil) -> nil sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) split(@l) -> split#1(@l) split#1(::(@x, @xs)) -> insert(@x, split(@xs)) split#1(nil) -> nil splitAndSort(@l) -> sortAll(split(@l)) splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) splitqs#1(nil, @pivot) -> tuple#2(nil, nil) splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #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) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #eq(nil, tuple#2(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), nil) -> #false #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) Rewrite Strategy: INNERMOST ---------------------------------------- (134) LowerBoundPropagationProof (FINISHED) Propagated lower bound. ---------------------------------------- (135) BOUNDS(n^1, INF) ---------------------------------------- (136) Obligation: Analyzing the following TRS for decreasing loops: The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3). The TRS R consists of the following rules: #equal(@x, @y) -> #eq(@x, @y) #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 insert(@x, @l) -> insert#1(@x, @l, @x) insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) quicksort(@l) -> quicksort#1(@l) quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) quicksort#1(nil) -> nil quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) sortAll(@l) -> sortAll#1(@l) sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) sortAll#1(nil) -> nil sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) split(@l) -> split#1(@l) split#1(::(@x, @xs)) -> insert(@x, split(@xs)) split#1(nil) -> nil splitAndSort(@l) -> sortAll(split(@l)) splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) splitqs#1(nil, @pivot) -> tuple#2(nil, nil) splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) The (relative) TRS S consists of the following rules: #and(#false, #false) -> #false #and(#false, #true) -> #false #and(#true, #false) -> #false #and(#true, #true) -> #true #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) #eq(#0, #0) -> #true #eq(#0, #neg(@y)) -> #false #eq(#0, #pos(@y)) -> #false #eq(#0, #s(@y)) -> #false #eq(#neg(@x), #0) -> #false #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) #eq(#neg(@x), #pos(@y)) -> #false #eq(#pos(@x), #0) -> #false #eq(#pos(@x), #neg(@y)) -> #false #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) #eq(#s(@x), #0) -> #false #eq(#s(@x), #s(@y)) -> #eq(@x, @y) #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) #eq(::(@x_1, @x_2), nil) -> #false #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false #eq(nil, ::(@y_1, @y_2)) -> #false #eq(nil, nil) -> #true #eq(nil, tuple#2(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false #eq(tuple#2(@x_1, @x_2), nil) -> #false #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) Rewrite Strategy: INNERMOST