222.91/58.11	WORST_CASE(Omega(n^2), O(n^2))
222.91/58.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
222.91/58.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
222.91/58.12	
222.91/58.12	
222.91/58.12	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, n^2).
222.91/58.12	
222.91/58.12	(0) CpxRelTRS
222.91/58.12	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 182 ms]
222.91/58.12	(2) CpxRelTRS
222.91/58.12	(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
222.91/58.12	(4) CpxWeightedTrs
222.91/58.12	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
222.91/58.12	    (6) CpxTypedWeightedTrs
222.91/58.12	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (8) CpxTypedWeightedCompleteTrs
222.91/58.12	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 5 ms]
222.91/58.12	    (10) CpxTypedWeightedCompleteTrs
222.91/58.12	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (12) CpxRNTS
222.91/58.12	    (13) InliningProof [UPPER BOUND(ID), 729 ms]
222.91/58.12	    (14) CpxRNTS
222.91/58.12	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 10 ms]
222.91/58.12	    (16) CpxRNTS
222.91/58.12	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms]
222.91/58.12	    (18) CpxRNTS
222.91/58.12	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (20) CpxRNTS
222.91/58.12	    (21) IntTrsBoundProof [UPPER BOUND(ID), 756 ms]
222.91/58.12	    (22) CpxRNTS
222.91/58.12	    (23) IntTrsBoundProof [UPPER BOUND(ID), 134 ms]
222.91/58.12	    (24) CpxRNTS
222.91/58.12	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (26) CpxRNTS
222.91/58.12	    (27) IntTrsBoundProof [UPPER BOUND(ID), 422 ms]
222.91/58.12	    (28) CpxRNTS
222.91/58.12	    (29) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
222.91/58.12	    (30) CpxRNTS
222.91/58.12	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (32) CpxRNTS
222.91/58.12	    (33) IntTrsBoundProof [UPPER BOUND(ID), 431 ms]
222.91/58.12	    (34) CpxRNTS
222.91/58.12	    (35) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
222.91/58.12	    (36) CpxRNTS
222.91/58.12	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (38) CpxRNTS
222.91/58.12	    (39) IntTrsBoundProof [UPPER BOUND(ID), 1176 ms]
222.91/58.12	    (40) CpxRNTS
222.91/58.12	    (41) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
222.91/58.12	    (42) CpxRNTS
222.91/58.12	    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (44) CpxRNTS
222.91/58.12	    (45) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
222.91/58.12	    (46) CpxRNTS
222.91/58.12	    (47) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
222.91/58.12	    (48) CpxRNTS
222.91/58.12	    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (50) CpxRNTS
222.91/58.12	    (51) IntTrsBoundProof [UPPER BOUND(ID), 672 ms]
222.91/58.12	    (52) CpxRNTS
222.91/58.12	    (53) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
222.91/58.12	    (54) CpxRNTS
222.91/58.12	    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (56) CpxRNTS
222.91/58.12	    (57) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
222.91/58.12	    (58) CpxRNTS
222.91/58.12	    (59) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
222.91/58.12	    (60) CpxRNTS
222.91/58.12	    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (62) CpxRNTS
222.91/58.12	    (63) IntTrsBoundProof [UPPER BOUND(ID), 261 ms]
222.91/58.12	    (64) CpxRNTS
222.91/58.12	    (65) IntTrsBoundProof [UPPER BOUND(ID), 44 ms]
222.91/58.12	    (66) CpxRNTS
222.91/58.12	    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (68) CpxRNTS
222.91/58.12	    (69) IntTrsBoundProof [UPPER BOUND(ID), 300 ms]
222.91/58.12	    (70) CpxRNTS
222.91/58.12	    (71) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
222.91/58.12	    (72) CpxRNTS
222.91/58.12	    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (74) CpxRNTS
222.91/58.12	    (75) IntTrsBoundProof [UPPER BOUND(ID), 618 ms]
222.91/58.12	    (76) CpxRNTS
222.91/58.12	    (77) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
222.91/58.12	    (78) CpxRNTS
222.91/58.12	    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (80) CpxRNTS
222.91/58.12	    (81) IntTrsBoundProof [UPPER BOUND(ID), 411 ms]
222.91/58.12	    (82) CpxRNTS
222.91/58.12	    (83) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
222.91/58.12	    (84) CpxRNTS
222.91/58.12	    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (86) CpxRNTS
222.91/58.12	    (87) IntTrsBoundProof [UPPER BOUND(ID), 338 ms]
222.91/58.12	    (88) CpxRNTS
222.91/58.12	    (89) IntTrsBoundProof [UPPER BOUND(ID), 134 ms]
222.91/58.12	    (90) CpxRNTS
222.91/58.12	    (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (92) CpxRNTS
222.91/58.12	    (93) IntTrsBoundProof [UPPER BOUND(ID), 389 ms]
222.91/58.12	    (94) CpxRNTS
222.91/58.12	    (95) IntTrsBoundProof [UPPER BOUND(ID), 134 ms]
222.91/58.12	    (96) CpxRNTS
222.91/58.12	    (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (98) CpxRNTS
222.91/58.12	    (99) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
222.91/58.12	    (100) CpxRNTS
222.91/58.12	    (101) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
222.91/58.12	    (102) CpxRNTS
222.91/58.12	    (103) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (104) CpxRNTS
222.91/58.12	    (105) IntTrsBoundProof [UPPER BOUND(ID), 891 ms]
222.91/58.12	    (106) CpxRNTS
222.91/58.12	    (107) IntTrsBoundProof [UPPER BOUND(ID), 309 ms]
222.91/58.12	    (108) CpxRNTS
222.91/58.12	    (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (110) CpxRNTS
222.91/58.12	    (111) IntTrsBoundProof [UPPER BOUND(ID), 1169 ms]
222.91/58.12	    (112) CpxRNTS
222.91/58.12	    (113) IntTrsBoundProof [UPPER BOUND(ID), 889 ms]
222.91/58.12	    (114) CpxRNTS
222.91/58.12	    (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (116) CpxRNTS
222.91/58.12	    (117) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
222.91/58.12	    (118) CpxRNTS
222.91/58.12	    (119) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
222.91/58.12	    (120) CpxRNTS
222.91/58.12	    (121) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (122) CpxRNTS
222.91/58.12	    (123) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
222.91/58.12	    (124) CpxRNTS
222.91/58.12	    (125) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
222.91/58.12	    (126) CpxRNTS
222.91/58.12	    (127) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (128) CpxRNTS
222.91/58.12	    (129) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
222.91/58.12	    (130) CpxRNTS
222.91/58.12	    (131) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
222.91/58.12	    (132) CpxRNTS
222.91/58.12	    (133) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (134) CpxRNTS
222.91/58.12	    (135) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
222.91/58.12	    (136) CpxRNTS
222.91/58.12	    (137) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
222.91/58.12	    (138) CpxRNTS
222.91/58.12	    (139) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
222.91/58.12	    (140) CpxRNTS
222.91/58.12	    (141) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
222.91/58.12	    (142) CpxRNTS
222.91/58.12	    (143) IntTrsBoundProof [UPPER BOUND(ID), 5 ms]
222.91/58.12	    (144) CpxRNTS
222.91/58.12	    (145) FinalProof [FINISHED, 0 ms]
222.91/58.12	    (146) BOUNDS(1, n^2)
222.91/58.12	(147) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
222.91/58.12	(148) CpxRelTRS
222.91/58.12	    (149) SlicingProof [LOWER BOUND(ID), 0 ms]
222.91/58.12	    (150) CpxRelTRS
222.91/58.12	    (151) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
222.91/58.12	    (152) typed CpxTrs
222.91/58.12	    (153) OrderProof [LOWER BOUND(ID), 5 ms]
222.91/58.12	    (154) typed CpxTrs
222.91/58.12	    (155) RewriteLemmaProof [LOWER BOUND(ID), 2007 ms]
222.91/58.12	    (156) typed CpxTrs
222.91/58.12	    (157) RewriteLemmaProof [LOWER BOUND(ID), 115 ms]
222.91/58.12	    (158) BEST
222.91/58.12	        (159) proven lower bound
222.91/58.12	            (160) LowerBoundPropagationProof [FINISHED, 0 ms]
222.91/58.12	            (161) BOUNDS(n^1, INF)
222.91/58.12	        (162) typed CpxTrs
222.91/58.12	            (163) RewriteLemmaProof [LOWER BOUND(ID), 147 ms]
222.91/58.12	            (164) typed CpxTrs
222.91/58.12	            (165) RewriteLemmaProof [LOWER BOUND(ID), 33 ms]
222.91/58.12	            (166) typed CpxTrs
222.91/58.12	            (167) RewriteLemmaProof [LOWER BOUND(ID), 83 ms]
222.91/58.12	            (168) BEST
222.91/58.12	                (169) proven lower bound
222.91/58.12	                    (170) LowerBoundPropagationProof [FINISHED, 0 ms]
222.91/58.12	                    (171) BOUNDS(n^2, INF)
222.91/58.12	                (172) typed CpxTrs
222.91/58.12	                    (173) RewriteLemmaProof [LOWER BOUND(ID), 48 ms]
222.91/58.12	                    (174) typed CpxTrs
222.91/58.12	                    (175) RewriteLemmaProof [LOWER BOUND(ID), 74 ms]
222.91/58.12	                    (176) typed CpxTrs
222.91/58.12	
222.91/58.12	
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(0)
222.91/58.12	Obligation:
222.91/58.12	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, n^2).
222.91/58.12	
222.91/58.12	
222.91/58.12	The TRS R consists of the following rules:
222.91/58.12	
222.91/58.12	   #abs(#0) -> #0
222.91/58.12	   #abs(#neg(@x)) -> #pos(@x)
222.91/58.12	   #abs(#pos(@x)) -> #pos(@x)
222.91/58.12	   #abs(#s(@x)) -> #pos(#s(@x))
222.91/58.12	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
222.91/58.12	   append(@l, @ys) -> append#1(@l, @ys)
222.91/58.12	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
222.91/58.12	   append#1(nil, @ys) -> @ys
222.91/58.12	   appendD(@l, @ys) -> appendD#1(@l, @ys)
222.91/58.12	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
222.91/58.12	   appendD#1(nil, @ys) -> @ys
222.91/58.12	   quicksort(@l) -> quicksort#1(@l)
222.91/58.12	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
222.91/58.12	   quicksort#1(nil) -> nil
222.91/58.12	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
222.91/58.12	   quicksortD(@l) -> quicksortD#1(@l)
222.91/58.12	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
222.91/58.12	   quicksortD#1(nil) -> nil
222.91/58.12	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
222.91/58.12	   split(@pivot, @l) -> split#1(@l, @pivot)
222.91/58.12	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
222.91/58.12	   split#1(nil, @pivot) -> tuple#2(nil, nil)
222.91/58.12	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
222.91/58.12	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
222.91/58.12	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
222.91/58.12	   splitD(@pivot, @l) -> splitD#1(@l, @pivot)
222.91/58.12	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
222.91/58.12	   splitD#1(nil, @pivot) -> tuple#2(nil, nil)
222.91/58.12	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
222.91/58.12	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
222.91/58.12	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
222.91/58.12	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
222.91/58.12	   testQuicksort(@x) -> quicksort(testList(#unit))
222.91/58.12	   testQuicksort2(@x) -> quicksort(testList(#unit))
222.91/58.12	
222.91/58.12	The (relative) TRS S consists of the following rules:
222.91/58.12	
222.91/58.12	   #ckgt(#EQ) -> #false
222.91/58.12	   #ckgt(#GT) -> #true
222.91/58.12	   #ckgt(#LT) -> #false
222.91/58.12	   #compare(#0, #0) -> #EQ
222.91/58.12	   #compare(#0, #neg(@y)) -> #GT
222.91/58.12	   #compare(#0, #pos(@y)) -> #LT
222.91/58.12	   #compare(#0, #s(@y)) -> #LT
222.91/58.12	   #compare(#neg(@x), #0) -> #LT
222.91/58.12	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
222.91/58.12	   #compare(#neg(@x), #pos(@y)) -> #LT
222.91/58.12	   #compare(#pos(@x), #0) -> #GT
222.91/58.12	   #compare(#pos(@x), #neg(@y)) -> #GT
222.91/58.12	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
222.91/58.12	   #compare(#s(@x), #0) -> #GT
222.91/58.12	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
222.91/58.12	
222.91/58.12	Rewrite Strategy: INNERMOST
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
222.91/58.12	proved termination of relative rules
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(2)
222.91/58.12	Obligation:
222.91/58.12	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, n^2).
222.91/58.12	
222.91/58.12	
222.91/58.12	The TRS R consists of the following rules:
222.91/58.12	
222.91/58.12	   #abs(#0) -> #0
222.91/58.12	   #abs(#neg(@x)) -> #pos(@x)
222.91/58.12	   #abs(#pos(@x)) -> #pos(@x)
222.91/58.12	   #abs(#s(@x)) -> #pos(#s(@x))
222.91/58.12	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
222.91/58.12	   append(@l, @ys) -> append#1(@l, @ys)
222.91/58.12	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
222.91/58.12	   append#1(nil, @ys) -> @ys
222.91/58.12	   appendD(@l, @ys) -> appendD#1(@l, @ys)
222.91/58.12	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
222.91/58.12	   appendD#1(nil, @ys) -> @ys
222.91/58.12	   quicksort(@l) -> quicksort#1(@l)
222.91/58.12	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
222.91/58.12	   quicksort#1(nil) -> nil
222.91/58.12	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
222.91/58.12	   quicksortD(@l) -> quicksortD#1(@l)
222.91/58.12	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
222.91/58.12	   quicksortD#1(nil) -> nil
222.91/58.12	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
222.91/58.12	   split(@pivot, @l) -> split#1(@l, @pivot)
222.91/58.12	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
222.91/58.12	   split#1(nil, @pivot) -> tuple#2(nil, nil)
222.91/58.12	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
222.91/58.12	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
222.91/58.12	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
222.91/58.12	   splitD(@pivot, @l) -> splitD#1(@l, @pivot)
222.91/58.12	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
222.91/58.12	   splitD#1(nil, @pivot) -> tuple#2(nil, nil)
222.91/58.12	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
222.91/58.12	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
222.91/58.12	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
222.91/58.12	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
222.91/58.12	   testQuicksort(@x) -> quicksort(testList(#unit))
222.91/58.12	   testQuicksort2(@x) -> quicksort(testList(#unit))
222.91/58.12	
222.91/58.12	The (relative) TRS S consists of the following rules:
222.91/58.12	
222.91/58.12	   #ckgt(#EQ) -> #false
222.91/58.12	   #ckgt(#GT) -> #true
222.91/58.12	   #ckgt(#LT) -> #false
222.91/58.12	   #compare(#0, #0) -> #EQ
222.91/58.12	   #compare(#0, #neg(@y)) -> #GT
222.91/58.12	   #compare(#0, #pos(@y)) -> #LT
222.91/58.12	   #compare(#0, #s(@y)) -> #LT
222.91/58.12	   #compare(#neg(@x), #0) -> #LT
222.91/58.12	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
222.91/58.12	   #compare(#neg(@x), #pos(@y)) -> #LT
222.91/58.12	   #compare(#pos(@x), #0) -> #GT
222.91/58.12	   #compare(#pos(@x), #neg(@y)) -> #GT
222.91/58.12	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
222.91/58.12	   #compare(#s(@x), #0) -> #GT
222.91/58.12	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
222.91/58.12	
222.91/58.12	Rewrite Strategy: INNERMOST
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
222.91/58.12	Transformed relative TRS to weighted TRS
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(4)
222.91/58.12	Obligation:
222.91/58.12	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
222.91/58.12	
222.91/58.12	
222.91/58.12	The TRS R consists of the following rules:
222.91/58.12	
222.91/58.12	   #abs(#0) -> #0 [1]
222.91/58.12	   #abs(#neg(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#pos(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
222.91/58.12	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
222.91/58.12	   append(@l, @ys) -> append#1(@l, @ys) [1]
222.91/58.12	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
222.91/58.12	   append#1(nil, @ys) -> @ys [1]
222.91/58.12	   appendD(@l, @ys) -> appendD#1(@l, @ys) [1]
222.91/58.12	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) [1]
222.91/58.12	   appendD#1(nil, @ys) -> @ys [1]
222.91/58.12	   quicksort(@l) -> quicksort#1(@l) [1]
222.91/58.12	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z) [1]
222.91/58.12	   quicksort#1(nil) -> nil [1]
222.91/58.12	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
222.91/58.12	   quicksortD(@l) -> quicksortD#1(@l) [1]
222.91/58.12	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z) [1]
222.91/58.12	   quicksortD#1(nil) -> nil [1]
222.91/58.12	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys))) [1]
222.91/58.12	   split(@pivot, @l) -> split#1(@l, @pivot) [1]
222.91/58.12	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   split#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   splitD(@pivot, @l) -> splitD#1(@l, @pivot) [1]
222.91/58.12	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   splitD#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
222.91/58.12	   testQuicksort(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   testQuicksort2(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   #ckgt(#EQ) -> #false [0]
222.91/58.12	   #ckgt(#GT) -> #true [0]
222.91/58.12	   #ckgt(#LT) -> #false [0]
222.91/58.12	   #compare(#0, #0) -> #EQ [0]
222.91/58.12	   #compare(#0, #neg(@y)) -> #GT [0]
222.91/58.12	   #compare(#0, #pos(@y)) -> #LT [0]
222.91/58.12	   #compare(#0, #s(@y)) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #0) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
222.91/58.12	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
222.91/58.12	   #compare(#pos(@x), #0) -> #GT [0]
222.91/58.12	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
222.91/58.12	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
222.91/58.12	   #compare(#s(@x), #0) -> #GT [0]
222.91/58.12	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
222.91/58.12	
222.91/58.12	Rewrite Strategy: INNERMOST
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
222.91/58.12	Infered types.
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(6)
222.91/58.12	Obligation:
222.91/58.12	Runtime Complexity Weighted TRS with Types.
222.91/58.12	The TRS R consists of the following rules:
222.91/58.12	
222.91/58.12	   #abs(#0) -> #0 [1]
222.91/58.12	   #abs(#neg(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#pos(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
222.91/58.12	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
222.91/58.12	   append(@l, @ys) -> append#1(@l, @ys) [1]
222.91/58.12	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
222.91/58.12	   append#1(nil, @ys) -> @ys [1]
222.91/58.12	   appendD(@l, @ys) -> appendD#1(@l, @ys) [1]
222.91/58.12	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) [1]
222.91/58.12	   appendD#1(nil, @ys) -> @ys [1]
222.91/58.12	   quicksort(@l) -> quicksort#1(@l) [1]
222.91/58.12	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z) [1]
222.91/58.12	   quicksort#1(nil) -> nil [1]
222.91/58.12	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
222.91/58.12	   quicksortD(@l) -> quicksortD#1(@l) [1]
222.91/58.12	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z) [1]
222.91/58.12	   quicksortD#1(nil) -> nil [1]
222.91/58.12	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys))) [1]
222.91/58.12	   split(@pivot, @l) -> split#1(@l, @pivot) [1]
222.91/58.12	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   split#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   splitD(@pivot, @l) -> splitD#1(@l, @pivot) [1]
222.91/58.12	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   splitD#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
222.91/58.12	   testQuicksort(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   testQuicksort2(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   #ckgt(#EQ) -> #false [0]
222.91/58.12	   #ckgt(#GT) -> #true [0]
222.91/58.12	   #ckgt(#LT) -> #false [0]
222.91/58.12	   #compare(#0, #0) -> #EQ [0]
222.91/58.12	   #compare(#0, #neg(@y)) -> #GT [0]
222.91/58.12	   #compare(#0, #pos(@y)) -> #LT [0]
222.91/58.12	   #compare(#0, #s(@y)) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #0) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
222.91/58.12	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
222.91/58.12	   #compare(#pos(@x), #0) -> #GT [0]
222.91/58.12	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
222.91/58.12	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
222.91/58.12	   #compare(#s(@x), #0) -> #GT [0]
222.91/58.12	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
222.91/58.12	
222.91/58.12	The TRS has the following type information:
222.91/58.12	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.12	#0 :: #0:#neg:#pos:#s
222.91/58.12	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.12	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.12	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.12	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
222.91/58.12	#ckgt :: #EQ:#GT:#LT -> #false:#true
222.91/58.12	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
222.91/58.12	append :: :::nil -> :::nil -> :::nil
222.91/58.12	append#1 :: :::nil -> :::nil -> :::nil
222.91/58.12	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
222.91/58.12	nil :: :::nil
222.91/58.12	appendD :: :::nil -> :::nil -> :::nil
222.91/58.12	appendD#1 :: :::nil -> :::nil -> :::nil
222.91/58.12	quicksort :: :::nil -> :::nil
222.91/58.12	quicksort#1 :: :::nil -> :::nil
222.91/58.12	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
222.91/58.12	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
222.91/58.12	tuple#2 :: :::nil -> :::nil -> tuple#2
222.91/58.12	quicksortD :: :::nil -> :::nil
222.91/58.12	quicksortD#1 :: :::nil -> :::nil
222.91/58.12	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
222.91/58.12	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
222.91/58.12	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	#false :: #false:#true
222.91/58.12	#true :: #false:#true
222.91/58.12	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
222.91/58.12	testList :: #unit -> :::nil
222.91/58.12	testQuicksort :: a -> :::nil
222.91/58.12	#unit :: #unit
222.91/58.12	testQuicksort2 :: b -> :::nil
222.91/58.12	#EQ :: #EQ:#GT:#LT
222.91/58.12	#GT :: #EQ:#GT:#LT
222.91/58.12	#LT :: #EQ:#GT:#LT
222.91/58.12	
222.91/58.12	Rewrite Strategy: INNERMOST
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(7) CompletionProof (UPPER BOUND(ID))
222.91/58.12	The transformation into a RNTS is sound, since:
222.91/58.12	
222.91/58.12	(a) The obligation is a constructor system where every type has a constant constructor,
222.91/58.12	
222.91/58.12	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
222.91/58.12	
222.91/58.12	   testQuicksort_1
222.91/58.12	   testQuicksort2_1
222.91/58.12	
222.91/58.12	(c) The following functions are completely defined:
222.91/58.12	
222.91/58.12	   testList_1
222.91/58.12	   quicksort_1
222.91/58.12	   splitD_2
222.91/58.12	   quicksortD_1
222.91/58.12	   #greater_2
222.91/58.12	   split_2
222.91/58.12	   split#1_2
222.91/58.12	   #abs_1
222.91/58.12	   quicksortD#1_1
222.91/58.12	   splitD#1_2
222.91/58.12	   split#2_3
222.91/58.12	   quicksort#1_1
222.91/58.12	   quicksortD#2_2
222.91/58.12	   appendD_2
222.91/58.12	   quicksort#2_2
222.91/58.12	   splitD#2_3
222.91/58.12	   appendD#1_2
222.91/58.12	   split#3_4
222.91/58.12	   append_2
222.91/58.12	   splitD#3_4
222.91/58.12	   append#1_2
222.91/58.12	   #ckgt_1
222.91/58.12	   #compare_2
222.91/58.12	
222.91/58.12	Due to the following rules being added:
222.91/58.12	
222.91/58.12	   #ckgt(v0) -> null_#ckgt [0]
222.91/58.12	   #compare(v0, v1) -> null_#compare [0]
222.91/58.12	   split#2(v0, v1, v2) -> const [0]
222.91/58.12	   quicksortD#2(v0, v1) -> nil [0]
222.91/58.12	   quicksort#2(v0, v1) -> nil [0]
222.91/58.12	   splitD#2(v0, v1, v2) -> const [0]
222.91/58.12	   split#3(v0, v1, v2, v3) -> const [0]
222.91/58.12	   splitD#3(v0, v1, v2, v3) -> const [0]
222.91/58.12	
222.91/58.12	And the following fresh constants:    null_#ckgt, null_#compare, const, const1, const2
222.91/58.12	
222.91/58.12	----------------------------------------
222.91/58.12	
222.91/58.12	(8)
222.91/58.12	Obligation:
222.91/58.12	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
222.91/58.12	
222.91/58.12	Runtime Complexity Weighted TRS with Types.
222.91/58.12	The TRS R consists of the following rules:
222.91/58.12	
222.91/58.12	   #abs(#0) -> #0 [1]
222.91/58.12	   #abs(#neg(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#pos(@x)) -> #pos(@x) [1]
222.91/58.12	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
222.91/58.12	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
222.91/58.12	   append(@l, @ys) -> append#1(@l, @ys) [1]
222.91/58.12	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
222.91/58.12	   append#1(nil, @ys) -> @ys [1]
222.91/58.12	   appendD(@l, @ys) -> appendD#1(@l, @ys) [1]
222.91/58.12	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) [1]
222.91/58.12	   appendD#1(nil, @ys) -> @ys [1]
222.91/58.12	   quicksort(@l) -> quicksort#1(@l) [1]
222.91/58.12	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z) [1]
222.91/58.12	   quicksort#1(nil) -> nil [1]
222.91/58.12	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
222.91/58.12	   quicksortD(@l) -> quicksortD#1(@l) [1]
222.91/58.12	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z) [1]
222.91/58.12	   quicksortD#1(nil) -> nil [1]
222.91/58.12	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys))) [1]
222.91/58.12	   split(@pivot, @l) -> split#1(@l, @pivot) [1]
222.91/58.12	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   split#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   splitD(@pivot, @l) -> splitD#1(@l, @pivot) [1]
222.91/58.12	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x) [1]
222.91/58.12	   splitD#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.12	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
222.91/58.12	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.12	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.12	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
222.91/58.12	   testQuicksort(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   testQuicksort2(@x) -> quicksort(testList(#unit)) [1]
222.91/58.12	   #ckgt(#EQ) -> #false [0]
222.91/58.12	   #ckgt(#GT) -> #true [0]
222.91/58.12	   #ckgt(#LT) -> #false [0]
222.91/58.12	   #compare(#0, #0) -> #EQ [0]
222.91/58.12	   #compare(#0, #neg(@y)) -> #GT [0]
222.91/58.12	   #compare(#0, #pos(@y)) -> #LT [0]
222.91/58.12	   #compare(#0, #s(@y)) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #0) -> #LT [0]
222.91/58.12	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
222.91/58.14	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
222.91/58.14	   #compare(#pos(@x), #0) -> #GT [0]
222.91/58.14	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
222.91/58.14	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
222.91/58.14	   #compare(#s(@x), #0) -> #GT [0]
222.91/58.14	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
222.91/58.14	   #ckgt(v0) -> null_#ckgt [0]
222.91/58.14	   #compare(v0, v1) -> null_#compare [0]
222.91/58.14	   split#2(v0, v1, v2) -> const [0]
222.91/58.14	   quicksortD#2(v0, v1) -> nil [0]
222.91/58.14	   quicksort#2(v0, v1) -> nil [0]
222.91/58.14	   splitD#2(v0, v1, v2) -> const [0]
222.91/58.14	   split#3(v0, v1, v2, v3) -> const [0]
222.91/58.14	   splitD#3(v0, v1, v2, v3) -> const [0]
222.91/58.14	
222.91/58.14	The TRS has the following type information:
222.91/58.14	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#0 :: #0:#neg:#pos:#s
222.91/58.14	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#ckgt
222.91/58.14	#ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#ckgt
222.91/58.14	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare
222.91/58.14	append :: :::nil -> :::nil -> :::nil
222.91/58.14	append#1 :: :::nil -> :::nil -> :::nil
222.91/58.14	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
222.91/58.14	nil :: :::nil
222.91/58.14	appendD :: :::nil -> :::nil -> :::nil
222.91/58.14	appendD#1 :: :::nil -> :::nil -> :::nil
222.91/58.14	quicksort :: :::nil -> :::nil
222.91/58.14	quicksort#1 :: :::nil -> :::nil
222.91/58.14	quicksort#2 :: tuple#2:const -> #0:#neg:#pos:#s -> :::nil
222.91/58.14	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2:const
222.91/58.14	tuple#2 :: :::nil -> :::nil -> tuple#2:const
222.91/58.14	quicksortD :: :::nil -> :::nil
222.91/58.14	quicksortD#1 :: :::nil -> :::nil
222.91/58.14	quicksortD#2 :: tuple#2:const -> #0:#neg:#pos:#s -> :::nil
222.91/58.14	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2:const
222.91/58.14	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	split#2 :: tuple#2:const -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	split#3 :: #false:#true:null_#ckgt -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	#false :: #false:#true:null_#ckgt
222.91/58.14	#true :: #false:#true:null_#ckgt
222.91/58.14	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	splitD#2 :: tuple#2:const -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	splitD#3 :: #false:#true:null_#ckgt -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2:const
222.91/58.14	testList :: #unit -> :::nil
222.91/58.14	testQuicksort :: a -> :::nil
222.91/58.14	#unit :: #unit
222.91/58.14	testQuicksort2 :: b -> :::nil
222.91/58.14	#EQ :: #EQ:#GT:#LT:null_#compare
222.91/58.14	#GT :: #EQ:#GT:#LT:null_#compare
222.91/58.14	#LT :: #EQ:#GT:#LT:null_#compare
222.91/58.14	null_#ckgt :: #false:#true:null_#ckgt
222.91/58.14	null_#compare :: #EQ:#GT:#LT:null_#compare
222.91/58.14	const :: tuple#2:const
222.91/58.14	const1 :: a
222.91/58.14	const2 :: b
222.91/58.14	
222.91/58.14	Rewrite Strategy: INNERMOST
222.91/58.14	----------------------------------------
222.91/58.14	
222.91/58.14	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
222.91/58.14	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
222.91/58.14	----------------------------------------
222.91/58.14	
222.91/58.14	(10)
222.91/58.14	Obligation:
222.91/58.14	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
222.91/58.14	
222.91/58.14	Runtime Complexity Weighted TRS with Types.
222.91/58.14	The TRS R consists of the following rules:
222.91/58.14	
222.91/58.14	   #abs(#0) -> #0 [1]
222.91/58.14	   #abs(#neg(@x)) -> #pos(@x) [1]
222.91/58.14	   #abs(#pos(@x)) -> #pos(@x) [1]
222.91/58.14	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
222.91/58.14	   #greater(#0, #0) -> #ckgt(#EQ) [1]
222.91/58.14	   #greater(#0, #neg(@y')) -> #ckgt(#GT) [1]
222.91/58.14	   #greater(#0, #pos(@y'')) -> #ckgt(#LT) [1]
222.91/58.14	   #greater(#0, #s(@y1)) -> #ckgt(#LT) [1]
222.91/58.14	   #greater(#neg(@x'), #0) -> #ckgt(#LT) [1]
222.91/58.14	   #greater(#neg(@x''), #neg(@y2)) -> #ckgt(#compare(@y2, @x'')) [1]
222.91/58.14	   #greater(#neg(@x1), #pos(@y3)) -> #ckgt(#LT) [1]
222.91/58.14	   #greater(#pos(@x2), #0) -> #ckgt(#GT) [1]
222.91/58.14	   #greater(#pos(@x3), #neg(@y4)) -> #ckgt(#GT) [1]
222.91/58.14	   #greater(#pos(@x4), #pos(@y5)) -> #ckgt(#compare(@x4, @y5)) [1]
222.91/58.14	   #greater(#s(@x5), #0) -> #ckgt(#GT) [1]
222.91/58.14	   #greater(#s(@x6), #s(@y6)) -> #ckgt(#compare(@x6, @y6)) [1]
222.91/58.14	   #greater(@x, @y) -> #ckgt(null_#compare) [1]
222.91/58.14	   append(@l, @ys) -> append#1(@l, @ys) [1]
222.91/58.14	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
222.91/58.14	   append#1(nil, @ys) -> @ys [1]
222.91/58.14	   appendD(@l, @ys) -> appendD#1(@l, @ys) [1]
222.91/58.14	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys)) [1]
222.91/58.14	   appendD#1(nil, @ys) -> @ys [1]
222.91/58.14	   quicksort(@l) -> quicksort#1(@l) [1]
222.91/58.14	   quicksort#1(::(@z, @zs)) -> quicksort#2(split#1(@zs, @z), @z) [2]
222.91/58.14	   quicksort#1(nil) -> nil [1]
222.91/58.14	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort#1(@xs), ::(@z, quicksort#1(@ys))) [3]
222.91/58.14	   quicksortD(@l) -> quicksortD#1(@l) [1]
222.91/58.14	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD#1(@zs, @z), @z) [2]
222.91/58.14	   quicksortD#1(nil) -> nil [1]
222.91/58.14	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD#1(@xs), ::(@z, quicksortD#1(@ys))) [3]
222.91/58.14	   split(@pivot, @l) -> split#1(@l, @pivot) [1]
222.91/58.14	   split#1(::(@x, @xs), @pivot) -> split#2(split#1(@xs, @pivot), @pivot, @x) [2]
222.91/58.14	   split#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.14	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) [2]
222.91/58.14	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.14	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.14	   splitD(@pivot, @l) -> splitD#1(@l, @pivot) [1]
222.91/58.14	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD#1(@xs, @pivot), @pivot, @x) [2]
222.91/58.14	   splitD#1(nil, @pivot) -> tuple#2(nil, nil) [1]
222.91/58.14	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) [2]
222.91/58.14	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
222.91/58.14	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
222.91/58.14	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
222.91/58.14	   testQuicksort(@x) -> quicksort(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2]
222.91/58.14	   testQuicksort2(@x) -> quicksort(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2]
222.91/58.14	   #ckgt(#EQ) -> #false [0]
222.91/58.14	   #ckgt(#GT) -> #true [0]
222.91/58.14	   #ckgt(#LT) -> #false [0]
222.91/58.14	   #compare(#0, #0) -> #EQ [0]
222.91/58.14	   #compare(#0, #neg(@y)) -> #GT [0]
222.91/58.14	   #compare(#0, #pos(@y)) -> #LT [0]
222.91/58.14	   #compare(#0, #s(@y)) -> #LT [0]
222.91/58.14	   #compare(#neg(@x), #0) -> #LT [0]
222.91/58.14	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
222.91/58.14	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
222.91/58.14	   #compare(#pos(@x), #0) -> #GT [0]
222.91/58.14	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
222.91/58.14	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
222.91/58.14	   #compare(#s(@x), #0) -> #GT [0]
222.91/58.14	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
222.91/58.14	   #ckgt(v0) -> null_#ckgt [0]
222.91/58.14	   #compare(v0, v1) -> null_#compare [0]
222.91/58.14	   split#2(v0, v1, v2) -> const [0]
222.91/58.14	   quicksortD#2(v0, v1) -> nil [0]
222.91/58.14	   quicksort#2(v0, v1) -> nil [0]
222.91/58.14	   splitD#2(v0, v1, v2) -> const [0]
222.91/58.14	   split#3(v0, v1, v2, v3) -> const [0]
222.91/58.14	   splitD#3(v0, v1, v2, v3) -> const [0]
222.91/58.14	
222.91/58.14	The TRS has the following type information:
222.91/58.14	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#0 :: #0:#neg:#pos:#s
222.91/58.14	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
222.91/58.14	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#ckgt
222.91/58.14	#ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#ckgt
222.91/58.14	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare
222.91/58.14	append :: :::nil -> :::nil -> :::nil
222.91/58.14	append#1 :: :::nil -> :::nil -> :::nil
222.91/58.14	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
222.91/58.14	nil :: :::nil
222.91/58.14	appendD :: :::nil -> :::nil -> :::nil
222.91/58.14	appendD#1 :: :::nil -> :::nil -> :::nil
222.91/58.14	quicksort :: :::nil -> :::nil
222.91/58.14	quicksort#1 :: :::nil -> :::nil
222.91/58.14	quicksort#2 :: tuple#2:const -> #0:#neg:#pos:#s -> :::nil
223.22/58.17	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2:const
223.22/58.17	tuple#2 :: :::nil -> :::nil -> tuple#2:const
223.22/58.17	quicksortD :: :::nil -> :::nil
223.22/58.17	quicksortD#1 :: :::nil -> :::nil
223.22/58.17	quicksortD#2 :: tuple#2:const -> #0:#neg:#pos:#s -> :::nil
223.22/58.17	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2:const
223.22/58.17	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	split#2 :: tuple#2:const -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	split#3 :: #false:#true:null_#ckgt -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	#false :: #false:#true:null_#ckgt
223.22/58.17	#true :: #false:#true:null_#ckgt
223.22/58.17	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	splitD#2 :: tuple#2:const -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	splitD#3 :: #false:#true:null_#ckgt -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2:const
223.22/58.17	testList :: #unit -> :::nil
223.22/58.17	testQuicksort :: a -> :::nil
223.22/58.17	#unit :: #unit
223.22/58.17	testQuicksort2 :: b -> :::nil
223.22/58.17	#EQ :: #EQ:#GT:#LT:null_#compare
223.22/58.17	#GT :: #EQ:#GT:#LT:null_#compare
223.22/58.17	#LT :: #EQ:#GT:#LT:null_#compare
223.22/58.17	null_#ckgt :: #false:#true:null_#ckgt
223.22/58.17	null_#compare :: #EQ:#GT:#LT:null_#compare
223.22/58.17	const :: tuple#2:const
223.22/58.17	const1 :: a
223.22/58.17	const2 :: b
223.22/58.17	
223.22/58.17	Rewrite Strategy: INNERMOST
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
223.22/58.17	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
223.22/58.17	The constant constructors are abstracted as follows:
223.22/58.17	
223.22/58.17	   #0 => 0
223.22/58.17	   nil => 0
223.22/58.17	   #false => 1
223.22/58.17	   #true => 2
223.22/58.17	   #unit => 0
223.22/58.17	   #EQ => 1
223.22/58.17	   #GT => 2
223.22/58.17	   #LT => 3
223.22/58.17	   null_#ckgt => 0
223.22/58.17	   null_#compare => 0
223.22/58.17	   const => 0
223.22/58.17	   const1 => 0
223.22/58.17	   const2 => 0
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(12)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z = 1 + @x', @x' >= 0, z' = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @y' >= 0, z' = 1 + @y', z = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(1) :|: z = 0, z' = 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 0, @ys >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 0, @ys >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   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
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(@l) :|: z = @l, @l >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + @z + quicksortD#1(@ys)) :|: z' = @z, z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, @z >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   split(z, z') -{ 1 }-> split#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' = @pivot, @pivot >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   split#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
223.22/58.17	   split#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
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' = @pivot, @pivot >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   splitD#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
223.22/58.17	   splitD#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
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z = @x, @x >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
223.22/58.17	
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(13) InliningProof (UPPER BOUND(ID))
223.22/58.17	Inlined the following terminating rules on right-hand sides where appropriate:
223.22/58.17	
223.22/58.17	   split#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
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   split#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
223.22/58.17	   splitD#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
223.22/58.17	   splitD#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
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(14)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 0, @ys >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 0, @ys >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   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
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(@l) :|: z = @l, @l >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + @z + quicksortD#1(@ys)) :|: z' = @z, z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, @z >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
223.22/58.17	   split(z, z') -{ 1 }-> split#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' = @pivot, @pivot >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   split#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
223.22/58.17	   split#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
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' = @pivot, @pivot >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
223.22/58.17	   splitD#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
223.22/58.17	   splitD#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
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z = @x, @x >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
223.22/58.17	
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
223.22/58.17	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(16)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
223.22/58.17	Found the following analysis order by SCC decomposition:
223.22/58.17	
223.22/58.17	   { appendD, appendD#1 }
223.22/58.17	   { split#3 }
223.22/58.17	   { splitD#3 }
223.22/58.17	   { #compare }
223.22/58.17	   { #ckgt }
223.22/58.17	   { append#1, append }
223.22/58.17	   { #abs }
223.22/58.17	   { split#2 }
223.22/58.17	   { splitD#2 }
223.22/58.17	   { #greater }
223.22/58.17	   { testList }
223.22/58.17	   { split#1 }
223.22/58.17	   { splitD#1 }
223.22/58.17	   { split }
223.22/58.17	   { quicksort#1, quicksort#2 }
223.22/58.17	   { quicksortD#2, quicksortD#1 }
223.22/58.17	   { splitD }
223.22/58.17	   { quicksort }
223.22/58.17	   { quicksortD }
223.22/58.17	   { testQuicksort2 }
223.22/58.17	   { testQuicksort }
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(18)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(19) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.17	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(20)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(21) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.17	
223.22/58.17	Computed SIZE bound using CoFloCo for: appendD
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(n^1) with polynomial bound: z + z'
223.22/58.17	
223.22/58.17	Computed SIZE bound using CoFloCo for: appendD#1
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(n^1) with polynomial bound: z + z'
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(22)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {appendD,appendD#1}, {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	Previous analysis results are:
223.22/58.17	  appendD: runtime: ?, size: O(n^1) [z + z']
223.22/58.17	  appendD#1: runtime: ?, size: O(n^1) [z + z']
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(23) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.17	
223.22/58.17	Computed RUNTIME bound using CoFloCo for: appendD
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(n^1) with polynomial bound: 2 + 2*z
223.22/58.17	
223.22/58.17	Computed RUNTIME bound using CoFloCo for: appendD#1
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(n^1) with polynomial bound: 1 + 2*z
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(24)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 1 }-> appendD#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> 1 + @x + appendD(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	Previous analysis results are:
223.22/58.17	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.17	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(25) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.17	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(26)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	Previous analysis results are:
223.22/58.17	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.17	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(27) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.17	
223.22/58.17	Computed SIZE bound using CoFloCo for: split#3
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(28)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.17	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.17	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.17	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.17	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.17	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.17	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.17	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.17	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.17	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.17	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.17	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.17	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.17	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.17	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.17	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.17	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.17	
223.22/58.17	Function symbols to be analyzed: {split#3}, {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.17	Previous analysis results are:
223.22/58.17	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.17	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.17	  split#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(29) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.17	
223.22/58.17	Computed RUNTIME bound using CoFloCo for: split#3
223.22/58.17	after applying outer abstraction to obtain an ITS,
223.22/58.17	resulting in: O(1) with polynomial bound: 1
223.22/58.17	
223.22/58.17	----------------------------------------
223.22/58.17	
223.22/58.17	(30)
223.22/58.17	Obligation:
223.22/58.17	Complexity RNTS consisting of the following rules:
223.22/58.17	
223.22/58.17	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.17	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.17	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.17	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.18	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.18	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.18	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.18	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	
223.22/58.18	Function symbols to be analyzed: {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.18	Previous analysis results are:
223.22/58.18	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(31) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.18	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(32)
223.22/58.18	Obligation:
223.22/58.18	Complexity RNTS consisting of the following rules:
223.22/58.18	
223.22/58.18	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.18	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.18	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.18	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.18	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.18	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	
223.22/58.18	Function symbols to be analyzed: {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.18	Previous analysis results are:
223.22/58.18	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(33) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.18	
223.22/58.18	Computed SIZE bound using CoFloCo for: splitD#3
223.22/58.18	after applying outer abstraction to obtain an ITS,
223.22/58.18	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(34)
223.22/58.18	Obligation:
223.22/58.18	Complexity RNTS consisting of the following rules:
223.22/58.18	
223.22/58.18	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.18	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.18	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.18	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.18	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.18	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	
223.22/58.18	Function symbols to be analyzed: {splitD#3}, {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.18	Previous analysis results are:
223.22/58.18	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	  splitD#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(35) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.18	
223.22/58.18	Computed RUNTIME bound using CoFloCo for: splitD#3
223.22/58.18	after applying outer abstraction to obtain an ITS,
223.22/58.18	resulting in: O(1) with polynomial bound: 1
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(36)
223.22/58.18	Obligation:
223.22/58.18	Complexity RNTS consisting of the following rules:
223.22/58.18	
223.22/58.18	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.18	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.18	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.18	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.18	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.18	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.18	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.18	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.18	
223.22/58.18	Function symbols to be analyzed: {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.18	Previous analysis results are:
223.22/58.18	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.18	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.18	
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(37) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.18	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.18	----------------------------------------
223.22/58.18	
223.22/58.18	(38)
223.22/58.18	Obligation:
223.22/58.18	Complexity RNTS consisting of the following rules:
223.22/58.18	
223.22/58.18	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.18	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.18	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.18	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.18	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.18	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.18	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.18	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.18	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.18	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.18	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.18	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.18	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.18	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.18	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.18	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.18	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.18	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.18	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(39) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed SIZE bound using CoFloCo for: #compare
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(1) with polynomial bound: 3
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(40)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#compare}, {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: ?, size: O(1) [3]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(41) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed RUNTIME bound using CoFloCo for: #compare
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(1) with polynomial bound: 0
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(42)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(43) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.19	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(44)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s2) :|: s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s3) :|: s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(s4), @ls, @rs, z'') :|: s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(s5), @ls, @rs, z'') :|: s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(45) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed SIZE bound using CoFloCo for: #ckgt
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(1) with polynomial bound: 2
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(46)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s2) :|: s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s3) :|: s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(s4), @ls, @rs, z'') :|: s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(s5), @ls, @rs, z'') :|: s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#ckgt}, {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	  #ckgt: runtime: ?, size: O(1) [2]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(47) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed RUNTIME bound using CoFloCo for: #ckgt
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(1) with polynomial bound: 0
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(48)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s2) :|: s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> #ckgt(s3) :|: s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 2 }-> split#3(#ckgt(s4), @ls, @rs, z'') :|: s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 2 }-> splitD#3(#ckgt(s5), @ls, @rs, z'') :|: s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(49) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.19	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(50)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(51) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed SIZE bound using CoFloCo for: append#1
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(n^1) with polynomial bound: z + z'
223.22/58.19	
223.22/58.19	Computed SIZE bound using CoFloCo for: append
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(n^1) with polynomial bound: z + z'
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(52)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {append#1,append}, {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.22/58.19	  append#1: runtime: ?, size: O(n^1) [z + z']
223.22/58.19	  append: runtime: ?, size: O(n^1) [z + z']
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(53) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.19	
223.22/58.19	Computed RUNTIME bound using CoFloCo for: append#1
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(n^1) with polynomial bound: 3 + 2*z
223.22/58.19	
223.22/58.19	Computed RUNTIME bound using CoFloCo for: append
223.22/58.19	after applying outer abstraction to obtain an ITS,
223.22/58.19	resulting in: O(n^1) with polynomial bound: 4 + 2*z
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(54)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.19	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.19	
223.22/58.19	Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.19	Previous analysis results are:
223.22/58.19	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.19	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.19	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.22/58.19	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.22/58.19	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.22/58.19	
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(55) ResultPropagationProof (UPPER BOUND(ID))
223.22/58.19	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.22/58.19	----------------------------------------
223.22/58.19	
223.22/58.19	(56)
223.22/58.19	Obligation:
223.22/58.19	Complexity RNTS consisting of the following rules:
223.22/58.19	
223.22/58.19	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.19	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.19	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.19	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.19	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.19	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.19	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.19	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.19	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.19	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.19	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.19	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.19	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.19	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.19	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.19	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.19	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.19	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.19	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.22/58.19	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.19	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.20	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.22/58.20	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.20	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.22/58.20	
223.22/58.20	Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.22/58.20	Previous analysis results are:
223.22/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.22/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.22/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.22/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.22/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.22/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.22/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.22/58.20	
223.22/58.20	----------------------------------------
223.22/58.20	
223.22/58.20	(57) IntTrsBoundProof (UPPER BOUND(ID))
223.22/58.20	
223.22/58.20	Computed SIZE bound using CoFloCo for: #abs
223.22/58.20	after applying outer abstraction to obtain an ITS,
223.22/58.20	resulting in: O(n^1) with polynomial bound: 1 + z
223.22/58.20	
223.22/58.20	----------------------------------------
223.22/58.20	
223.22/58.20	(58)
223.22/58.20	Obligation:
223.22/58.20	Complexity RNTS consisting of the following rules:
223.22/58.20	
223.22/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.22/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.22/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.22/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.22/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.22/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.22/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.22/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.22/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.22/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.22/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.22/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.22/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.22/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.22/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.22/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.22/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.22/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.22/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.22/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.22/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.22/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.22/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.22/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.22/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.22/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.22/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.22/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.22/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.22/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.22/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.22/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.22/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.22/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.39/58.20	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.39/58.20	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {#abs}, {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: ?, size: O(n^1) [1 + z]
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(59) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.20	
223.39/58.20	Computed RUNTIME bound using CoFloCo for: #abs
223.39/58.20	after applying outer abstraction to obtain an ITS,
223.39/58.20	resulting in: O(1) with polynomial bound: 1
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(60)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
223.39/58.20	   testQuicksort(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.39/58.20	   testQuicksort2(z) -{ 2 }-> quicksort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(61) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(62)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(63) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.20	
223.39/58.20	Computed SIZE bound using CoFloCo for: split#2
223.39/58.20	after applying outer abstraction to obtain an ITS,
223.39/58.20	resulting in: O(n^1) with polynomial bound: 1 + z + z''
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(64)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {split#2}, {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.20	  split#2: runtime: ?, size: O(n^1) [1 + z + z'']
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(65) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.20	
223.39/58.20	Computed RUNTIME bound using CoFloCo for: split#2
223.39/58.20	after applying outer abstraction to obtain an ITS,
223.39/58.20	resulting in: O(1) with polynomial bound: 3
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(66)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.20	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(67) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.20	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(68)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.20	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.20	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.20	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.20	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.20	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.20	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.20	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.20	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.20	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.20	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.20	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.20	
223.39/58.20	Function symbols to be analyzed: {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.20	Previous analysis results are:
223.39/58.20	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.20	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.20	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.20	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.20	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.20	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(69) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.20	
223.39/58.20	Computed SIZE bound using CoFloCo for: splitD#2
223.39/58.20	after applying outer abstraction to obtain an ITS,
223.39/58.20	resulting in: O(n^1) with polynomial bound: 1 + z + z''
223.39/58.20	
223.39/58.20	----------------------------------------
223.39/58.20	
223.39/58.20	(70)
223.39/58.20	Obligation:
223.39/58.20	Complexity RNTS consisting of the following rules:
223.39/58.20	
223.39/58.20	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.20	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.20	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.20	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.20	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.20	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.20	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.20	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {splitD#2}, {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: ?, size: O(n^1) [1 + z + z'']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(71) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: splitD#2
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(1) with polynomial bound: 3
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(72)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(73) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(74)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(75) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: #greater
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(1) with polynomial bound: 2
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(76)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {#greater}, {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: ?, size: O(1) [2]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(77) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: #greater
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(1) with polynomial bound: 1
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(78)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(79) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(80)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(81) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: testList
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(1) with polynomial bound: 74
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(82)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {testList}, {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: ?, size: O(1) [74]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(83) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: testList
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(1) with polynomial bound: 11
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(84)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(85) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(86)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(87) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: split#1
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 1 + z
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(88)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split#1}, {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: ?, size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(89) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: split#1
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 1 + 5*z
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(90)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 2 }-> quicksort#2(split#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 1 }-> split#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 2 }-> split#2(split#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(91) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(92)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(93) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: splitD#1
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 1 + z
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(94)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {splitD#1}, {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: ?, size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(95) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: splitD#1
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 1 + 5*z
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(96)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 2 }-> quicksortD#2(splitD#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 1 }-> splitD#1(z', z) :|: z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 2 }-> splitD#2(splitD#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(97) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(98)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(99) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: split
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 1 + z'
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(100)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {split}, {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  split: runtime: ?, size: O(n^1) [1 + z']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(101) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed RUNTIME bound using CoFloCo for: split
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: 2 + 5*z'
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(102)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(103) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.21	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(104)
223.39/58.21	Obligation:
223.39/58.21	Complexity RNTS consisting of the following rules:
223.39/58.21	
223.39/58.21	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.21	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.21	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.21	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.21	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.21	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.21	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.21	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.21	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.21	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.21	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.21	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.21	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.21	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.21	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.21	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.21	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.21	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.21	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.21	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.21	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.21	
223.39/58.21	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.21	Previous analysis results are:
223.39/58.21	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.21	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.21	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.21	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.21	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.21	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.21	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.21	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.21	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.21	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.21	
223.39/58.21	----------------------------------------
223.39/58.21	
223.39/58.21	(105) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.21	
223.39/58.21	Computed SIZE bound using CoFloCo for: quicksort#1
223.39/58.21	after applying outer abstraction to obtain an ITS,
223.39/58.21	resulting in: O(n^1) with polynomial bound: z
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: quicksort#2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: z + z'
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(106)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.22	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.22	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: ?, size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: ?, size: O(n^1) [z + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(107) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using CoFloCo for: quicksort#1
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 5 + 48*z + 35*z^2
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using KoAT for: quicksort#2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 17 + 98*z + 70*z^2
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(108)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
223.39/58.22	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s44, @z) :|: s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.22	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(109) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.22	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(110)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.22	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(111) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: quicksortD#2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: 1 + 3*z + 3*z'
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: quicksortD#1
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: 1 + 3*z
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(112)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.22	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD#2,quicksortD#1}, {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: ?, size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: ?, size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(113) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using KoAT for: quicksortD#2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 11 + 55*z + 44*z^2
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using KoAT for: quicksortD#1
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 114 + 148*z + 44*z^2
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(114)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 1 }-> quicksortD#1(z) :|: z >= 0
223.39/58.22	   quicksortD#1(z) -{ 3 + 5*@zs }-> quicksortD#2(s48, @z) :|: s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 3 }-> appendD(quicksortD#1(@xs), 1 + z' + quicksortD#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(115) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.22	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(116)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(117) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: splitD
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: 1 + z'
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(118)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {splitD}, {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: ?, size: O(n^1) [1 + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(119) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using CoFloCo for: splitD
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: 2 + 5*z'
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(120)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(121) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.22	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(122)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(123) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: quicksort
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: z
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(124)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksort}, {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: ?, size: O(n^1) [z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(125) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using KoAT for: quicksort
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 6 + 48*z + 35*z^2
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(126)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 12 }-> quicksort(1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0)))))))))) :|: s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 12 }-> quicksort(1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0)))))))))) :|: s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(127) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.22	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(128)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(129) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: quicksortD
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^1) with polynomial bound: 1 + 3*z
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(130)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {quicksortD}, {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksortD: runtime: ?, size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(131) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using KoAT for: quicksortD
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(n^2) with polynomial bound: 115 + 148*z + 44*z^2
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(132)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(133) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.22	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(134)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(135) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed SIZE bound using CoFloCo for: testQuicksort2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(1) with polynomial bound: 74
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(136)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.22	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.22	
223.39/58.22	Function symbols to be analyzed: {testQuicksort2}, {testQuicksort}
223.39/58.22	Previous analysis results are:
223.39/58.22	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.22	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.22	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.22	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.22	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.22	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.22	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.22	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.22	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.22	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.22	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.22	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.22	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.22	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.22	  testQuicksort2: runtime: ?, size: O(1) [74]
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(137) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.22	
223.39/58.22	Computed RUNTIME bound using CoFloCo for: testQuicksort2
223.39/58.22	after applying outer abstraction to obtain an ITS,
223.39/58.22	resulting in: O(1) with polynomial bound: 195230
223.39/58.22	
223.39/58.22	----------------------------------------
223.39/58.22	
223.39/58.22	(138)
223.39/58.22	Obligation:
223.39/58.22	Complexity RNTS consisting of the following rules:
223.39/58.22	
223.39/58.22	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.22	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.22	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.22	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.22	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.22	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.22	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.22	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.22	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.22	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.22	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.22	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.22	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.22	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.22	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.22	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.22	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.22	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.22	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.22	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	
223.39/58.23	Function symbols to be analyzed: {testQuicksort}
223.39/58.23	Previous analysis results are:
223.39/58.23	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.23	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.23	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.23	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.23	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.23	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.23	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.23	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  testQuicksort2: runtime: O(1) [195230], size: O(1) [74]
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(139) ResultPropagationProof (UPPER BOUND(ID))
223.39/58.23	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(140)
223.39/58.23	Obligation:
223.39/58.23	Complexity RNTS consisting of the following rules:
223.39/58.23	
223.39/58.23	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.23	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.23	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.23	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.23	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.23	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	
223.39/58.23	Function symbols to be analyzed: {testQuicksort}
223.39/58.23	Previous analysis results are:
223.39/58.23	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.23	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.23	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.23	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.23	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.23	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.23	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.23	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  testQuicksort2: runtime: O(1) [195230], size: O(1) [74]
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(141) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.23	
223.39/58.23	Computed SIZE bound using CoFloCo for: testQuicksort
223.39/58.23	after applying outer abstraction to obtain an ITS,
223.39/58.23	resulting in: O(1) with polynomial bound: 74
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(142)
223.39/58.23	Obligation:
223.39/58.23	Complexity RNTS consisting of the following rules:
223.39/58.23	
223.39/58.23	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.23	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.23	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.23	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.23	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.23	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	
223.39/58.23	Function symbols to be analyzed: {testQuicksort}
223.39/58.23	Previous analysis results are:
223.39/58.23	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.23	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.23	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.23	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.23	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.23	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.23	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.23	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  testQuicksort2: runtime: O(1) [195230], size: O(1) [74]
223.39/58.23	  testQuicksort: runtime: ?, size: O(1) [74]
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(143) IntTrsBoundProof (UPPER BOUND(ID))
223.39/58.23	
223.39/58.23	Computed RUNTIME bound using CoFloCo for: testQuicksort
223.39/58.23	after applying outer abstraction to obtain an ITS,
223.39/58.23	resulting in: O(1) with polynomial bound: 195230
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(144)
223.39/58.23	Obligation:
223.39/58.23	Complexity RNTS consisting of the following rules:
223.39/58.23	
223.39/58.23	   #abs(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
223.39/58.23	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
223.39/58.23	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
223.39/58.23	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
223.39/58.23	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 3, z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
223.39/58.23	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s6 :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> s7 :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z' - 1 >= 0, z - 1 >= 0
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
223.39/58.23	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
223.39/58.23	   append(z, z') -{ 4 + 2*z }-> s12 :|: s12 >= 0, s12 <= z + z', z >= 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s13 :|: s13 >= 0, s13 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   appendD(z, z') -{ 2 + 2*z }-> s :|: s >= 0, s <= z + z', z >= 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 1 }-> z' :|: z = 0, z' >= 0
223.39/58.23	   appendD#1(z, z') -{ 3 + 2*@xs }-> 1 + @x + s' :|: s' >= 0, s' <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   quicksort(z) -{ 6 + 48*z + 35*z^2 }-> s52 :|: s52 >= 0, s52 <= z, z >= 0
223.39/58.23	   quicksort#1(z) -{ 20 + 5*@zs + 98*s44 + 70*s44^2 }-> s53 :|: s53 >= 0, s53 <= s44 + @z, s44 >= 0, s44 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksort#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksort#2(z, z') -{ 17 + 48*@xs + 35*@xs^2 + 48*@ys + 35*@ys^2 + 2*s54 }-> s56 :|: s54 >= 0, s54 <= @xs, s55 >= 0, s55 <= @ys, s56 >= 0, s56 <= s54 + (1 + z' + s55), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   quicksortD(z) -{ 115 + 148*z + 44*z^2 }-> s57 :|: s57 >= 0, s57 <= 3 * z + 1, z >= 0
223.39/58.23	   quicksortD#1(z) -{ 14 + 5*@zs + 55*s48 + 44*s48^2 }-> s58 :|: s58 >= 0, s58 <= 3 * @z + 1 + 3 * s48, s48 >= 0, s48 <= @zs + 1, z = 1 + @z + @zs, @zs >= 0, @z >= 0
223.39/58.23	   quicksortD#1(z) -{ 1 }-> 0 :|: z = 0
223.39/58.23	   quicksortD#2(z, z') -{ 233 + 148*@xs + 44*@xs^2 + 148*@ys + 44*@ys^2 + 2*s59 }-> s61 :|: s59 >= 0, s59 <= 3 * @xs + 1, s60 >= 0, s60 <= 3 * @ys + 1, s61 >= 0, s61 <= s59 + (1 + z' + s60), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
223.39/58.23	   quicksortD#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
223.39/58.23	   split(z, z') -{ 2 + 5*z' }-> s45 :|: s45 >= 0, s45 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   split#1(z, z') -{ 6 + 5*@xs }-> s47 :|: s46 >= 0, s46 <= @xs + 1, s47 >= 0, s47 <= s46 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   split#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   split#2(z, z', z'') -{ 3 }-> s9 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= @ls + @rs + z'' + 2, s4 >= 0, s4 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   split#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   split#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   splitD(z, z') -{ 2 + 5*z' }-> s49 :|: s49 >= 0, s49 <= z' + 1, z' >= 0, z >= 0
223.39/58.23	   splitD#1(z, z') -{ 6 + 5*@xs }-> s51 :|: s50 >= 0, s50 <= @xs + 1, s51 >= 0, s51 <= s50 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
223.39/58.23	   splitD#1(z, z') -{ 1 }-> 1 + 0 + 0 :|: z' >= 0, z = 0
223.39/58.23	   splitD#2(z, z', z'') -{ 3 }-> s11 :|: s10 >= 0, s10 <= 2, s11 >= 0, s11 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
223.39/58.23	   splitD#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
223.39/58.23	   splitD#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
223.39/58.23	   testList(z) -{ 11 }-> 1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + (1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + 0))))))))) :|: s14 >= 0, s14 <= 0 + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s18 >= 0, s18 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s19 >= 0, s19 <= 1 + (1 + 0) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + 0)) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s23 >= 0, s23 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort(z) -{ 3998 + 748*s24 + 70*s24*s25 + 70*s24*s26 + 70*s24*s27 + 70*s24*s28 + 70*s24*s29 + 70*s24*s30 + 70*s24*s31 + 70*s24*s32 + 70*s24*s33 + 35*s24^2 + 748*s25 + 70*s25*s26 + 70*s25*s27 + 70*s25*s28 + 70*s25*s29 + 70*s25*s30 + 70*s25*s31 + 70*s25*s32 + 70*s25*s33 + 35*s25^2 + 748*s26 + 70*s26*s27 + 70*s26*s28 + 70*s26*s29 + 70*s26*s30 + 70*s26*s31 + 70*s26*s32 + 70*s26*s33 + 35*s26^2 + 748*s27 + 70*s27*s28 + 70*s27*s29 + 70*s27*s30 + 70*s27*s31 + 70*s27*s32 + 70*s27*s33 + 35*s27^2 + 748*s28 + 70*s28*s29 + 70*s28*s30 + 70*s28*s31 + 70*s28*s32 + 70*s28*s33 + 35*s28^2 + 748*s29 + 70*s29*s30 + 70*s29*s31 + 70*s29*s32 + 70*s29*s33 + 35*s29^2 + 748*s30 + 70*s30*s31 + 70*s30*s32 + 70*s30*s33 + 35*s30^2 + 748*s31 + 70*s31*s32 + 70*s31*s33 + 35*s31^2 + 748*s32 + 70*s32*s33 + 35*s32^2 + 748*s33 + 35*s33^2 }-> s62 :|: s62 >= 0, s62 <= 1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + (1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + 0))))))))), s24 >= 0, s24 <= 0 + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s28 >= 0, s28 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s29 >= 0, s29 <= 1 + (1 + 0) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + 0)) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s33 >= 0, s33 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	   testQuicksort2(z) -{ 3998 + 748*s34 + 70*s34*s35 + 70*s34*s36 + 70*s34*s37 + 70*s34*s38 + 70*s34*s39 + 70*s34*s40 + 70*s34*s41 + 70*s34*s42 + 70*s34*s43 + 35*s34^2 + 748*s35 + 70*s35*s36 + 70*s35*s37 + 70*s35*s38 + 70*s35*s39 + 70*s35*s40 + 70*s35*s41 + 70*s35*s42 + 70*s35*s43 + 35*s35^2 + 748*s36 + 70*s36*s37 + 70*s36*s38 + 70*s36*s39 + 70*s36*s40 + 70*s36*s41 + 70*s36*s42 + 70*s36*s43 + 35*s36^2 + 748*s37 + 70*s37*s38 + 70*s37*s39 + 70*s37*s40 + 70*s37*s41 + 70*s37*s42 + 70*s37*s43 + 35*s37^2 + 748*s38 + 70*s38*s39 + 70*s38*s40 + 70*s38*s41 + 70*s38*s42 + 70*s38*s43 + 35*s38^2 + 748*s39 + 70*s39*s40 + 70*s39*s41 + 70*s39*s42 + 70*s39*s43 + 35*s39^2 + 748*s40 + 70*s40*s41 + 70*s40*s42 + 70*s40*s43 + 35*s40^2 + 748*s41 + 70*s41*s42 + 70*s41*s43 + 35*s41^2 + 748*s42 + 70*s42*s43 + 35*s42^2 + 748*s43 + 35*s43^2 }-> s63 :|: s63 >= 0, s63 <= 1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + (1 + s38 + (1 + s39 + (1 + s40 + (1 + s41 + (1 + s42 + (1 + s43 + 0))))))))), s34 >= 0, s34 <= 0 + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s38 >= 0, s38 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s39 >= 0, s39 <= 1 + (1 + 0) + 1, s40 >= 0, s40 <= 1 + (1 + (1 + 0)) + 1, s41 >= 0, s41 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s42 >= 0, s42 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s43 >= 0, s43 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
223.39/58.23	
223.39/58.23	Function symbols to be analyzed: 
223.39/58.23	Previous analysis results are:
223.39/58.23	  appendD: runtime: O(n^1) [2 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  appendD#1: runtime: O(n^1) [1 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  split#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  splitD#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
223.39/58.23	  #compare: runtime: O(1) [0], size: O(1) [3]
223.39/58.23	  #ckgt: runtime: O(1) [0], size: O(1) [2]
223.39/58.23	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
223.39/58.23	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
223.39/58.23	  split#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  splitD#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
223.39/58.23	  #greater: runtime: O(1) [1], size: O(1) [2]
223.39/58.23	  testList: runtime: O(1) [11], size: O(1) [74]
223.39/58.23	  split#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  splitD#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [1 + z]
223.39/58.23	  split: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort#1: runtime: O(n^2) [5 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksort#2: runtime: O(n^2) [17 + 98*z + 70*z^2], size: O(n^1) [z + z']
223.39/58.23	  quicksortD#2: runtime: O(n^2) [11 + 55*z + 44*z^2], size: O(n^1) [1 + 3*z + 3*z']
223.39/58.23	  quicksortD#1: runtime: O(n^2) [114 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  splitD: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [1 + z']
223.39/58.23	  quicksort: runtime: O(n^2) [6 + 48*z + 35*z^2], size: O(n^1) [z]
223.39/58.23	  quicksortD: runtime: O(n^2) [115 + 148*z + 44*z^2], size: O(n^1) [1 + 3*z]
223.39/58.23	  testQuicksort2: runtime: O(1) [195230], size: O(1) [74]
223.39/58.23	  testQuicksort: runtime: O(1) [195230], size: O(1) [74]
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(145) FinalProof (FINISHED)
223.39/58.23	Computed overall runtime complexity
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(146)
223.39/58.23	BOUNDS(1, n^2)
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(147) RenamingProof (BOTH BOUNDS(ID, ID))
223.39/58.23	Renamed function symbols to avoid clashes with predefined symbol.
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(148)
223.39/58.23	Obligation:
223.39/58.23	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).
223.39/58.23	
223.39/58.23	
223.39/58.23	The TRS R consists of the following rules:
223.39/58.23	
223.39/58.23	   #abs(#0) -> #0
223.39/58.23	   #abs(#neg(@x)) -> #pos(@x)
223.39/58.23	   #abs(#pos(@x)) -> #pos(@x)
223.39/58.23	   #abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	   append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	   append#1(nil, @ys) -> @ys
223.39/58.23	   appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	   appendD#1(nil, @ys) -> @ys
223.39/58.23	   quicksort(@l) -> quicksort#1(@l)
223.39/58.23	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	   quicksort#1(nil) -> nil
223.39/58.23	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	   quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	   quicksortD#1(nil) -> nil
223.39/58.23	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	   split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	   split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	   splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	   splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	   testList(@x) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	   testQuicksort(@x) -> quicksort(testList(#unit))
223.39/58.23	   testQuicksort2(@x) -> quicksort(testList(#unit))
223.39/58.23	
223.39/58.23	The (relative) TRS S consists of the following rules:
223.39/58.23	
223.39/58.23	   #ckgt(#EQ) -> #false
223.39/58.23	   #ckgt(#GT) -> #true
223.39/58.23	   #ckgt(#LT) -> #false
223.39/58.23	   #compare(#0, #0) -> #EQ
223.39/58.23	   #compare(#0, #neg(@y)) -> #GT
223.39/58.23	   #compare(#0, #pos(@y)) -> #LT
223.39/58.23	   #compare(#0, #s(@y)) -> #LT
223.39/58.23	   #compare(#neg(@x), #0) -> #LT
223.39/58.23	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	   #compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	   #compare(#pos(@x), #0) -> #GT
223.39/58.23	   #compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	   #compare(#s(@x), #0) -> #GT
223.39/58.23	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Rewrite Strategy: INNERMOST
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(149) SlicingProof (LOWER BOUND(ID))
223.39/58.23	Sliced the following arguments:
223.39/58.23	testList/0
223.39/58.23	testQuicksort/0
223.39/58.23	testQuicksort2/0
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(150)
223.39/58.23	Obligation:
223.39/58.23	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^2, INF).
223.39/58.23	
223.39/58.23	
223.39/58.23	The TRS R consists of the following rules:
223.39/58.23	
223.39/58.23	   #abs(#0) -> #0
223.39/58.23	   #abs(#neg(@x)) -> #pos(@x)
223.39/58.23	   #abs(#pos(@x)) -> #pos(@x)
223.39/58.23	   #abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	   append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	   append#1(nil, @ys) -> @ys
223.39/58.23	   appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	   appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	   appendD#1(nil, @ys) -> @ys
223.39/58.23	   quicksort(@l) -> quicksort#1(@l)
223.39/58.23	   quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	   quicksort#1(nil) -> nil
223.39/58.23	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	   quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	   quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	   quicksortD#1(nil) -> nil
223.39/58.23	   quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	   split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	   split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	   split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	   split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	   split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	   split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	   splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	   splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	   splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	   splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	   splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	   splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	   testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	   testQuicksort -> quicksort(testList)
223.39/58.23	   testQuicksort2 -> quicksort(testList)
223.39/58.23	
223.39/58.23	The (relative) TRS S consists of the following rules:
223.39/58.23	
223.39/58.23	   #ckgt(#EQ) -> #false
223.39/58.23	   #ckgt(#GT) -> #true
223.39/58.23	   #ckgt(#LT) -> #false
223.39/58.23	   #compare(#0, #0) -> #EQ
223.39/58.23	   #compare(#0, #neg(@y)) -> #GT
223.39/58.23	   #compare(#0, #pos(@y)) -> #LT
223.39/58.23	   #compare(#0, #s(@y)) -> #LT
223.39/58.23	   #compare(#neg(@x), #0) -> #LT
223.39/58.23	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	   #compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	   #compare(#pos(@x), #0) -> #GT
223.39/58.23	   #compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	   #compare(#s(@x), #0) -> #GT
223.39/58.23	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Rewrite Strategy: INNERMOST
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(151) TypeInferenceProof (BOTH BOUNDS(ID, ID))
223.39/58.23	Infered types.
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(152)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(153) OrderProof (LOWER BOUND(ID))
223.39/58.23	Heuristically decided to analyse the following defined symbols:
223.39/58.23	#compare, append, append#1, appendD, appendD#1, quicksort, quicksort#1, split, quicksortD, quicksortD#1, splitD, split#1, splitD#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	splitD < quicksortD#1
223.39/58.23	splitD = splitD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(154)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	#compare, append, append#1, appendD, appendD#1, quicksort, quicksort#1, split, quicksortD, quicksortD#1, splitD, split#1, splitD#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	splitD < quicksortD#1
223.39/58.23	splitD = splitD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(155) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(0), gen_#0:#neg:#pos:#s6_4(0)) ->_R^Omega(0)
223.39/58.23	#EQ
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(+(n9_4, 1)), gen_#0:#neg:#pos:#s6_4(+(n9_4, 1))) ->_R^Omega(0)
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) ->_IH
223.39/58.23	#EQ
223.39/58.23	
223.39/58.23	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(156)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	splitD#1, append, append#1, appendD, appendD#1, quicksort, quicksort#1, split, quicksortD, quicksortD#1, splitD, split#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	splitD < quicksortD#1
223.39/58.23	splitD = splitD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(157) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	splitD#1(gen_:::nil7_4(0), gen_#0:#neg:#pos:#s6_4(0)) ->_R^Omega(1)
223.39/58.23	tuple#2(nil, nil)
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	splitD#1(gen_:::nil7_4(+(n318812_4, 1)), gen_#0:#neg:#pos:#s6_4(0)) ->_R^Omega(1)
223.39/58.23	splitD#2(splitD(gen_#0:#neg:#pos:#s6_4(0), gen_:::nil7_4(n318812_4)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_R^Omega(1)
223.39/58.23	splitD#2(splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_IH
223.39/58.23	splitD#2(tuple#2(gen_:::nil7_4(c318813_4), gen_:::nil7_4(0)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_R^Omega(1)
223.39/58.23	splitD#3(#greater(#0, gen_#0:#neg:#pos:#s6_4(0)), gen_:::nil7_4(n318812_4), gen_:::nil7_4(0), #0) ->_R^Omega(1)
223.39/58.23	splitD#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s6_4(0))), gen_:::nil7_4(n318812_4), gen_:::nil7_4(0), #0) ->_L^Omega(0)
223.39/58.23	splitD#3(#ckgt(#EQ), gen_:::nil7_4(n318812_4), gen_:::nil7_4(0), #0) ->_R^Omega(0)
223.39/58.23	splitD#3(#false, gen_:::nil7_4(n318812_4), gen_:::nil7_4(0), #0) ->_R^Omega(1)
223.39/58.23	tuple#2(::(#0, gen_:::nil7_4(n318812_4)), gen_:::nil7_4(0))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(158)
223.39/58.23	Complex Obligation (BEST)
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(159)
223.39/58.23	Obligation:
223.39/58.23	Proved the lower bound n^1 for the following obligation:
223.39/58.23	
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	splitD#1, append, append#1, appendD, appendD#1, quicksort, quicksort#1, split, quicksortD, quicksortD#1, splitD, split#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	splitD < quicksortD#1
223.39/58.23	splitD = splitD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(160) LowerBoundPropagationProof (FINISHED)
223.39/58.23	Propagated lower bound.
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(161)
223.39/58.23	BOUNDS(n^1, INF)
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(162)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	splitD, append, append#1, appendD, appendD#1, quicksort, quicksort#1, split, quicksortD, quicksortD#1, split#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	splitD < quicksortD#1
223.39/58.23	splitD = splitD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(163) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	split#1(gen_:::nil7_4(0), gen_#0:#neg:#pos:#s6_4(0)) ->_R^Omega(1)
223.39/58.23	tuple#2(nil, nil)
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	split#1(gen_:::nil7_4(+(n320997_4, 1)), gen_#0:#neg:#pos:#s6_4(0)) ->_R^Omega(1)
223.39/58.23	split#2(split(gen_#0:#neg:#pos:#s6_4(0), gen_:::nil7_4(n320997_4)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_R^Omega(1)
223.39/58.23	split#2(split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_IH
223.39/58.23	split#2(tuple#2(gen_:::nil7_4(c320998_4), gen_:::nil7_4(0)), gen_#0:#neg:#pos:#s6_4(0), #0) ->_R^Omega(1)
223.39/58.23	split#3(#greater(#0, gen_#0:#neg:#pos:#s6_4(0)), gen_:::nil7_4(n320997_4), gen_:::nil7_4(0), #0) ->_R^Omega(1)
223.39/58.23	split#3(#ckgt(#compare(#0, gen_#0:#neg:#pos:#s6_4(0))), gen_:::nil7_4(n320997_4), gen_:::nil7_4(0), #0) ->_L^Omega(0)
223.39/58.23	split#3(#ckgt(#EQ), gen_:::nil7_4(n320997_4), gen_:::nil7_4(0), #0) ->_R^Omega(0)
223.39/58.23	split#3(#false, gen_:::nil7_4(n320997_4), gen_:::nil7_4(0), #0) ->_R^Omega(1)
223.39/58.23	tuple#2(::(#0, gen_:::nil7_4(n320997_4)), gen_:::nil7_4(0))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(164)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	split, append, append#1, appendD, appendD#1, quicksort, quicksort#1, quicksortD, quicksortD#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	split < quicksort#1
223.39/58.23	split = split#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(165) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	appendD#1(gen_:::nil7_4(0), gen_:::nil7_4(b)) ->_R^Omega(1)
223.39/58.23	gen_:::nil7_4(b)
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	appendD#1(gen_:::nil7_4(+(n323205_4, 1)), gen_:::nil7_4(b)) ->_R^Omega(1)
223.39/58.23	::(#0, appendD(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b))) ->_R^Omega(1)
223.39/58.23	::(#0, appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b))) ->_IH
223.39/58.23	::(#0, gen_:::nil7_4(+(b, c323206_4)))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(166)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	appendD, append, append#1, quicksort, quicksort#1, quicksortD, quicksortD#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	appendD = appendD#1
223.39/58.23	appendD < quicksortD#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(167) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	quicksortD#1(gen_:::nil7_4(n324930_4)) -> gen_:::nil7_4(n324930_4), rt in Omega(1 + n324930_4 + n324930_4^2)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	quicksortD#1(gen_:::nil7_4(0)) ->_R^Omega(1)
223.39/58.23	nil
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	quicksortD#1(gen_:::nil7_4(+(n324930_4, 1))) ->_R^Omega(1)
223.39/58.23	quicksortD#2(splitD(#0, gen_:::nil7_4(n324930_4)), #0) ->_R^Omega(1)
223.39/58.23	quicksortD#2(splitD#1(gen_:::nil7_4(n324930_4), #0), #0) ->_L^Omega(1 + n324930_4)
223.39/58.23	quicksortD#2(tuple#2(gen_:::nil7_4(n324930_4), gen_:::nil7_4(0)), #0) ->_R^Omega(1)
223.39/58.23	appendD(quicksortD(gen_:::nil7_4(n324930_4)), ::(#0, quicksortD(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	appendD(quicksortD#1(gen_:::nil7_4(n324930_4)), ::(#0, quicksortD(gen_:::nil7_4(0)))) ->_IH
223.39/58.23	appendD(gen_:::nil7_4(c324931_4), ::(#0, quicksortD(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	appendD(gen_:::nil7_4(n324930_4), ::(#0, quicksortD#1(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	appendD(gen_:::nil7_4(n324930_4), ::(#0, nil)) ->_R^Omega(1)
223.39/58.23	appendD#1(gen_:::nil7_4(n324930_4), ::(#0, nil)) ->_L^Omega(1 + n324930_4)
223.39/58.23	gen_:::nil7_4(+(n324930_4, +(0, 1)))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(168)
223.39/58.23	Complex Obligation (BEST)
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(169)
223.39/58.23	Obligation:
223.39/58.23	Proved the lower bound n^2 for the following obligation:
223.39/58.23	
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	quicksortD#1, append, append#1, quicksort, quicksort#1, quicksortD
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(170) LowerBoundPropagationProof (FINISHED)
223.39/58.23	Propagated lower bound.
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(171)
223.39/58.23	BOUNDS(n^2, INF)
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(172)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.23	quicksortD#1(gen_:::nil7_4(n324930_4)) -> gen_:::nil7_4(n324930_4), rt in Omega(1 + n324930_4 + n324930_4^2)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	quicksortD, append, append#1, quicksort, quicksort#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	quicksortD = quicksortD#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(173) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	append#1(gen_:::nil7_4(n326285_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n326285_4, b)), rt in Omega(1 + n326285_4)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	append#1(gen_:::nil7_4(0), gen_:::nil7_4(b)) ->_R^Omega(1)
223.39/58.23	gen_:::nil7_4(b)
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	append#1(gen_:::nil7_4(+(n326285_4, 1)), gen_:::nil7_4(b)) ->_R^Omega(1)
223.39/58.23	::(#0, append(gen_:::nil7_4(n326285_4), gen_:::nil7_4(b))) ->_R^Omega(1)
223.39/58.23	::(#0, append#1(gen_:::nil7_4(n326285_4), gen_:::nil7_4(b))) ->_IH
223.39/58.23	::(#0, gen_:::nil7_4(+(b, c326286_4)))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(174)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.23	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.23	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.23	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.23	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.23	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.23	testQuicksort -> quicksort(testList)
223.39/58.23	testQuicksort2 -> quicksort(testList)
223.39/58.23	#ckgt(#EQ) -> #false
223.39/58.23	#ckgt(#GT) -> #true
223.39/58.23	#ckgt(#LT) -> #false
223.39/58.23	#compare(#0, #0) -> #EQ
223.39/58.23	#compare(#0, #neg(@y)) -> #GT
223.39/58.23	#compare(#0, #pos(@y)) -> #LT
223.39/58.23	#compare(#0, #s(@y)) -> #LT
223.39/58.23	#compare(#neg(@x), #0) -> #LT
223.39/58.23	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.23	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.23	#compare(#pos(@x), #0) -> #GT
223.39/58.23	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.23	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.23	#compare(#s(@x), #0) -> #GT
223.39/58.23	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.23	
223.39/58.23	Types:
223.39/58.23	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#0 :: #0:#neg:#pos:#s
223.39/58.23	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.23	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.23	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.23	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.23	append :: :::nil -> :::nil -> :::nil
223.39/58.23	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.23	nil :: :::nil
223.39/58.23	appendD :: :::nil -> :::nil -> :::nil
223.39/58.23	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.23	quicksort :: :::nil -> :::nil
223.39/58.23	quicksort#1 :: :::nil -> :::nil
223.39/58.23	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.23	quicksortD :: :::nil -> :::nil
223.39/58.23	quicksortD#1 :: :::nil -> :::nil
223.39/58.23	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.23	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.23	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	#false :: #false:#true
223.39/58.23	#true :: #false:#true
223.39/58.23	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.23	testList :: :::nil
223.39/58.23	testQuicksort :: :::nil
223.39/58.23	testQuicksort2 :: :::nil
223.39/58.23	#EQ :: #EQ:#GT:#LT
223.39/58.23	#GT :: #EQ:#GT:#LT
223.39/58.23	#LT :: #EQ:#GT:#LT
223.39/58.23	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.23	hole_#false:#true2_4 :: #false:#true
223.39/58.23	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.23	hole_:::nil4_4 :: :::nil
223.39/58.23	hole_tuple#25_4 :: tuple#2
223.39/58.23	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.23	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.23	
223.39/58.23	
223.39/58.23	Lemmas:
223.39/58.23	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.23	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.23	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.23	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.23	quicksortD#1(gen_:::nil7_4(n324930_4)) -> gen_:::nil7_4(n324930_4), rt in Omega(1 + n324930_4 + n324930_4^2)
223.39/58.23	append#1(gen_:::nil7_4(n326285_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n326285_4, b)), rt in Omega(1 + n326285_4)
223.39/58.23	
223.39/58.23	
223.39/58.23	Generator Equations:
223.39/58.23	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.23	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.23	gen_:::nil7_4(0) <=> nil
223.39/58.23	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.23	
223.39/58.23	
223.39/58.23	The following defined symbols remain to be analysed:
223.39/58.23	append, quicksort, quicksort#1
223.39/58.23	
223.39/58.23	They will be analysed ascendingly in the following order:
223.39/58.23	append = append#1
223.39/58.23	append < quicksort#1
223.39/58.23	quicksort = quicksort#1
223.39/58.23	
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(175) RewriteLemmaProof (LOWER BOUND(ID))
223.39/58.23	Proved the following rewrite lemma:
223.39/58.23	quicksort#1(gen_:::nil7_4(n328054_4)) -> gen_:::nil7_4(n328054_4), rt in Omega(1 + n328054_4 + n328054_4^2)
223.39/58.23	
223.39/58.23	Induction Base:
223.39/58.23	quicksort#1(gen_:::nil7_4(0)) ->_R^Omega(1)
223.39/58.23	nil
223.39/58.23	
223.39/58.23	Induction Step:
223.39/58.23	quicksort#1(gen_:::nil7_4(+(n328054_4, 1))) ->_R^Omega(1)
223.39/58.23	quicksort#2(split(#0, gen_:::nil7_4(n328054_4)), #0) ->_R^Omega(1)
223.39/58.23	quicksort#2(split#1(gen_:::nil7_4(n328054_4), #0), #0) ->_L^Omega(1 + n328054_4)
223.39/58.23	quicksort#2(tuple#2(gen_:::nil7_4(n328054_4), gen_:::nil7_4(0)), #0) ->_R^Omega(1)
223.39/58.23	append(quicksort(gen_:::nil7_4(n328054_4)), ::(#0, quicksort(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	append(quicksort#1(gen_:::nil7_4(n328054_4)), ::(#0, quicksort(gen_:::nil7_4(0)))) ->_IH
223.39/58.23	append(gen_:::nil7_4(c328055_4), ::(#0, quicksort(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	append(gen_:::nil7_4(n328054_4), ::(#0, quicksort#1(gen_:::nil7_4(0)))) ->_R^Omega(1)
223.39/58.23	append(gen_:::nil7_4(n328054_4), ::(#0, nil)) ->_R^Omega(1)
223.39/58.23	append#1(gen_:::nil7_4(n328054_4), ::(#0, nil)) ->_L^Omega(1 + n328054_4)
223.39/58.23	gen_:::nil7_4(+(n328054_4, +(0, 1)))
223.39/58.23	
223.39/58.23	We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
223.39/58.23	----------------------------------------
223.39/58.23	
223.39/58.23	(176)
223.39/58.23	Obligation:
223.39/58.23	Innermost TRS:
223.39/58.23	Rules:
223.39/58.23	#abs(#0) -> #0
223.39/58.23	#abs(#neg(@x)) -> #pos(@x)
223.39/58.23	#abs(#pos(@x)) -> #pos(@x)
223.39/58.23	#abs(#s(@x)) -> #pos(#s(@x))
223.39/58.23	#greater(@x, @y) -> #ckgt(#compare(@x, @y))
223.39/58.23	append(@l, @ys) -> append#1(@l, @ys)
223.39/58.23	append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
223.39/58.23	append#1(nil, @ys) -> @ys
223.39/58.23	appendD(@l, @ys) -> appendD#1(@l, @ys)
223.39/58.23	appendD#1(::(@x, @xs), @ys) -> ::(@x, appendD(@xs, @ys))
223.39/58.23	appendD#1(nil, @ys) -> @ys
223.39/58.23	quicksort(@l) -> quicksort#1(@l)
223.39/58.23	quicksort#1(::(@z, @zs)) -> quicksort#2(split(@z, @zs), @z)
223.39/58.23	quicksort#1(nil) -> nil
223.39/58.23	quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
223.39/58.23	quicksortD(@l) -> quicksortD#1(@l)
223.39/58.23	quicksortD#1(::(@z, @zs)) -> quicksortD#2(splitD(@z, @zs), @z)
223.39/58.23	quicksortD#1(nil) -> nil
223.39/58.23	quicksortD#2(tuple#2(@xs, @ys), @z) -> appendD(quicksortD(@xs), ::(@z, quicksortD(@ys)))
223.39/58.23	split(@pivot, @l) -> split#1(@l, @pivot)
223.39/58.23	split#1(::(@x, @xs), @pivot) -> split#2(split(@pivot, @xs), @pivot, @x)
223.39/58.23	split#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.23	split#2(tuple#2(@ls, @rs), @pivot, @x) -> split#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.24	split#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.24	split#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.24	splitD(@pivot, @l) -> splitD#1(@l, @pivot)
223.39/58.24	splitD#1(::(@x, @xs), @pivot) -> splitD#2(splitD(@pivot, @xs), @pivot, @x)
223.39/58.24	splitD#1(nil, @pivot) -> tuple#2(nil, nil)
223.39/58.24	splitD#2(tuple#2(@ls, @rs), @pivot, @x) -> splitD#3(#greater(@x, @pivot), @ls, @rs, @x)
223.39/58.24	splitD#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
223.39/58.24	splitD#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
223.39/58.24	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
223.39/58.24	testQuicksort -> quicksort(testList)
223.39/58.24	testQuicksort2 -> quicksort(testList)
223.39/58.24	#ckgt(#EQ) -> #false
223.39/58.24	#ckgt(#GT) -> #true
223.39/58.24	#ckgt(#LT) -> #false
223.39/58.24	#compare(#0, #0) -> #EQ
223.39/58.24	#compare(#0, #neg(@y)) -> #GT
223.39/58.24	#compare(#0, #pos(@y)) -> #LT
223.39/58.24	#compare(#0, #s(@y)) -> #LT
223.39/58.24	#compare(#neg(@x), #0) -> #LT
223.39/58.24	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
223.39/58.24	#compare(#neg(@x), #pos(@y)) -> #LT
223.39/58.24	#compare(#pos(@x), #0) -> #GT
223.39/58.24	#compare(#pos(@x), #neg(@y)) -> #GT
223.39/58.24	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
223.39/58.24	#compare(#s(@x), #0) -> #GT
223.39/58.24	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
223.39/58.24	
223.39/58.24	Types:
223.39/58.24	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.24	#0 :: #0:#neg:#pos:#s
223.39/58.24	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.24	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.24	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
223.39/58.24	#greater :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
223.39/58.24	#ckgt :: #EQ:#GT:#LT -> #false:#true
223.39/58.24	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
223.39/58.24	append :: :::nil -> :::nil -> :::nil
223.39/58.24	append#1 :: :::nil -> :::nil -> :::nil
223.39/58.24	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
223.39/58.24	nil :: :::nil
223.39/58.24	appendD :: :::nil -> :::nil -> :::nil
223.39/58.24	appendD#1 :: :::nil -> :::nil -> :::nil
223.39/58.24	quicksort :: :::nil -> :::nil
223.39/58.24	quicksort#1 :: :::nil -> :::nil
223.39/58.24	quicksort#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.24	split :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.24	tuple#2 :: :::nil -> :::nil -> tuple#2
223.39/58.24	quicksortD :: :::nil -> :::nil
223.39/58.24	quicksortD#1 :: :::nil -> :::nil
223.39/58.24	quicksortD#2 :: tuple#2 -> #0:#neg:#pos:#s -> :::nil
223.39/58.24	splitD :: #0:#neg:#pos:#s -> :::nil -> tuple#2
223.39/58.24	split#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	split#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	split#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	#false :: #false:#true
223.39/58.24	#true :: #false:#true
223.39/58.24	splitD#1 :: :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	splitD#2 :: tuple#2 -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	splitD#3 :: #false:#true -> :::nil -> :::nil -> #0:#neg:#pos:#s -> tuple#2
223.39/58.24	testList :: :::nil
223.39/58.24	testQuicksort :: :::nil
223.39/58.24	testQuicksort2 :: :::nil
223.39/58.24	#EQ :: #EQ:#GT:#LT
223.39/58.24	#GT :: #EQ:#GT:#LT
223.39/58.24	#LT :: #EQ:#GT:#LT
223.39/58.24	hole_#0:#neg:#pos:#s1_4 :: #0:#neg:#pos:#s
223.39/58.24	hole_#false:#true2_4 :: #false:#true
223.39/58.24	hole_#EQ:#GT:#LT3_4 :: #EQ:#GT:#LT
223.39/58.24	hole_:::nil4_4 :: :::nil
223.39/58.24	hole_tuple#25_4 :: tuple#2
223.39/58.24	gen_#0:#neg:#pos:#s6_4 :: Nat -> #0:#neg:#pos:#s
223.39/58.24	gen_:::nil7_4 :: Nat -> :::nil
223.39/58.24	
223.39/58.24	
223.39/58.24	Lemmas:
223.39/58.24	#compare(gen_#0:#neg:#pos:#s6_4(n9_4), gen_#0:#neg:#pos:#s6_4(n9_4)) -> #EQ, rt in Omega(0)
223.39/58.24	splitD#1(gen_:::nil7_4(n318812_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n318812_4), gen_:::nil7_4(0)), rt in Omega(1 + n318812_4)
223.39/58.24	split#1(gen_:::nil7_4(n320997_4), gen_#0:#neg:#pos:#s6_4(0)) -> tuple#2(gen_:::nil7_4(n320997_4), gen_:::nil7_4(0)), rt in Omega(1 + n320997_4)
223.39/58.24	appendD#1(gen_:::nil7_4(n323205_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n323205_4, b)), rt in Omega(1 + n323205_4)
223.39/58.24	quicksortD#1(gen_:::nil7_4(n324930_4)) -> gen_:::nil7_4(n324930_4), rt in Omega(1 + n324930_4 + n324930_4^2)
223.39/58.24	append#1(gen_:::nil7_4(n326285_4), gen_:::nil7_4(b)) -> gen_:::nil7_4(+(n326285_4, b)), rt in Omega(1 + n326285_4)
223.39/58.24	quicksort#1(gen_:::nil7_4(n328054_4)) -> gen_:::nil7_4(n328054_4), rt in Omega(1 + n328054_4 + n328054_4^2)
223.39/58.24	
223.39/58.24	
223.39/58.24	Generator Equations:
223.39/58.24	gen_#0:#neg:#pos:#s6_4(0) <=> #0
223.39/58.24	gen_#0:#neg:#pos:#s6_4(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s6_4(x))
223.39/58.24	gen_:::nil7_4(0) <=> nil
223.39/58.24	gen_:::nil7_4(+(x, 1)) <=> ::(#0, gen_:::nil7_4(x))
223.39/58.24	
223.39/58.24	
223.39/58.24	The following defined symbols remain to be analysed:
223.39/58.24	quicksort
223.39/58.24	
223.39/58.24	They will be analysed ascendingly in the following order:
223.39/58.24	quicksort = quicksort#1
223.50/58.29	EOF