1173.46/298.92	WORST_CASE(Omega(n^1), O(n^3))
1173.46/298.94	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1173.46/298.94	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1173.46/298.94	
1173.46/298.94	
1173.46/298.94	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1173.46/298.94	
1173.46/298.94	(0) CpxRelTRS
1173.46/298.94	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 260 ms]
1173.46/298.94	(2) CpxRelTRS
1173.46/298.94	(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1173.46/298.94	(4) CpxWeightedTrs
1173.46/298.94	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1173.46/298.94	    (6) CpxTypedWeightedTrs
1173.46/298.94	    (7) CompletionProof [UPPER BOUND(ID), 97 ms]
1173.46/298.94	    (8) CpxTypedWeightedCompleteTrs
1173.46/298.94	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 531 ms]
1173.46/298.94	    (10) CpxTypedWeightedCompleteTrs
1173.46/298.94	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (12) CpxRNTS
1173.46/298.94	    (13) InliningProof [UPPER BOUND(ID), 708 ms]
1173.46/298.94	    (14) CpxRNTS
1173.46/298.94	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
1173.46/298.94	    (16) CpxRNTS
1173.46/298.94	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
1173.46/298.94	    (18) CpxRNTS
1173.46/298.94	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (20) CpxRNTS
1173.46/298.94	    (21) IntTrsBoundProof [UPPER BOUND(ID), 292 ms]
1173.46/298.94	    (22) CpxRNTS
1173.46/298.94	    (23) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
1173.46/298.94	    (24) CpxRNTS
1173.46/298.94	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (26) CpxRNTS
1173.46/298.94	    (27) IntTrsBoundProof [UPPER BOUND(ID), 998 ms]
1173.46/298.94	    (28) CpxRNTS
1173.46/298.94	    (29) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
1173.46/298.94	    (30) CpxRNTS
1173.46/298.94	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (32) CpxRNTS
1173.46/298.94	    (33) IntTrsBoundProof [UPPER BOUND(ID), 220 ms]
1173.46/298.94	    (34) CpxRNTS
1173.46/298.94	    (35) IntTrsBoundProof [UPPER BOUND(ID), 34 ms]
1173.46/298.94	    (36) CpxRNTS
1173.46/298.94	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (38) CpxRNTS
1173.46/298.94	    (39) IntTrsBoundProof [UPPER BOUND(ID), 343 ms]
1173.46/298.94	    (40) CpxRNTS
1173.46/298.94	    (41) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
1173.46/298.94	    (42) CpxRNTS
1173.46/298.94	    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (44) CpxRNTS
1173.46/298.94	    (45) IntTrsBoundProof [UPPER BOUND(ID), 544 ms]
1173.46/298.94	    (46) CpxRNTS
1173.46/298.94	    (47) IntTrsBoundProof [UPPER BOUND(ID), 210 ms]
1173.46/298.94	    (48) CpxRNTS
1173.46/298.94	    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (50) CpxRNTS
1173.46/298.94	    (51) IntTrsBoundProof [UPPER BOUND(ID), 835 ms]
1173.46/298.94	    (52) CpxRNTS
1173.46/298.94	    (53) IntTrsBoundProof [UPPER BOUND(ID), 397 ms]
1173.46/298.94	    (54) CpxRNTS
1173.46/298.94	    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (56) CpxRNTS
1173.46/298.94	    (57) IntTrsBoundProof [UPPER BOUND(ID), 278 ms]
1173.46/298.94	    (58) CpxRNTS
1173.46/298.94	    (59) IntTrsBoundProof [UPPER BOUND(ID), 62 ms]
1173.46/298.94	    (60) CpxRNTS
1173.46/298.94	    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (62) CpxRNTS
1173.46/298.94	    (63) IntTrsBoundProof [UPPER BOUND(ID), 707 ms]
1173.46/298.94	    (64) CpxRNTS
1173.46/298.94	    (65) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
1173.46/298.94	    (66) CpxRNTS
1173.46/298.94	    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (68) CpxRNTS
1173.46/298.94	    (69) IntTrsBoundProof [UPPER BOUND(ID), 3541 ms]
1173.46/298.94	    (70) CpxRNTS
1173.46/298.94	    (71) IntTrsBoundProof [UPPER BOUND(ID), 996 ms]
1173.46/298.94	    (72) CpxRNTS
1173.46/298.94	    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (74) CpxRNTS
1173.46/298.94	    (75) IntTrsBoundProof [UPPER BOUND(ID), 206 ms]
1173.46/298.94	    (76) CpxRNTS
1173.46/298.94	    (77) IntTrsBoundProof [UPPER BOUND(ID), 51 ms]
1173.46/298.94	    (78) CpxRNTS
1173.46/298.94	    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (80) CpxRNTS
1173.46/298.94	    (81) IntTrsBoundProof [UPPER BOUND(ID), 439 ms]
1173.46/298.94	    (82) CpxRNTS
1173.46/298.94	    (83) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
1173.46/298.94	    (84) CpxRNTS
1173.46/298.94	    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (86) CpxRNTS
1173.46/298.94	    (87) IntTrsBoundProof [UPPER BOUND(ID), 348 ms]
1173.46/298.94	    (88) CpxRNTS
1173.46/298.94	    (89) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
1173.46/298.94	    (90) CpxRNTS
1173.46/298.94	    (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (92) CpxRNTS
1173.46/298.94	    (93) IntTrsBoundProof [UPPER BOUND(ID), 185 ms]
1173.46/298.94	    (94) CpxRNTS
1173.46/298.94	    (95) IntTrsBoundProof [UPPER BOUND(ID), 41 ms]
1173.46/298.94	    (96) CpxRNTS
1173.46/298.94	    (97) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (98) CpxRNTS
1173.46/298.94	    (99) IntTrsBoundProof [UPPER BOUND(ID), 2589 ms]
1173.46/298.94	    (100) CpxRNTS
1173.46/298.94	    (101) IntTrsBoundProof [UPPER BOUND(ID), 698 ms]
1173.46/298.94	    (102) CpxRNTS
1173.46/298.94	    (103) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
1173.46/298.94	    (104) CpxRNTS
1173.46/298.94	    (105) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
1173.46/298.94	    (106) CpxRNTS
1173.46/298.94	    (107) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (108) CpxRNTS
1173.46/298.94	    (109) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (110) CpxRNTS
1173.46/298.94	    (111) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
1173.46/298.94	    (112) CpxRNTS
1173.46/298.94	    (113) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (114) CpxRNTS
1173.46/298.94	    (115) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (116) CpxRNTS
1173.46/298.94	    (117) IntTrsBoundProof [UPPER BOUND(ID), 1147 ms]
1173.46/298.94	    (118) CpxRNTS
1173.46/298.94	    (119) IntTrsBoundProof [UPPER BOUND(ID), 70 ms]
1173.46/298.94	    (120) CpxRNTS
1173.46/298.94	    (121) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1173.46/298.94	    (122) CpxRNTS
1173.46/298.94	    (123) IntTrsBoundProof [UPPER BOUND(ID), 178 ms]
1173.46/298.94	    (124) CpxRNTS
1173.46/298.94	    (125) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
1173.46/298.94	    (126) CpxRNTS
1173.46/298.94	    (127) FinalProof [FINISHED, 0 ms]
1173.46/298.94	    (128) BOUNDS(1, n^3)
1173.46/298.94	(129) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1173.46/298.94	(130) TRS for Loop Detection
1173.46/298.94	    (131) DecreasingLoopProof [LOWER BOUND(ID), 65 ms]
1173.46/298.94	    (132) BEST
1173.46/298.94	        (133) proven lower bound
1173.46/298.94	            (134) LowerBoundPropagationProof [FINISHED, 0 ms]
1173.46/298.94	            (135) BOUNDS(n^1, INF)
1173.46/298.94	        (136) TRS for Loop Detection
1173.46/298.94	
1173.46/298.94	
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(0)
1173.46/298.94	Obligation:
1173.46/298.94	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1173.46/298.94	
1173.46/298.94	
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y)
1173.46/298.94	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys)
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
1173.46/298.94	   append#1(nil, @ys) -> @ys
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x)
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x)
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x)
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil)
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls))
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls)
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l)
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z)
1173.46/298.94	   quicksort#1(nil) -> nil
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l)
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs)
1173.46/298.94	   sortAll#1(nil) -> nil
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs))
1173.46/298.94	   split(@l) -> split#1(@l)
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split(@xs))
1173.46/298.94	   split#1(nil) -> nil
1173.46/298.94	   splitAndSort(@l) -> sortAll(split(@l))
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot)
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x)
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil)
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
1173.46/298.94	
1173.46/298.94	The (relative) TRS S consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #and(#false, #false) -> #false
1173.46/298.94	   #and(#false, #true) -> #false
1173.46/298.94	   #and(#true, #false) -> #false
1173.46/298.94	   #and(#true, #true) -> #true
1173.46/298.94	   #ckgt(#EQ) -> #false
1173.46/298.94	   #ckgt(#GT) -> #true
1173.46/298.94	   #ckgt(#LT) -> #false
1173.46/298.94	   #compare(#0, #0) -> #EQ
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1173.46/298.94	   #compare(#s(@x), #0) -> #GT
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1173.46/298.94	   #eq(#0, #0) -> #true
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false
1173.46/298.94	   #eq(#0, #s(@y)) -> #false
1173.46/298.94	   #eq(#neg(@x), #0) -> #false
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false
1173.46/298.94	   #eq(#pos(@x), #0) -> #false
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(#s(@x), #0) -> #false
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(nil, nil) -> #true
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
1173.46/298.94	proved termination of relative rules
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(2)
1173.46/298.94	Obligation:
1173.46/298.94	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1173.46/298.94	
1173.46/298.94	
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y)
1173.46/298.94	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys)
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
1173.46/298.94	   append#1(nil, @ys) -> @ys
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x)
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x)
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x)
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil)
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls))
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls)
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l)
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z)
1173.46/298.94	   quicksort#1(nil) -> nil
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l)
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs)
1173.46/298.94	   sortAll#1(nil) -> nil
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs))
1173.46/298.94	   split(@l) -> split#1(@l)
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split(@xs))
1173.46/298.94	   split#1(nil) -> nil
1173.46/298.94	   splitAndSort(@l) -> sortAll(split(@l))
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot)
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x)
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil)
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
1173.46/298.94	
1173.46/298.94	The (relative) TRS S consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #and(#false, #false) -> #false
1173.46/298.94	   #and(#false, #true) -> #false
1173.46/298.94	   #and(#true, #false) -> #false
1173.46/298.94	   #and(#true, #true) -> #true
1173.46/298.94	   #ckgt(#EQ) -> #false
1173.46/298.94	   #ckgt(#GT) -> #true
1173.46/298.94	   #ckgt(#LT) -> #false
1173.46/298.94	   #compare(#0, #0) -> #EQ
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1173.46/298.94	   #compare(#s(@x), #0) -> #GT
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1173.46/298.94	   #eq(#0, #0) -> #true
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false
1173.46/298.94	   #eq(#0, #s(@y)) -> #false
1173.46/298.94	   #eq(#neg(@x), #0) -> #false
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false
1173.46/298.94	   #eq(#pos(@x), #0) -> #false
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(#s(@x), #0) -> #false
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(nil, nil) -> #true
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1173.46/298.94	Transformed relative TRS to weighted TRS
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(4)
1173.46/298.94	Obligation:
1173.46/298.94	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).
1173.46/298.94	
1173.46/298.94	
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y) [1]
1173.46/298.94	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys) [1]
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
1173.46/298.94	   append#1(nil, @ys) -> @ys [1]
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x) [1]
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1]
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1]
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1]
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1]
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1]
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1]
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l) [1]
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1]
1173.46/298.94	   quicksort#1(nil) -> nil [1]
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l) [1]
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1]
1173.46/298.94	   sortAll#1(nil) -> nil [1]
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1]
1173.46/298.94	   split(@l) -> split#1(@l) [1]
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1]
1173.46/298.94	   split#1(nil) -> nil [1]
1173.46/298.94	   splitAndSort(@l) -> sortAll(split(@l)) [1]
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1]
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1]
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1]
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
1173.46/298.94	   #and(#false, #false) -> #false [0]
1173.46/298.94	   #and(#false, #true) -> #false [0]
1173.46/298.94	   #and(#true, #false) -> #false [0]
1173.46/298.94	   #and(#true, #true) -> #true [0]
1173.46/298.94	   #ckgt(#EQ) -> #false [0]
1173.46/298.94	   #ckgt(#GT) -> #true [0]
1173.46/298.94	   #ckgt(#LT) -> #false [0]
1173.46/298.94	   #compare(#0, #0) -> #EQ [0]
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #compare(#s(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #eq(#0, #0) -> #true [0]
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #s(@y)) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#s(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, nil) -> #true [0]
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1173.46/298.94	Infered types.
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(6)
1173.46/298.94	Obligation:
1173.46/298.94	Runtime Complexity Weighted TRS with Types.
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y) [1]
1173.46/298.94	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys) [1]
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
1173.46/298.94	   append#1(nil, @ys) -> @ys [1]
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x) [1]
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1]
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1]
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1]
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1]
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1]
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1]
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l) [1]
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1]
1173.46/298.94	   quicksort#1(nil) -> nil [1]
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l) [1]
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1]
1173.46/298.94	   sortAll#1(nil) -> nil [1]
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1]
1173.46/298.94	   split(@l) -> split#1(@l) [1]
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1]
1173.46/298.94	   split#1(nil) -> nil [1]
1173.46/298.94	   splitAndSort(@l) -> sortAll(split(@l)) [1]
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1]
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1]
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1]
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
1173.46/298.94	   #and(#false, #false) -> #false [0]
1173.46/298.94	   #and(#false, #true) -> #false [0]
1173.46/298.94	   #and(#true, #false) -> #false [0]
1173.46/298.94	   #and(#true, #true) -> #true [0]
1173.46/298.94	   #ckgt(#EQ) -> #false [0]
1173.46/298.94	   #ckgt(#GT) -> #true [0]
1173.46/298.94	   #ckgt(#LT) -> #false [0]
1173.46/298.94	   #compare(#0, #0) -> #EQ [0]
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #compare(#s(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #eq(#0, #0) -> #true [0]
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #s(@y)) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#s(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, nil) -> #true [0]
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	
1173.46/298.94	The TRS has the following type information:
1173.46/298.94	#equal :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true
1173.46/298.94	#eq :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true
1173.46/298.94	#greater :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #false:#true
1173.46/298.94	#ckgt :: #EQ:#GT:#LT -> #false:#true
1173.46/298.94	#compare :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> #EQ:#GT:#LT
1173.46/298.94	append :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	:: :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	nil :: :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	insert :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	insert#4 :: #false:#true -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	#false :: #false:#true
1173.46/298.94	#true :: #false:#true
1173.46/298.94	quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	split :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	splitqs#3 :: #false:#true -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	#and :: #false:#true -> #false:#true -> #false:#true
1173.46/298.94	#EQ :: #EQ:#GT:#LT
1173.46/298.94	#GT :: #EQ:#GT:#LT
1173.46/298.94	#LT :: #EQ:#GT:#LT
1173.46/298.94	#0 :: :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	#neg :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	#pos :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	#s :: :::nil:tuple#2:#0:#neg:#pos:#s -> :::nil:tuple#2:#0:#neg:#pos:#s
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(7) CompletionProof (UPPER BOUND(ID))
1173.46/298.94	The transformation into a RNTS is sound, since:
1173.46/298.94	
1173.46/298.94	(a) The obligation is a constructor system where every type has a constant constructor,
1173.46/298.94	
1173.46/298.94	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
1173.46/298.94	
1173.46/298.94	   sortAll_1
1173.46/298.94	   sortAll#1_1
1173.46/298.94	   sortAll#2_2
1173.46/298.94	   splitAndSort_1
1173.46/298.94	
1173.46/298.94	(c) The following functions are completely defined:
1173.46/298.94	
1173.46/298.94	   quicksort_1
1173.46/298.94	   split_1
1173.46/298.94	   #greater_2
1173.46/298.94	   #equal_2
1173.46/298.94	   splitqs_2
1173.46/298.94	   split#1_1
1173.46/298.94	   insert_2
1173.46/298.94	   splitqs#1_2
1173.46/298.94	   splitqs#2_3
1173.46/298.94	   insert#1_3
1173.46/298.94	   quicksort#1_1
1173.46/298.94	   insert#2_4
1173.46/298.94	   insert#3_5
1173.46/298.94	   splitqs#3_4
1173.46/298.94	   insert#4_6
1173.46/298.94	   quicksort#2_2
1173.46/298.94	   append_2
1173.46/298.94	   append#1_2
1173.46/298.94	   #and_2
1173.46/298.94	   #ckgt_1
1173.46/298.94	   #compare_2
1173.46/298.94	   #eq_2
1173.46/298.94	
1173.46/298.94	Due to the following rules being added:
1173.46/298.94	
1173.46/298.94	   #and(v0, v1) -> null_#and [0]
1173.46/298.94	   #ckgt(v0) -> null_#ckgt [0]
1173.46/298.94	   #compare(v0, v1) -> null_#compare [0]
1173.46/298.94	   #eq(v0, v1) -> null_#eq [0]
1173.46/298.94	   split#1(v0) -> null_split#1 [0]
1173.46/298.94	   splitqs#1(v0, v1) -> null_splitqs#1 [0]
1173.46/298.94	   splitqs#2(v0, v1, v2) -> null_splitqs#2 [0]
1173.46/298.94	   insert#1(v0, v1, v2) -> null_insert#1 [0]
1173.46/298.94	   quicksort#1(v0) -> null_quicksort#1 [0]
1173.46/298.94	   insert#2(v0, v1, v2, v3) -> null_insert#2 [0]
1173.46/298.94	   insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0]
1173.46/298.94	   splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0]
1173.46/298.94	   insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0]
1173.46/298.94	   quicksort#2(v0, v1) -> null_quicksort#2 [0]
1173.46/298.94	   append#1(v0, v1) -> null_append#1 [0]
1173.46/298.94	
1173.46/298.94	And the following fresh constants:    null_#and, null_#ckgt, null_#compare, null_#eq, null_split#1, null_splitqs#1, null_splitqs#2, null_insert#1, null_quicksort#1, null_insert#2, null_insert#3, null_splitqs#3, null_insert#4, null_quicksort#2, null_append#1
1173.46/298.94	
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(8)
1173.46/298.94	Obligation:
1173.46/298.94	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1173.46/298.94	
1173.46/298.94	Runtime Complexity Weighted TRS with Types.
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y) [1]
1173.46/298.94	   #greater(@x, @y) -> #ckgt(#compare(@x, @y)) [1]
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys) [1]
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
1173.46/298.94	   append#1(nil, @ys) -> @ys [1]
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x) [1]
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1]
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1]
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1]
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [1]
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1]
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1]
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l) [1]
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) [1]
1173.46/298.94	   quicksort#1(nil) -> nil [1]
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys))) [1]
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l) [1]
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1]
1173.46/298.94	   sortAll#1(nil) -> nil [1]
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1]
1173.46/298.94	   split(@l) -> split#1(@l) [1]
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split(@xs)) [1]
1173.46/298.94	   split#1(nil) -> nil [1]
1173.46/298.94	   splitAndSort(@l) -> sortAll(split(@l)) [1]
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1]
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x) [1]
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1]
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) [1]
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
1173.46/298.94	   #and(#false, #false) -> #false [0]
1173.46/298.94	   #and(#false, #true) -> #false [0]
1173.46/298.94	   #and(#true, #false) -> #false [0]
1173.46/298.94	   #and(#true, #true) -> #true [0]
1173.46/298.94	   #ckgt(#EQ) -> #false [0]
1173.46/298.94	   #ckgt(#GT) -> #true [0]
1173.46/298.94	   #ckgt(#LT) -> #false [0]
1173.46/298.94	   #compare(#0, #0) -> #EQ [0]
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #compare(#s(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #eq(#0, #0) -> #true [0]
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #s(@y)) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#s(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, nil) -> #true [0]
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #and(v0, v1) -> null_#and [0]
1173.46/298.94	   #ckgt(v0) -> null_#ckgt [0]
1173.46/298.94	   #compare(v0, v1) -> null_#compare [0]
1173.46/298.94	   #eq(v0, v1) -> null_#eq [0]
1173.46/298.94	   split#1(v0) -> null_split#1 [0]
1173.46/298.94	   splitqs#1(v0, v1) -> null_splitqs#1 [0]
1173.46/298.94	   splitqs#2(v0, v1, v2) -> null_splitqs#2 [0]
1173.46/298.94	   insert#1(v0, v1, v2) -> null_insert#1 [0]
1173.46/298.94	   quicksort#1(v0) -> null_quicksort#1 [0]
1173.46/298.94	   insert#2(v0, v1, v2, v3) -> null_insert#2 [0]
1173.46/298.94	   insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0]
1173.46/298.94	   splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0]
1173.46/298.94	   insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0]
1173.46/298.94	   quicksort#2(v0, v1) -> null_quicksort#2 [0]
1173.46/298.94	   append#1(v0, v1) -> null_append#1 [0]
1173.46/298.94	
1173.46/298.94	The TRS has the following type information:
1173.46/298.94	#equal :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#eq :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#greater :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#compare :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #EQ:#GT:#LT:null_#compare
1173.46/298.94	append :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	:: :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	nil :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#4 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#false :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#true :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	split :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#3 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#EQ :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#GT :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#LT :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#0 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#neg :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#pos :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#s :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_#compare :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#4 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
1173.46/298.94	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(10)
1173.46/298.94	Obligation:
1173.46/298.94	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1173.46/298.94	
1173.46/298.94	Runtime Complexity Weighted TRS with Types.
1173.46/298.94	The TRS R consists of the following rules:
1173.46/298.94	
1173.46/298.94	   #equal(@x, @y) -> #eq(@x, @y) [1]
1173.46/298.94	   #greater(#0, #0) -> #ckgt(#EQ) [1]
1173.46/298.94	   #greater(#0, #neg(@y')) -> #ckgt(#GT) [1]
1173.46/298.94	   #greater(#0, #pos(@y'')) -> #ckgt(#LT) [1]
1173.46/298.94	   #greater(#0, #s(@y1)) -> #ckgt(#LT) [1]
1173.46/298.94	   #greater(#neg(@x'), #0) -> #ckgt(#LT) [1]
1173.46/298.94	   #greater(#neg(@x''), #neg(@y2)) -> #ckgt(#compare(@y2, @x'')) [1]
1173.46/298.94	   #greater(#neg(@x1), #pos(@y3)) -> #ckgt(#LT) [1]
1173.46/298.94	   #greater(#pos(@x2), #0) -> #ckgt(#GT) [1]
1173.46/298.94	   #greater(#pos(@x3), #neg(@y4)) -> #ckgt(#GT) [1]
1173.46/298.94	   #greater(#pos(@x4), #pos(@y5)) -> #ckgt(#compare(@x4, @y5)) [1]
1173.46/298.94	   #greater(#s(@x5), #0) -> #ckgt(#GT) [1]
1173.46/298.94	   #greater(#s(@x6), #s(@y6)) -> #ckgt(#compare(@x6, @y6)) [1]
1173.46/298.94	   #greater(@x, @y) -> #ckgt(null_#compare) [1]
1173.46/298.94	   append(@l, @ys) -> append#1(@l, @ys) [1]
1173.46/298.94	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) [1]
1173.46/298.94	   append#1(nil, @ys) -> @ys [1]
1173.46/298.94	   insert(@x, @l) -> insert#1(@x, @l, @x) [1]
1173.46/298.94	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x) [1]
1173.46/298.94	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x) [1]
1173.46/298.94	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil) [1]
1173.46/298.94	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) [2]
1173.46/298.94	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls)) [1]
1173.46/298.94	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls) [1]
1173.46/298.94	   quicksort(@l) -> quicksort#1(@l) [1]
1173.46/298.94	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs#1(@zs, @z), @z) [2]
1173.46/298.94	   quicksort#1(nil) -> nil [1]
1173.46/298.94	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort#1(@xs), ::(@z, quicksort#1(@ys))) [3]
1173.46/298.94	   sortAll(@l) -> sortAll#1(@l) [1]
1173.46/298.94	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) [1]
1173.46/298.94	   sortAll#1(nil) -> nil [1]
1173.46/298.94	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) [1]
1173.46/298.94	   split(@l) -> split#1(@l) [1]
1173.46/298.94	   split#1(::(@x, @xs)) -> insert(@x, split#1(@xs)) [2]
1173.46/298.94	   split#1(nil) -> nil [1]
1173.46/298.94	   splitAndSort(@l) -> sortAll(split#1(@l)) [2]
1173.46/298.94	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) [1]
1173.46/298.94	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) [2]
1173.46/298.94	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil) [1]
1173.46/298.94	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) [2]
1173.46/298.94	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) [1]
1173.46/298.94	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) [1]
1173.46/298.94	   #and(#false, #false) -> #false [0]
1173.46/298.94	   #and(#false, #true) -> #false [0]
1173.46/298.94	   #and(#true, #false) -> #false [0]
1173.46/298.94	   #and(#true, #true) -> #true [0]
1173.46/298.94	   #ckgt(#EQ) -> #false [0]
1173.46/298.94	   #ckgt(#GT) -> #true [0]
1173.46/298.94	   #ckgt(#LT) -> #false [0]
1173.46/298.94	   #compare(#0, #0) -> #EQ [0]
1173.46/298.94	   #compare(#0, #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#0, #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#0, #s(@y)) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #0) -> #LT [0]
1173.46/298.94	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1173.46/298.94	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1173.46/298.94	   #compare(#pos(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1173.46/298.94	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #compare(#s(@x), #0) -> #GT [0]
1173.46/298.94	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1173.46/298.94	   #eq(#0, #0) -> #true [0]
1173.46/298.94	   #eq(#0, #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#0, #s(@y)) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#neg(@x), #pos(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #neg(@y)) -> #false [0]
1173.46/298.94	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(#s(@x), #0) -> #false [0]
1173.46/298.94	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(nil, nil) -> #true [0]
1173.46/298.94	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), nil) -> #false [0]
1173.46/298.94	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
1173.46/298.94	   #and(v0, v1) -> null_#and [0]
1173.46/298.94	   #ckgt(v0) -> null_#ckgt [0]
1173.46/298.94	   #compare(v0, v1) -> null_#compare [0]
1173.46/298.94	   #eq(v0, v1) -> null_#eq [0]
1173.46/298.94	   split#1(v0) -> null_split#1 [0]
1173.46/298.94	   splitqs#1(v0, v1) -> null_splitqs#1 [0]
1173.46/298.94	   splitqs#2(v0, v1, v2) -> null_splitqs#2 [0]
1173.46/298.94	   insert#1(v0, v1, v2) -> null_insert#1 [0]
1173.46/298.94	   quicksort#1(v0) -> null_quicksort#1 [0]
1173.46/298.94	   insert#2(v0, v1, v2, v3) -> null_insert#2 [0]
1173.46/298.94	   insert#3(v0, v1, v2, v3, v4) -> null_insert#3 [0]
1173.46/298.94	   splitqs#3(v0, v1, v2, v3) -> null_splitqs#3 [0]
1173.46/298.94	   insert#4(v0, v1, v2, v3, v4, v5) -> null_insert#4 [0]
1173.46/298.94	   quicksort#2(v0, v1) -> null_quicksort#2 [0]
1173.46/298.94	   append#1(v0, v1) -> null_append#1 [0]
1173.46/298.94	
1173.46/298.94	The TRS has the following type information:
1173.46/298.94	#equal :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#eq :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#greater :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#ckgt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#compare :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> #EQ:#GT:#LT:null_#compare
1173.46/298.94	append :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	:: :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	nil :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	tuple#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	insert#4 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#false :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#true :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	quicksort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	sortAll#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	split :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitAndSort :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	splitqs#3 :: #false:#true:null_#and:null_#ckgt:null_#eq -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#and :: #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq -> #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	#EQ :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#GT :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#LT :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	#0 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#neg :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#pos :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	#s :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1 -> :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_#and :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_#ckgt :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_#compare :: #EQ:#GT:#LT:null_#compare
1173.46/298.94	null_#eq :: #false:#true:null_#and:null_#ckgt:null_#eq
1173.46/298.94	null_split#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_quicksort#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_splitqs#3 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_insert#4 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_quicksort#2 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	null_append#1 :: :::nil:tuple#2:#0:#neg:#pos:#s:null_split#1:null_splitqs#1:null_splitqs#2:null_insert#1:null_quicksort#1:null_insert#2:null_insert#3:null_splitqs#3:null_insert#4:null_quicksort#2:null_append#1
1173.46/298.94	
1173.46/298.94	Rewrite Strategy: INNERMOST
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1173.46/298.94	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1173.46/298.94	The constant constructors are abstracted as follows:
1173.46/298.94	
1173.46/298.94	   nil => 1
1173.46/298.94	   #false => 1
1173.46/298.94	   #true => 2
1173.46/298.94	   #EQ => 1
1173.46/298.94	   #GT => 2
1173.46/298.94	   #LT => 3
1173.46/298.94	   #0 => 0
1173.46/298.94	   null_#and => 0
1173.46/298.94	   null_#ckgt => 0
1173.46/298.94	   null_#compare => 0
1173.46/298.94	   null_#eq => 0
1173.46/298.94	   null_split#1 => 0
1173.46/298.94	   null_splitqs#1 => 0
1173.46/298.94	   null_splitqs#2 => 0
1173.46/298.94	   null_insert#1 => 0
1173.46/298.94	   null_quicksort#1 => 0
1173.46/298.94	   null_insert#2 => 0
1173.46/298.94	   null_insert#3 => 0
1173.46/298.94	   null_splitqs#3 => 0
1173.46/298.94	   null_insert#4 => 0
1173.46/298.94	   null_quicksort#2 => 0
1173.46/298.94	   null_append#1 => 0
1173.46/298.94	
1173.46/298.94	----------------------------------------
1173.46/298.94	
1173.46/298.94	(12)
1173.46/298.94	Obligation:
1173.46/298.94	Complexity RNTS consisting of the following rules:
1173.46/298.94	
1173.46/298.94	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.46/298.94	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.46/298.94	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.46/298.94	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.46/298.94	   #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.46/298.94	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.46/298.94	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.46/298.94	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.46/298.94	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.46/298.94	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.46/298.94	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.46/298.94	   #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.46/298.94	   #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.46/298.94	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.46/298.94	   #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: z = 1 + @x', @x' >= 0, z' = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @y' >= 0, z' = 1 + @y', z = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(1) :|: z = 0, z' = 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
1173.46/298.94	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
1173.46/298.94	   append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
1173.46/298.94	   append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 1, @ys >= 0
1173.46/298.94	   append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.46/298.94	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
1173.46/298.94	   insert(z, z') -{ 1 }-> insert#1(@x, @l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1173.46/298.94	   insert#1(z, z', z'') -{ 1 }-> insert#2(@l, @keyX, @valX, @x) :|: @valX >= 0, @keyX >= 0, @l >= 0, z = 1 + @valX + @keyX, @x >= 0, z' = @l, z'' = @x
1173.46/298.94	   insert#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1173.46/298.94	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, @keyX, @ls, @valX, @x) :|: z1 = @x, @ls >= 0, @keyX >= 0, @valX >= 0, @l1 >= 0, @x >= 0, z = 1 + @l1 + @ls, z'' = @valX, z' = @keyX
1173.46/298.94	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1173.46/298.94	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + @valX + 1) + @keyX) + 1 :|: z1 = @x, @keyX >= 0, @valX >= 0, @x >= 0, z = 1, z'' = @valX, z' = @keyX
1173.46/298.94	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) :|: @key1 >= 0, @keyX >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @x, z = 1 + @vals1 + @key1, z' = @keyX, z'' = @ls
1173.46/298.94	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
1173.46/298.94	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z1 = v3, z3 = v5, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, v5 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
1173.46/298.94	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + @vals1 + @key1) + insert(@x, @ls) :|: @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z = 1, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls
1173.46/298.94	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + @valX + @vals1) + @key1) + @ls :|: z = 2, @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls
1173.46/298.94	   quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0
1173.46/298.94	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.95	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	   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
1173.57/298.95	   quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   sortAll(z) -{ 1 }-> sortAll#1(@l) :|: z = @l, @l >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(@xs) :|: z' = @xs, @vals >= 0, z = 1 + @vals + @key, @xs >= 0, @key >= 0
1173.57/298.95	   split(z) -{ 1 }-> split#1(@l) :|: z = @l, @l >= 0
1173.57/298.95	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   split#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	   splitAndSort(z) -{ 2 }-> sortAll(split#1(@l)) :|: z = @l, @l >= 0
1173.57/298.95	   splitqs(z, z') -{ 1 }-> splitqs#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' = @pivot, @pivot >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	
1173.57/298.95	
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(13) InliningProof (UPPER BOUND(ID))
1173.57/298.95	Inlined the following terminating rules on right-hand sides where appropriate:
1173.57/298.95	
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.95	   #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.95	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(14)
1173.57/298.95	Obligation:
1173.57/298.95	Complexity RNTS consisting of the following rules:
1173.57/298.95	
1173.57/298.95	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.95	   #ckgt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' = 1 + @y, @y >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #eq(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   #eq(z, z') -{ 0 }-> #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #equal(z, z') -{ 1 }-> #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
1173.57/298.95	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
1173.57/298.95	   #greater(z, z') -{ 1 }-> #ckgt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
1173.57/298.95	   append(z, z') -{ 1 }-> append#1(@l, @ys) :|: z = @l, @l >= 0, z' = @ys, @ys >= 0
1173.57/298.95	   append#1(z, z') -{ 1 }-> @ys :|: z' = @ys, z = 1, @ys >= 0
1173.57/298.95	   append#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, @ys) :|: z' = @ys, @x >= 0, z = 1 + @x + @xs, @xs >= 0, @ys >= 0
1173.57/298.95	   insert(z, z') -{ 1 }-> insert#1(@x, @l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1173.57/298.95	   insert#1(z, z', z'') -{ 1 }-> insert#2(@l, @keyX, @valX, @x) :|: @valX >= 0, @keyX >= 0, @l >= 0, z = 1 + @valX + @keyX, @x >= 0, z' = @l, z'' = @x
1173.57/298.95	   insert#1(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, @keyX, @ls, @valX, @x) :|: z1 = @x, @ls >= 0, @keyX >= 0, @valX >= 0, @l1 >= 0, @x >= 0, z = 1 + @l1 + @ls, z'' = @valX, z' = @keyX
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + @valX + 1) + @keyX) + 1 :|: z1 = @x, @keyX >= 0, @valX >= 0, @x >= 0, z = 1, z'' = @valX, z' = @keyX
1173.57/298.95	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) :|: @key1 >= 0, @keyX >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @x, z = 1 + @vals1 + @key1, z' = @keyX, z'' = @ls
1173.57/298.95	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z1 = v3, z3 = v5, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, v5 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + @vals1 + @key1) + insert(@x, @ls) :|: @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z = 1, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + @valX + @vals1) + @key1) + @ls :|: z = 2, @key1 >= 0, @ls >= 0, @valX >= 0, @x >= 0, @vals1 >= 0, z1 = @valX, z2 = @vals1, z' = @key1, z3 = @x, z'' = @ls
1173.57/298.95	   quicksort(z) -{ 1 }-> quicksort#1(@l) :|: z = @l, @l >= 0
1173.57/298.95	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.95	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   quicksort#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	   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
1173.57/298.95	   quicksort#2(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   sortAll(z) -{ 1 }-> sortAll#1(@l) :|: z = @l, @l >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(@xs) :|: z' = @xs, @vals >= 0, z = 1 + @vals + @key, @xs >= 0, @key >= 0
1173.57/298.95	   split(z) -{ 1 }-> split#1(@l) :|: z = @l, @l >= 0
1173.57/298.95	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   split#1(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1173.57/298.95	   splitAndSort(z) -{ 2 }-> sortAll(split#1(@l)) :|: z = @l, @l >= 0
1173.57/298.95	   splitqs(z, z') -{ 1 }-> splitqs#1(@l, @pivot) :|: @l >= 0, z = @pivot, z' = @l, @pivot >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, @pivot), @pivot, @x) :|: @x >= 0, z = 1 + @x + @xs, z' = @pivot, @xs >= 0, @pivot >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1173.57/298.95	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' = @pivot, @pivot >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(@x, @pivot)), @ls, @rs, @x) :|: @ls >= 0, z = 1 + @ls + @rs, @x >= 0, z' = @pivot, @pivot >= 0, z'' = @x, @rs >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + @ls + (1 + @x + @rs) :|: z = 2, z1 = @x, @ls >= 0, @x >= 0, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + @x + @ls) + @rs :|: z1 = @x, @ls >= 0, @x >= 0, z = 1, z' = @ls, @rs >= 0, z'' = @rs
1173.57/298.95	
1173.57/298.95	
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
1173.57/298.95	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(16)
1173.57/298.95	Obligation:
1173.57/298.95	Complexity RNTS consisting of the following rules:
1173.57/298.95	
1173.57/298.95	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.95	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.95	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.95	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.95	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.95	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.95	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.95	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.95	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.95	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.95	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.95	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.95	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.95	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.95	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.95	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.95	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.95	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.95	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.95	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.95	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.95	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.95	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.95	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.95	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.95	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.95	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.95	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.95	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.95	
1173.57/298.95	
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
1173.57/298.95	Found the following analysis order by SCC decomposition:
1173.57/298.95	
1173.57/298.95	   { #and }
1173.57/298.95	   { #compare }
1173.57/298.95	   { #ckgt }
1173.57/298.95	   { splitqs#3 }
1173.57/298.95	   { append#1, append }
1173.57/298.95	   { #eq }
1173.57/298.95	   { splitqs#2 }
1173.57/298.95	   { #greater }
1173.57/298.95	   { insert#4, insert#2, insert, insert#3, insert#1 }
1173.57/298.95	   { #equal }
1173.57/298.95	   { splitqs#1 }
1173.57/298.95	   { split#1 }
1173.57/298.95	   { splitqs }
1173.57/298.95	   { quicksort#1, quicksort#2 }
1173.57/298.95	   { split }
1173.57/298.95	   { quicksort }
1173.57/298.95	   { sortAll#2, sortAll, sortAll#1 }
1173.57/298.95	   { splitAndSort }
1173.57/298.95	
1173.57/298.95	----------------------------------------
1173.57/298.95	
1173.57/298.95	(18)
1173.57/298.95	Obligation:
1173.57/298.95	Complexity RNTS consisting of the following rules:
1173.57/298.95	
1173.57/298.95	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.95	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.95	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.95	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.95	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.96	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.96	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.96	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.96	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.96	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.96	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.96	
1173.57/298.96	Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(19) ResultPropagationProof (UPPER BOUND(ID))
1173.57/298.96	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(20)
1173.57/298.96	Obligation:
1173.57/298.96	Complexity RNTS consisting of the following rules:
1173.57/298.96	
1173.57/298.96	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.96	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.96	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.96	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.96	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.96	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.96	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.96	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.96	
1173.57/298.96	Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(21) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.96	
1173.57/298.96	Computed SIZE bound using CoFloCo for: #and
1173.57/298.96	after applying outer abstraction to obtain an ITS,
1173.57/298.96	resulting in: O(1) with polynomial bound: 2
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(22)
1173.57/298.96	Obligation:
1173.57/298.96	Complexity RNTS consisting of the following rules:
1173.57/298.96	
1173.57/298.96	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.96	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.96	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.96	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.96	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.96	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.96	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.96	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.96	
1173.57/298.96	Function symbols to be analyzed: {#and}, {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.96	Previous analysis results are:
1173.57/298.96	  #and: runtime: ?, size: O(1) [2]
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(23) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.96	
1173.57/298.96	Computed RUNTIME bound using CoFloCo for: #and
1173.57/298.96	after applying outer abstraction to obtain an ITS,
1173.57/298.96	resulting in: O(1) with polynomial bound: 0
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(24)
1173.57/298.96	Obligation:
1173.57/298.96	Complexity RNTS consisting of the following rules:
1173.57/298.96	
1173.57/298.96	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.96	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.96	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.96	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.96	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.96	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.96	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.96	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.96	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.96	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.96	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.96	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.96	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.96	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.96	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.96	
1173.57/298.96	Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.96	Previous analysis results are:
1173.57/298.96	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.96	
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(25) ResultPropagationProof (UPPER BOUND(ID))
1173.57/298.96	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/298.96	----------------------------------------
1173.57/298.96	
1173.57/298.96	(26)
1173.57/298.96	Obligation:
1173.57/298.96	Complexity RNTS consisting of the following rules:
1173.57/298.96	
1173.57/298.96	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.96	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.96	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.96	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.96	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.96	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.96	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.96	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.96	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.96	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.96	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.97	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.97	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.97	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.97	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.97	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.97	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.97	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.97	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.97	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.97	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.97	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.97	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.97	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.97	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.97	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.97	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.97	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.97	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.97	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.97	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.97	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.97	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.97	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.97	
1173.57/298.97	Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.97	Previous analysis results are:
1173.57/298.97	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.97	
1173.57/298.97	----------------------------------------
1173.57/298.97	
1173.57/298.97	(27) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.97	
1173.57/298.97	Computed SIZE bound using CoFloCo for: #compare
1173.57/298.97	after applying outer abstraction to obtain an ITS,
1173.57/298.97	resulting in: O(1) with polynomial bound: 3
1173.57/298.97	
1173.57/298.97	----------------------------------------
1173.57/298.97	
1173.57/298.97	(28)
1173.57/298.97	Obligation:
1173.57/298.97	Complexity RNTS consisting of the following rules:
1173.57/298.97	
1173.57/298.97	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.97	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.97	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.97	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.97	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.97	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.97	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.97	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.97	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.97	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.97	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.97	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.97	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.97	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.97	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.97	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.97	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.97	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.97	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.97	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.97	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.97	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.97	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.97	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.97	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.97	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.97	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.97	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.97	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.97	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.97	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.97	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.97	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.97	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.97	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.97	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.97	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.97	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.98	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.98	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.98	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.98	
1173.57/298.98	Function symbols to be analyzed: {#compare}, {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.98	Previous analysis results are:
1173.57/298.98	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.98	  #compare: runtime: ?, size: O(1) [3]
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(29) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.98	
1173.57/298.98	Computed RUNTIME bound using CoFloCo for: #compare
1173.57/298.98	after applying outer abstraction to obtain an ITS,
1173.57/298.98	resulting in: O(1) with polynomial bound: 0
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(30)
1173.57/298.98	Obligation:
1173.57/298.98	Complexity RNTS consisting of the following rules:
1173.57/298.98	
1173.57/298.98	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.98	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.98	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.98	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.98	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.98	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.98	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.98	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(#compare(z'', z')), @ls, @rs, z'') :|: @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.98	
1173.57/298.98	Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.98	Previous analysis results are:
1173.57/298.98	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.98	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(31) ResultPropagationProof (UPPER BOUND(ID))
1173.57/298.98	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(32)
1173.57/298.98	Obligation:
1173.57/298.98	Complexity RNTS consisting of the following rules:
1173.57/298.98	
1173.57/298.98	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.98	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.98	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.98	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.98	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.98	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.98	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.98	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.98	
1173.57/298.98	Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.98	Previous analysis results are:
1173.57/298.98	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.98	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(33) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.98	
1173.57/298.98	Computed SIZE bound using CoFloCo for: #ckgt
1173.57/298.98	after applying outer abstraction to obtain an ITS,
1173.57/298.98	resulting in: O(1) with polynomial bound: 2
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(34)
1173.57/298.98	Obligation:
1173.57/298.98	Complexity RNTS consisting of the following rules:
1173.57/298.98	
1173.57/298.98	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.98	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/298.98	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/298.98	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/298.98	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/298.98	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/298.98	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/298.98	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/298.98	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/298.98	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/298.98	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/298.98	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/298.98	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/298.98	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/298.98	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/298.98	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/298.98	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/298.98	
1173.57/298.98	Function symbols to be analyzed: {#ckgt}, {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/298.98	Previous analysis results are:
1173.57/298.98	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/298.98	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/298.98	  #ckgt: runtime: ?, size: O(1) [2]
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(35) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/298.98	
1173.57/298.98	Computed RUNTIME bound using CoFloCo for: #ckgt
1173.57/298.98	after applying outer abstraction to obtain an ITS,
1173.57/298.98	resulting in: O(1) with polynomial bound: 0
1173.57/298.98	
1173.57/298.98	----------------------------------------
1173.57/298.98	
1173.57/298.98	(36)
1173.57/298.98	Obligation:
1173.57/298.98	Complexity RNTS consisting of the following rules:
1173.57/298.98	
1173.57/298.98	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/298.98	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/298.98	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/298.98	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/298.98	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/298.98	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/298.98	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> #ckgt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> #ckgt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(#ckgt(s2), @ls, @rs, z'') :|: s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(37) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.00	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(38)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(39) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed SIZE bound using CoFloCo for: splitqs#3
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(40)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {splitqs#3}, {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(41) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed RUNTIME bound using CoFloCo for: splitqs#3
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(1) with polynomial bound: 1
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(42)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 2 }-> splitqs#3(s5, @ls, @rs, z'') :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(43) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.00	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(44)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(45) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed SIZE bound using KoAT for: append#1
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(n^1) with polynomial bound: z + z'
1173.57/299.00	
1173.57/299.00	Computed SIZE bound using CoFloCo for: append
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(n^1) with polynomial bound: z + z'
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(46)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {append#1,append}, {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	  append#1: runtime: ?, size: O(n^1) [z + z']
1173.57/299.00	  append: runtime: ?, size: O(n^1) [z + z']
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(47) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed RUNTIME bound using CoFloCo for: append#1
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(n^1) with polynomial bound: 3 + 2*z
1173.57/299.00	
1173.57/299.00	Computed RUNTIME bound using CoFloCo for: append
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(n^1) with polynomial bound: 4 + 2*z
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(48)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 1 }-> append#1(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> 1 + @x + append(@xs, z') :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(49) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.00	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(50)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(51) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed SIZE bound using CoFloCo for: #eq
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(1) with polynomial bound: 2
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(52)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.00	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.00	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.00	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.00	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.00	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.00	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.00	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.00	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.00	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.00	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.00	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.00	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.00	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.00	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.00	
1173.57/299.00	Function symbols to be analyzed: {#eq}, {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.00	Previous analysis results are:
1173.57/299.00	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.00	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.00	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.00	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.00	  #eq: runtime: ?, size: O(1) [2]
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(53) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.00	
1173.57/299.00	Computed RUNTIME bound using CoFloCo for: #eq
1173.57/299.00	after applying outer abstraction to obtain an ITS,
1173.57/299.00	resulting in: O(1) with polynomial bound: 0
1173.57/299.00	
1173.57/299.00	----------------------------------------
1173.57/299.00	
1173.57/299.00	(54)
1173.57/299.00	Obligation:
1173.57/299.00	Complexity RNTS consisting of the following rules:
1173.57/299.00	
1173.57/299.00	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.00	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.00	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.00	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.00	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.00	   #eq(z, z') -{ 0 }-> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.00	   #equal(z, z') -{ 1 }-> #eq(z, z') :|: z >= 0, z' >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.00	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.00	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.00	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(#eq(@key1, z'), @key1, z'', z1, @vals1, z2) :|: @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.01	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.01	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.01	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.01	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.01	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.01	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.01	
1173.57/299.01	Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.01	Previous analysis results are:
1173.57/299.01	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.01	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.01	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(55) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.01	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(56)
1173.57/299.01	Obligation:
1173.57/299.01	Complexity RNTS consisting of the following rules:
1173.57/299.01	
1173.57/299.01	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.01	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.01	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.01	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.01	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.01	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.01	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.01	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.01	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.01	
1173.57/299.01	Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.01	Previous analysis results are:
1173.57/299.01	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.01	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.01	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(57) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.01	
1173.57/299.01	Computed SIZE bound using CoFloCo for: splitqs#2
1173.57/299.01	after applying outer abstraction to obtain an ITS,
1173.57/299.01	resulting in: O(n^1) with polynomial bound: 1 + z + z''
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(58)
1173.57/299.01	Obligation:
1173.57/299.01	Complexity RNTS consisting of the following rules:
1173.57/299.01	
1173.57/299.01	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.01	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.01	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.01	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.01	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.01	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.01	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.01	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.01	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.01	
1173.57/299.01	Function symbols to be analyzed: {splitqs#2}, {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.01	Previous analysis results are:
1173.57/299.01	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.01	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.01	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#2: runtime: ?, size: O(n^1) [1 + z + z'']
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(59) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.01	
1173.57/299.01	Computed RUNTIME bound using CoFloCo for: splitqs#2
1173.57/299.01	after applying outer abstraction to obtain an ITS,
1173.57/299.01	resulting in: O(1) with polynomial bound: 3
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(60)
1173.57/299.01	Obligation:
1173.57/299.01	Complexity RNTS consisting of the following rules:
1173.57/299.01	
1173.57/299.01	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.01	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.01	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.01	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.01	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.01	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.01	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.01	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.01	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.01	
1173.57/299.01	Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.01	Previous analysis results are:
1173.57/299.01	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.01	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.01	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.01	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.01	
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(61) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.01	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.01	----------------------------------------
1173.57/299.01	
1173.57/299.01	(62)
1173.57/299.01	Obligation:
1173.57/299.01	Complexity RNTS consisting of the following rules:
1173.57/299.01	
1173.57/299.01	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.01	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.01	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.01	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.01	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.01	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.01	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.01	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.01	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.01	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.01	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.01	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.01	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.01	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.01	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.01	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.01	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.01	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.01	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.01	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.01	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.01	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.01	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.01	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.01	
1173.57/299.01	Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.01	Previous analysis results are:
1173.57/299.01	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.01	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.01	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.02	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(63) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: #greater
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(1) with polynomial bound: 2
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(64)
1173.57/299.02	Obligation:
1173.57/299.02	Complexity RNTS consisting of the following rules:
1173.57/299.02	
1173.57/299.02	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.02	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.02	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.02	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.02	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.02	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.02	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.02	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.02	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.02	
1173.57/299.02	Function symbols to be analyzed: {#greater}, {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.02	Previous analysis results are:
1173.57/299.02	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.02	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.02	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.02	  #greater: runtime: ?, size: O(1) [2]
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(65) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: #greater
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(1) with polynomial bound: 1
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(66)
1173.57/299.02	Obligation:
1173.57/299.02	Complexity RNTS consisting of the following rules:
1173.57/299.02	
1173.57/299.02	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.02	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.02	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.02	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.02	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.02	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.02	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.02	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.02	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.02	
1173.57/299.02	Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.02	Previous analysis results are:
1173.57/299.02	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.02	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.02	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.02	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(67) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.02	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(68)
1173.57/299.02	Obligation:
1173.57/299.02	Complexity RNTS consisting of the following rules:
1173.57/299.02	
1173.57/299.02	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.02	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.02	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.02	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.02	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.02	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.02	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.02	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.02	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.02	
1173.57/299.02	Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.02	Previous analysis results are:
1173.57/299.02	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.02	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.02	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.02	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(69) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: insert#4
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 6 + z' + z'' + z1 + z2 + z3
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: insert#2
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 5 + z + z' + z'' + z1
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: insert
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 4 + 2*z + z'
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: insert#3
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 5 + z + z'' + z1 + z2
1173.57/299.02	
1173.57/299.02	Computed SIZE bound using CoFloCo for: insert#1
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 4 + z + z' + z''
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(70)
1173.57/299.02	Obligation:
1173.57/299.02	Complexity RNTS consisting of the following rules:
1173.57/299.02	
1173.57/299.02	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.02	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.02	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.02	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.02	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.02	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.02	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.02	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.02	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.02	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.02	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.02	
1173.57/299.02	Function symbols to be analyzed: {insert#4,insert#2,insert,insert#3,insert#1}, {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.02	Previous analysis results are:
1173.57/299.02	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.02	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.02	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.02	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.02	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.02	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.57/299.02	  insert#4: runtime: ?, size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.57/299.02	  insert#2: runtime: ?, size: O(n^1) [5 + z + z' + z'' + z1]
1173.57/299.02	  insert: runtime: ?, size: O(n^1) [4 + 2*z + z']
1173.57/299.02	  insert#3: runtime: ?, size: O(n^1) [5 + z + z'' + z1 + z2]
1173.57/299.02	  insert#1: runtime: ?, size: O(n^1) [4 + z + z' + z'']
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(71) IntTrsBoundProof (UPPER BOUND(ID))
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: insert#4
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 10 + 6*z''
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: insert#2
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 1 + 6*z
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: insert
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 3 + 6*z'
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: insert#3
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 12 + 6*z''
1173.57/299.02	
1173.57/299.02	Computed RUNTIME bound using CoFloCo for: insert#1
1173.57/299.02	after applying outer abstraction to obtain an ITS,
1173.57/299.02	resulting in: O(n^1) with polynomial bound: 2 + 6*z'
1173.57/299.02	
1173.57/299.02	----------------------------------------
1173.57/299.02	
1173.57/299.02	(72)
1173.57/299.02	Obligation:
1173.57/299.02	Complexity RNTS consisting of the following rules:
1173.57/299.02	
1173.57/299.02	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.02	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.02	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.02	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.02	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.02	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.57/299.02	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.57/299.02	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.02	   insert(z, z') -{ 1 }-> insert#1(z, z', z) :|: z' >= 0, z >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 1 }-> insert#2(z', @keyX, @valX, z'') :|: @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.57/299.02	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> insert#3(@l1, z', @ls, z'', z1) :|: @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 2 }-> insert#4(s10, @key1, z'', z1, @vals1, z2) :|: s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.57/299.02	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + z2 + z') + insert(z3, z'') :|: z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.57/299.02	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.57/299.02	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.57/299.02	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.57/299.02	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.57/299.02	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.02	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.02	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.02	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.57/299.03	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.57/299.03	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.57/299.03	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.57/299.03	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.03	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.57/299.03	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.57/299.03	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.57/299.03	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.03	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.57/299.03	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.57/299.03	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.57/299.03	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.57/299.03	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.57/299.03	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.57/299.03	
1173.57/299.03	Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.57/299.03	Previous analysis results are:
1173.57/299.03	  #and: runtime: O(1) [0], size: O(1) [2]
1173.57/299.03	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.57/299.03	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.57/299.03	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.57/299.03	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.57/299.03	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.57/299.03	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.57/299.03	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.57/299.03	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.57/299.03	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.57/299.03	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.57/299.03	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.57/299.03	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.57/299.03	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.57/299.03	
1173.57/299.03	----------------------------------------
1173.57/299.03	
1173.57/299.03	(73) ResultPropagationProof (UPPER BOUND(ID))
1173.57/299.03	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.57/299.03	----------------------------------------
1173.57/299.03	
1173.57/299.03	(74)
1173.57/299.03	Obligation:
1173.57/299.03	Complexity RNTS consisting of the following rules:
1173.57/299.03	
1173.57/299.03	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.57/299.03	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.57/299.03	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.57/299.03	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.57/299.03	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.03	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.57/299.03	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.57/299.03	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.57/299.03	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.57/299.03	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.03	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.57/299.03	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.04	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.04	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.04	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.04	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.04	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.04	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.04	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.04	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.04	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.04	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.04	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.04	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.04	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.04	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.04	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.04	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.04	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.04	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.04	
1173.93/299.04	Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.04	Previous analysis results are:
1173.93/299.04	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.04	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.04	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.04	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.04	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.04	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.04	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.04	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.04	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.04	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.04	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.04	
1173.93/299.04	----------------------------------------
1173.93/299.04	
1173.93/299.04	(75) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.04	
1173.93/299.04	Computed SIZE bound using CoFloCo for: #equal
1173.93/299.04	after applying outer abstraction to obtain an ITS,
1173.93/299.04	resulting in: O(1) with polynomial bound: 2
1173.93/299.04	
1173.93/299.04	----------------------------------------
1173.93/299.04	
1173.93/299.04	(76)
1173.93/299.04	Obligation:
1173.93/299.04	Complexity RNTS consisting of the following rules:
1173.93/299.04	
1173.93/299.04	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.04	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.04	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.04	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.04	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.04	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.04	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.04	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.04	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.04	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.04	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.04	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.04	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.04	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.04	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.04	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.04	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.04	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.04	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.04	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.04	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.04	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.04	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.04	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.04	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.04	
1173.93/299.04	Function symbols to be analyzed: {#equal}, {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.04	Previous analysis results are:
1173.93/299.04	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.04	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.04	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.04	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.04	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.04	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.04	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.04	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.04	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.04	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.04	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.04	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.04	  #equal: runtime: ?, size: O(1) [2]
1173.93/299.04	
1173.93/299.04	----------------------------------------
1173.93/299.04	
1173.93/299.04	(77) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.04	
1173.93/299.04	Computed RUNTIME bound using CoFloCo for: #equal
1173.93/299.04	after applying outer abstraction to obtain an ITS,
1173.93/299.04	resulting in: O(1) with polynomial bound: 1
1173.93/299.04	
1173.93/299.04	----------------------------------------
1173.93/299.04	
1173.93/299.04	(78)
1173.93/299.04	Obligation:
1173.93/299.04	Complexity RNTS consisting of the following rules:
1173.93/299.04	
1173.93/299.04	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.04	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.04	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.04	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.04	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.04	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.04	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.04	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.04	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(79) ResultPropagationProof (UPPER BOUND(ID))
1173.93/299.05	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(80)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(81) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.05	
1173.93/299.05	Computed SIZE bound using KoAT for: splitqs#1
1173.93/299.05	after applying outer abstraction to obtain an ITS,
1173.93/299.05	resulting in: O(n^1) with polynomial bound: 2 + z
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(82)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {splitqs#1}, {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  splitqs#1: runtime: ?, size: O(n^1) [2 + z]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(83) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.05	
1173.93/299.05	Computed RUNTIME bound using KoAT for: splitqs#1
1173.93/299.05	after applying outer abstraction to obtain an ITS,
1173.93/299.05	resulting in: O(n^1) with polynomial bound: 1 + 5*z
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(84)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 2 }-> quicksort#2(splitqs#1(@zs, @z), @z) :|: z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 1 }-> splitqs#1(z', z) :|: z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 2 }-> splitqs#2(splitqs#1(@xs, z'), z', @x) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(85) ResultPropagationProof (UPPER BOUND(ID))
1173.93/299.05	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(86)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(87) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.05	
1173.93/299.05	Computed SIZE bound using KoAT for: split#1
1173.93/299.05	after applying outer abstraction to obtain an ITS,
1173.93/299.05	resulting in: O(n^1) with polynomial bound: 4*z
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(88)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.05	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.05	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.05	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.05	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.05	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.05	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.05	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.05	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.05	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.05	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.05	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.05	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.05	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.05	
1173.93/299.05	Function symbols to be analyzed: {split#1}, {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.05	Previous analysis results are:
1173.93/299.05	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.05	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.05	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.05	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.05	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.05	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.05	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.05	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.05	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.05	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.05	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.05	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.05	  split#1: runtime: ?, size: O(n^1) [4*z]
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(89) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.05	
1173.93/299.05	Computed RUNTIME bound using KoAT for: split#1
1173.93/299.05	after applying outer abstraction to obtain an ITS,
1173.93/299.05	resulting in: O(n^2) with polynomial bound: 1 + 5*z + 24*z^2
1173.93/299.05	
1173.93/299.05	----------------------------------------
1173.93/299.05	
1173.93/299.05	(90)
1173.93/299.05	Obligation:
1173.93/299.05	Complexity RNTS consisting of the following rules:
1173.93/299.05	
1173.93/299.05	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.05	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.05	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.05	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.05	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.05	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.05	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.05	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.05	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.05	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.05	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.05	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.06	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.06	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.06	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.06	   split(z) -{ 1 }-> split#1(z) :|: z >= 0
1173.93/299.06	   split#1(z) -{ 2 }-> insert(@x, split#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   splitAndSort(z) -{ 2 }-> sortAll(split#1(z)) :|: z >= 0
1173.93/299.06	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.06	
1173.93/299.06	Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.06	Previous analysis results are:
1173.93/299.06	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.06	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.06	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.06	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.06	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.06	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.06	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.06	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.06	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.06	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(91) ResultPropagationProof (UPPER BOUND(ID))
1173.93/299.06	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(92)
1173.93/299.06	Obligation:
1173.93/299.06	Complexity RNTS consisting of the following rules:
1173.93/299.06	
1173.93/299.06	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.06	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.06	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.06	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.06	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.06	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.06	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1173.93/299.06	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1173.93/299.06	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.06	
1173.93/299.06	Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.06	Previous analysis results are:
1173.93/299.06	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.06	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.06	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.06	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.06	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.06	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.06	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.06	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.06	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.06	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(93) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.06	
1173.93/299.06	Computed SIZE bound using CoFloCo for: splitqs
1173.93/299.06	after applying outer abstraction to obtain an ITS,
1173.93/299.06	resulting in: O(n^1) with polynomial bound: 2 + z'
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(94)
1173.93/299.06	Obligation:
1173.93/299.06	Complexity RNTS consisting of the following rules:
1173.93/299.06	
1173.93/299.06	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.06	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.06	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.06	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.06	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.06	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.06	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1173.93/299.06	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1173.93/299.06	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.06	
1173.93/299.06	Function symbols to be analyzed: {splitqs}, {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.06	Previous analysis results are:
1173.93/299.06	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.06	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.06	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.06	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.06	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.06	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.06	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.06	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.06	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.06	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.06	  splitqs: runtime: ?, size: O(n^1) [2 + z']
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(95) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.06	
1173.93/299.06	Computed RUNTIME bound using CoFloCo for: splitqs
1173.93/299.06	after applying outer abstraction to obtain an ITS,
1173.93/299.06	resulting in: O(n^1) with polynomial bound: 2 + 5*z'
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(96)
1173.93/299.06	Obligation:
1173.93/299.06	Complexity RNTS consisting of the following rules:
1173.93/299.06	
1173.93/299.06	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.06	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.06	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.06	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.06	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.06	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.06	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.06	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.06	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1173.93/299.06	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.06	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.06	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1173.93/299.06	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.06	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.06	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.06	
1173.93/299.06	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.06	Previous analysis results are:
1173.93/299.06	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.06	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.06	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.06	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.06	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.06	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.06	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.06	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.06	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.06	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.06	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.06	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.06	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.06	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1173.93/299.06	
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(97) ResultPropagationProof (UPPER BOUND(ID))
1173.93/299.06	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1173.93/299.06	----------------------------------------
1173.93/299.06	
1173.93/299.06	(98)
1173.93/299.06	Obligation:
1173.93/299.06	Complexity RNTS consisting of the following rules:
1173.93/299.06	
1173.93/299.06	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.06	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.06	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.06	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.06	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.06	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.06	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.06	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.06	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.06	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.06	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.06	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.06	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.07	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.07	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.07	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.07	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.07	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.07	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.07	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.07	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1173.93/299.07	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.07	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1173.93/299.07	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.07	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.07	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.07	
1173.93/299.07	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.07	Previous analysis results are:
1173.93/299.07	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.07	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.07	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.07	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.07	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.07	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.07	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.07	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.07	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.07	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.07	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.07	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.07	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.07	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.07	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1173.93/299.07	
1173.93/299.07	----------------------------------------
1173.93/299.07	
1173.93/299.07	(99) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.07	
1173.93/299.07	Computed SIZE bound using CoFloCo for: quicksort#1
1173.93/299.07	after applying outer abstraction to obtain an ITS,
1173.93/299.07	resulting in: O(n^1) with polynomial bound: 1 + 2*z
1173.93/299.07	
1173.93/299.07	Computed SIZE bound using CoFloCo for: quicksort#2
1173.93/299.07	after applying outer abstraction to obtain an ITS,
1173.93/299.07	resulting in: O(n^1) with polynomial bound: 1 + 2*z + z'
1173.93/299.07	
1173.93/299.07	----------------------------------------
1173.93/299.07	
1173.93/299.07	(100)
1173.93/299.07	Obligation:
1173.93/299.07	Complexity RNTS consisting of the following rules:
1173.93/299.07	
1173.93/299.07	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.07	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.07	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.07	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.07	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.07	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.07	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.07	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.07	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1173.93/299.07	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1173.93/299.07	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1173.93/299.07	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1173.93/299.07	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1173.93/299.07	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1173.93/299.07	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1173.93/299.07	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1173.93/299.07	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1173.93/299.07	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.07	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1173.93/299.07	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1173.93/299.07	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1173.93/299.07	   split#1(z) -{ 1 }-> 1 :|: z = 1
1173.93/299.07	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1173.93/299.07	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1173.93/299.07	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1173.93/299.07	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1173.93/299.07	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1173.93/299.07	
1173.93/299.07	Function symbols to be analyzed: {quicksort#1,quicksort#2}, {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1173.93/299.07	Previous analysis results are:
1173.93/299.07	  #and: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  #compare: runtime: O(1) [0], size: O(1) [3]
1173.93/299.07	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1173.93/299.07	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1173.93/299.07	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1173.93/299.07	  #eq: runtime: O(1) [0], size: O(1) [2]
1173.93/299.07	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1173.93/299.07	  #greater: runtime: O(1) [1], size: O(1) [2]
1173.93/299.07	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1173.93/299.07	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1173.93/299.07	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1173.93/299.07	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1173.93/299.07	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1173.93/299.07	  #equal: runtime: O(1) [1], size: O(1) [2]
1173.93/299.07	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1173.93/299.07	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1173.93/299.07	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1173.93/299.07	  quicksort#1: runtime: ?, size: O(n^1) [1 + 2*z]
1173.93/299.07	  quicksort#2: runtime: ?, size: O(n^1) [1 + 2*z + z']
1173.93/299.07	
1173.93/299.07	----------------------------------------
1173.93/299.07	
1173.93/299.07	(101) IntTrsBoundProof (UPPER BOUND(ID))
1173.93/299.07	
1173.93/299.07	Computed RUNTIME bound using CoFloCo for: quicksort#1
1173.93/299.07	after applying outer abstraction to obtain an ITS,
1173.93/299.07	resulting in: O(n^2) with polynomial bound: 69 + 184*z + 90*z^2
1173.93/299.07	
1173.93/299.07	Computed RUNTIME bound using KoAT for: quicksort#2
1173.93/299.07	after applying outer abstraction to obtain an ITS,
1173.93/299.07	resulting in: O(n^2) with polynomial bound: 147 + 372*z + 180*z^2
1173.93/299.07	
1173.93/299.07	----------------------------------------
1173.93/299.07	
1173.93/299.07	(102)
1173.93/299.07	Obligation:
1173.93/299.07	Complexity RNTS consisting of the following rules:
1173.93/299.07	
1173.93/299.07	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1173.93/299.07	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1173.93/299.07	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1173.93/299.07	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1173.93/299.07	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1173.93/299.07	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1173.93/299.07	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1173.93/299.07	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1173.93/299.07	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1173.93/299.07	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1173.93/299.07	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1173.93/299.07	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1173.93/299.07	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1173.93/299.07	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 1 }-> quicksort#1(z) :|: z >= 0
1174.14/299.10	   quicksort#1(z) -{ 3 + 5*@zs }-> quicksort#2(s20, @z) :|: s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 3 }-> append(quicksort#1(@xs), 1 + z' + quicksort#1(@ys)) :|: z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.10	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.10	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.10	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.10	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.10	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.10	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.10	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.10	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.10	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.10	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(103) ResultPropagationProof (UPPER BOUND(ID))
1174.14/299.10	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(104)
1174.14/299.10	Obligation:
1174.14/299.10	Complexity RNTS consisting of the following rules:
1174.14/299.10	
1174.14/299.10	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.10	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.10	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.10	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.10	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.10	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.10	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.10	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.10	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.10	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.10	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.10	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.10	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.10	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(105) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.10	
1174.14/299.10	Computed SIZE bound using CoFloCo for: split
1174.14/299.10	after applying outer abstraction to obtain an ITS,
1174.14/299.10	resulting in: O(n^1) with polynomial bound: 4*z
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(106)
1174.14/299.10	Obligation:
1174.14/299.10	Complexity RNTS consisting of the following rules:
1174.14/299.10	
1174.14/299.10	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.10	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.10	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.10	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {split}, {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.10	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.10	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.10	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.10	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.10	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.10	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.10	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.10	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.10	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.10	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.10	  split: runtime: ?, size: O(n^1) [4*z]
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(107) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.10	
1174.14/299.10	Computed RUNTIME bound using KoAT for: split
1174.14/299.10	after applying outer abstraction to obtain an ITS,
1174.14/299.10	resulting in: O(n^2) with polynomial bound: 2 + 5*z + 24*z^2
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(108)
1174.14/299.10	Obligation:
1174.14/299.10	Complexity RNTS consisting of the following rules:
1174.14/299.10	
1174.14/299.10	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.10	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.10	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.10	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.10	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.10	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.10	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.10	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.10	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.10	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.10	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.10	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.10	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.10	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.10	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(109) ResultPropagationProof (UPPER BOUND(ID))
1174.14/299.10	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(110)
1174.14/299.10	Obligation:
1174.14/299.10	Complexity RNTS consisting of the following rules:
1174.14/299.10	
1174.14/299.10	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.10	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.10	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.10	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.10	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.10	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.10	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.10	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.10	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.10	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.10	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.10	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.10	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.10	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.10	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.10	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.10	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(111) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.10	
1174.14/299.10	Computed SIZE bound using CoFloCo for: quicksort
1174.14/299.10	after applying outer abstraction to obtain an ITS,
1174.14/299.10	resulting in: O(n^1) with polynomial bound: 1 + 2*z
1174.14/299.10	
1174.14/299.10	----------------------------------------
1174.14/299.10	
1174.14/299.10	(112)
1174.14/299.10	Obligation:
1174.14/299.10	Complexity RNTS consisting of the following rules:
1174.14/299.10	
1174.14/299.10	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.10	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.10	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.10	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.10	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.10	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.10	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.10	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.10	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.10	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.10	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.10	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.10	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.10	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.10	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.10	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.10	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.10	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.10	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.10	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.10	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.10	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.10	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.10	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.10	
1174.14/299.10	Function symbols to be analyzed: {quicksort}, {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.10	Previous analysis results are:
1174.14/299.10	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.10	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.10	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: ?, size: O(n^1) [1 + 2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(113) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.11	
1174.14/299.11	Computed RUNTIME bound using KoAT for: quicksort
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^2) with polynomial bound: 70 + 184*z + 90*z^2
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(114)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 1 }-> 1 + (1 + quicksort(@vals) + @key) + sortAll(z') :|: @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(115) ResultPropagationProof (UPPER BOUND(ID))
1174.14/299.11	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(116)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(117) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.11	
1174.14/299.11	Computed SIZE bound using KoAT for: sortAll#2
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^1) with polynomial bound: 1 + 2*z + 2*z'
1174.14/299.11	
1174.14/299.11	Computed SIZE bound using KoAT for: sortAll
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^1) with polynomial bound: 2*z
1174.14/299.11	
1174.14/299.11	Computed SIZE bound using KoAT for: sortAll#1
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^1) with polynomial bound: 2*z
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(118)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {sortAll#2,sortAll,sortAll#1}, {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  sortAll#2: runtime: ?, size: O(n^1) [1 + 2*z + 2*z']
1174.14/299.11	  sortAll: runtime: ?, size: O(n^1) [2*z]
1174.14/299.11	  sortAll#1: runtime: ?, size: O(n^1) [2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(119) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.11	
1174.14/299.11	Computed RUNTIME bound using KoAT for: sortAll#2
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^3) with polynomial bound: 147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3
1174.14/299.11	
1174.14/299.11	Computed RUNTIME bound using KoAT for: sortAll
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^3) with polynomial bound: 150 + 955*z + 1464*z^2 + 540*z^3
1174.14/299.11	
1174.14/299.11	Computed RUNTIME bound using KoAT for: sortAll#1
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^3) with polynomial bound: 149 + 955*z + 1464*z^2 + 540*z^3
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(120)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 1 }-> sortAll#1(z) :|: z >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> sortAll#2(@x, @xs) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 71 + 184*@vals + 90*@vals^2 }-> 1 + (1 + s33 + @key) + sortAll(z') :|: s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 3 + 5*z + 24*z^2 }-> sortAll(s27) :|: s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z']
1174.14/299.11	  sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(121) ResultPropagationProof (UPPER BOUND(ID))
1174.14/299.11	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(122)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0
1174.14/299.11	   sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z']
1174.14/299.11	  sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(123) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.11	
1174.14/299.11	Computed SIZE bound using CoFloCo for: splitAndSort
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^1) with polynomial bound: 8*z
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(124)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0
1174.14/299.11	   sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: {splitAndSort}
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z']
1174.14/299.11	  sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  splitAndSort: runtime: ?, size: O(n^1) [8*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(125) IntTrsBoundProof (UPPER BOUND(ID))
1174.14/299.11	
1174.14/299.11	Computed RUNTIME bound using KoAT for: splitAndSort
1174.14/299.11	after applying outer abstraction to obtain an ITS,
1174.14/299.11	resulting in: O(n^3) with polynomial bound: 153 + 3825*z + 23448*z^2 + 34560*z^3
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(126)
1174.14/299.11	Obligation:
1174.14/299.11	Complexity RNTS consisting of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(z, z') -{ 0 }-> 2 :|: z = 2, z' = 2
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 1, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z' = 2, z = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 1 :|: z = 2, z' = 1
1174.14/299.11	   #and(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #ckgt(z) -{ 0 }-> 2 :|: z = 2
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 1
1174.14/299.11	   #ckgt(z) -{ 0 }-> 1 :|: z = 3
1174.14/299.11	   #ckgt(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1174.14/299.11	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2, z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> s14 :|: s12 >= 0, s12 <= 2, s13 >= 0, s13 <= 2, s14 >= 0, s14 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 2 :|: z = 1, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' = 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
1174.14/299.11	   #eq(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   #equal(z, z') -{ 1 }-> s9 :|: s9 >= 0, s9 <= 2, z >= 0, z' >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1174.14/299.11	   #greater(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1174.14/299.11	   append(z, z') -{ 4 + 2*z }-> s7 :|: s7 >= 0, s7 <= z + z', z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 1 }-> z' :|: z = 1, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   append#1(z, z') -{ 5 + 2*@xs }-> 1 + @x + s8 :|: s8 >= 0, s8 <= @xs + z', @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   insert(z, z') -{ 3 + 6*z' }-> s15 :|: s15 >= 0, s15 <= z + z' + z + 4, z' >= 0, z >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 2 + 6*z' }-> s16 :|: s16 >= 0, s16 <= z' + @valX + z'' + 5 + @keyX, @valX >= 0, @keyX >= 0, z' >= 0, z = 1 + @valX + @keyX, z'' >= 0
1174.14/299.11	   insert#1(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 13 + 6*@ls }-> s17 :|: s17 >= 0, s17 <= @l1 + @ls + z'' + z1 + 5, @ls >= 0, z' >= 0, z'' >= 0, @l1 >= 0, z1 >= 0, z = 1 + @l1 + @ls
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + (1 + (1 + z'' + 1) + z') + 1 :|: z' >= 0, z'' >= 0, z1 >= 0, z = 1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 12 + 6*z'' }-> s18 :|: s18 >= 0, s18 <= @key1 + z'' + @vals1 + z2 + 6 + z1, s10 >= 0, s10 <= 2, @key1 >= 0, z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, @vals1 >= 0, z = 1 + @vals1 + @key1
1174.14/299.11	   insert#3(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z3 >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 4 + 6*z'' }-> 1 + (1 + z2 + z') + s19 :|: s19 >= 0, s19 <= 2 * z3 + z'' + 4, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0, z = 1
1174.14/299.11	   insert#4(z, z', z'', z1, z2, z3) -{ 1 }-> 1 + (1 + (1 + z1 + z2) + z') + z'' :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0, z3 >= 0, z2 >= 0
1174.14/299.11	   quicksort(z) -{ 70 + 184*z + 90*z^2 }-> s28 :|: s28 >= 0, s28 <= 1 + 2 * z, z >= 0
1174.14/299.11	   quicksort#1(z) -{ 150 + 5*@zs + 372*s20 + 180*s20^2 }-> s29 :|: s29 >= 0, s29 <= 2 * s20 + @z + 1, s20 >= 0, s20 <= @zs + 2, z = 1 + @z + @zs, @zs >= 0, @z >= 0
1174.14/299.11	   quicksort#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   quicksort#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 145 + 184*@xs + 90*@xs^2 + 184*@ys + 90*@ys^2 + 2*s30 }-> s32 :|: s30 >= 0, s30 <= 1 + 2 * @xs, s31 >= 0, s31 <= 1 + 2 * @ys, s32 >= 0, s32 <= s30 + (1 + z' + s31), z = 1 + @xs + @ys, @xs >= 0, @ys >= 0, z' >= 0
1174.14/299.11	   quicksort#2(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   sortAll(z) -{ 150 + 955*z + 1464*z^2 + 540*z^3 }-> s34 :|: s34 >= 0, s34 <= 2 * z, z >= 0
1174.14/299.11	   sortAll#1(z) -{ 148 + 368*@x + 552*@x*@xs + 180*@x^2 + 270*@x^2*@xs + 587*@xs + 732*@xs^2 + 270*@xs^3 }-> s35 :|: s35 >= 0, s35 <= 2 * @x + 2 * @xs + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   sortAll#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   sortAll#2(z, z') -{ 221 + 184*@vals + 90*@vals^2 + 955*z' + 1464*z'^2 + 540*z'^3 }-> 1 + (1 + s33 + @key) + s36 :|: s36 >= 0, s36 <= 2 * z', s33 >= 0, s33 <= 2 * @vals + 1, @vals >= 0, z = 1 + @vals + @key, z' >= 0, @key >= 0
1174.14/299.11	   split(z) -{ 2 + 5*z + 24*z^2 }-> s24 :|: s24 >= 0, s24 <= 4 * z, z >= 0
1174.14/299.11	   split#1(z) -{ 6 + 5*@xs + 24*@xs^2 + 6*s25 }-> s26 :|: s25 >= 0, s25 <= 4 * @xs, s26 >= 0, s26 <= 2 * @x + s25 + 4, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1174.14/299.11	   split#1(z) -{ 1 }-> 1 :|: z = 1
1174.14/299.11	   split#1(z) -{ 0 }-> 0 :|: z >= 0
1174.14/299.11	   splitAndSort(z) -{ 153 + 955*s27 + 1464*s27^2 + 540*s27^3 + 5*z + 24*z^2 }-> s37 :|: s37 >= 0, s37 <= 2 * s27, s27 >= 0, s27 <= 4 * z, z >= 0
1174.14/299.11	   splitqs(z, z') -{ 2 + 5*z' }-> s21 :|: s21 >= 0, s21 <= z' + 2, z' >= 0, z >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 6 + 5*@xs }-> s23 :|: s22 >= 0, s22 <= @xs + 2, s23 >= 0, s23 <= s22 + @x + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1174.14/299.11	   splitqs#1(z, z') -{ 1 }-> 1 + 1 + 1 :|: z = 1, z' >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 3 }-> s6 :|: s6 >= 0, s6 <= @ls + @rs + z'' + 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, @ls >= 0, z = 1 + @ls + @rs, z'' >= 0, z' >= 0, @rs >= 0
1174.14/299.11	   splitqs#2(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z1 + z'') :|: z = 2, z' >= 0, z1 >= 0, z'' >= 0
1174.14/299.11	   splitqs#3(z, z', z'', z1) -{ 1 }-> 1 + (1 + z1 + z') + z'' :|: z' >= 0, z1 >= 0, z = 1, z'' >= 0
1174.14/299.11	
1174.14/299.11	Function symbols to be analyzed: 
1174.14/299.11	Previous analysis results are:
1174.14/299.11	  #and: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  #compare: runtime: O(1) [0], size: O(1) [3]
1174.14/299.11	  #ckgt: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#3: runtime: O(1) [1], size: O(n^1) [2 + z' + z'' + z1]
1174.14/299.11	  append#1: runtime: O(n^1) [3 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  append: runtime: O(n^1) [4 + 2*z], size: O(n^1) [z + z']
1174.14/299.11	  #eq: runtime: O(1) [0], size: O(1) [2]
1174.14/299.11	  splitqs#2: runtime: O(1) [3], size: O(n^1) [1 + z + z'']
1174.14/299.11	  #greater: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  insert#4: runtime: O(n^1) [10 + 6*z''], size: O(n^1) [6 + z' + z'' + z1 + z2 + z3]
1174.14/299.11	  insert#2: runtime: O(n^1) [1 + 6*z], size: O(n^1) [5 + z + z' + z'' + z1]
1174.14/299.11	  insert: runtime: O(n^1) [3 + 6*z'], size: O(n^1) [4 + 2*z + z']
1174.14/299.11	  insert#3: runtime: O(n^1) [12 + 6*z''], size: O(n^1) [5 + z + z'' + z1 + z2]
1174.14/299.11	  insert#1: runtime: O(n^1) [2 + 6*z'], size: O(n^1) [4 + z + z' + z'']
1174.14/299.11	  #equal: runtime: O(1) [1], size: O(1) [2]
1174.14/299.11	  splitqs#1: runtime: O(n^1) [1 + 5*z], size: O(n^1) [2 + z]
1174.14/299.11	  split#1: runtime: O(n^2) [1 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  splitqs: runtime: O(n^1) [2 + 5*z'], size: O(n^1) [2 + z']
1174.14/299.11	  quicksort#1: runtime: O(n^2) [69 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  quicksort#2: runtime: O(n^2) [147 + 372*z + 180*z^2], size: O(n^1) [1 + 2*z + z']
1174.14/299.11	  split: runtime: O(n^2) [2 + 5*z + 24*z^2], size: O(n^1) [4*z]
1174.14/299.11	  quicksort: runtime: O(n^2) [70 + 184*z + 90*z^2], size: O(n^1) [1 + 2*z]
1174.14/299.11	  sortAll#2: runtime: O(n^3) [147 + 368*z + 552*z*z' + 180*z^2 + 270*z^2*z' + 587*z' + 732*z'^2 + 270*z'^3], size: O(n^1) [1 + 2*z + 2*z']
1174.14/299.11	  sortAll: runtime: O(n^3) [150 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  sortAll#1: runtime: O(n^3) [149 + 955*z + 1464*z^2 + 540*z^3], size: O(n^1) [2*z]
1174.14/299.11	  splitAndSort: runtime: O(n^3) [153 + 3825*z + 23448*z^2 + 34560*z^3], size: O(n^1) [8*z]
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(127) FinalProof (FINISHED)
1174.14/299.11	Computed overall runtime complexity
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(128)
1174.14/299.11	BOUNDS(1, n^3)
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(129) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1174.14/299.11	Transformed a relative TRS into a decreasing-loop problem.
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(130)
1174.14/299.11	Obligation:
1174.14/299.11	Analyzing the following TRS for decreasing loops:
1174.14/299.11	
1174.14/299.11	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	The TRS R consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #equal(@x, @y) -> #eq(@x, @y)
1174.14/299.11	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
1174.14/299.11	   append(@l, @ys) -> append#1(@l, @ys)
1174.14/299.11	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
1174.14/299.11	   append#1(nil, @ys) -> @ys
1174.14/299.11	   insert(@x, @l) -> insert#1(@x, @l, @x)
1174.14/299.11	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x)
1174.14/299.11	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x)
1174.14/299.11	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil)
1174.14/299.11	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)
1174.14/299.11	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls))
1174.14/299.11	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls)
1174.14/299.11	   quicksort(@l) -> quicksort#1(@l)
1174.14/299.11	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z)
1174.14/299.11	   quicksort#1(nil) -> nil
1174.14/299.11	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
1174.14/299.11	   sortAll(@l) -> sortAll#1(@l)
1174.14/299.11	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs)
1174.14/299.11	   sortAll#1(nil) -> nil
1174.14/299.11	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs))
1174.14/299.11	   split(@l) -> split#1(@l)
1174.14/299.11	   split#1(::(@x, @xs)) -> insert(@x, split(@xs))
1174.14/299.11	   split#1(nil) -> nil
1174.14/299.11	   splitAndSort(@l) -> sortAll(split(@l))
1174.14/299.11	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot)
1174.14/299.11	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x)
1174.14/299.11	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil)
1174.14/299.11	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)
1174.14/299.11	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
1174.14/299.11	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
1174.14/299.11	
1174.14/299.11	The (relative) TRS S consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(#false, #false) -> #false
1174.14/299.11	   #and(#false, #true) -> #false
1174.14/299.11	   #and(#true, #false) -> #false
1174.14/299.11	   #and(#true, #true) -> #true
1174.14/299.11	   #ckgt(#EQ) -> #false
1174.14/299.11	   #ckgt(#GT) -> #true
1174.14/299.11	   #ckgt(#LT) -> #false
1174.14/299.11	   #compare(#0, #0) -> #EQ
1174.14/299.11	   #compare(#0, #neg(@y)) -> #GT
1174.14/299.11	   #compare(#0, #pos(@y)) -> #LT
1174.14/299.11	   #compare(#0, #s(@y)) -> #LT
1174.14/299.11	   #compare(#neg(@x), #0) -> #LT
1174.14/299.11	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1174.14/299.11	   #compare(#neg(@x), #pos(@y)) -> #LT
1174.14/299.11	   #compare(#pos(@x), #0) -> #GT
1174.14/299.11	   #compare(#pos(@x), #neg(@y)) -> #GT
1174.14/299.11	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1174.14/299.11	   #compare(#s(@x), #0) -> #GT
1174.14/299.11	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1174.14/299.11	   #eq(#0, #0) -> #true
1174.14/299.11	   #eq(#0, #neg(@y)) -> #false
1174.14/299.11	   #eq(#0, #pos(@y)) -> #false
1174.14/299.11	   #eq(#0, #s(@y)) -> #false
1174.14/299.11	   #eq(#neg(@x), #0) -> #false
1174.14/299.11	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#neg(@x), #pos(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #0) -> #false
1174.14/299.11	   #eq(#pos(@x), #neg(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#s(@x), #0) -> #false
1174.14/299.11	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	   #eq(::(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, nil) -> #true
1174.14/299.11	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	
1174.14/299.11	Rewrite Strategy: INNERMOST
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(131) DecreasingLoopProof (LOWER BOUND(ID))
1174.14/299.11	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1174.14/299.11	
1174.14/299.11	The rewrite sequence
1174.14/299.11	
1174.14/299.11	splitqs(@pivot, ::(@x1_0, @xs2_0)) ->^+ splitqs#2(splitqs(@pivot, @xs2_0), @pivot, @x1_0)
1174.14/299.11	
1174.14/299.11	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1174.14/299.11	
1174.14/299.11	The pumping substitution is [@xs2_0 / ::(@x1_0, @xs2_0)].
1174.14/299.11	
1174.14/299.11	The result substitution is [ ].
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(132)
1174.14/299.11	Complex Obligation (BEST)
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(133)
1174.14/299.11	Obligation:
1174.14/299.11	Proved the lower bound n^1 for the following obligation:
1174.14/299.11	
1174.14/299.11	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	The TRS R consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #equal(@x, @y) -> #eq(@x, @y)
1174.14/299.11	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
1174.14/299.11	   append(@l, @ys) -> append#1(@l, @ys)
1174.14/299.11	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
1174.14/299.11	   append#1(nil, @ys) -> @ys
1174.14/299.11	   insert(@x, @l) -> insert#1(@x, @l, @x)
1174.14/299.11	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x)
1174.14/299.11	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x)
1174.14/299.11	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil)
1174.14/299.11	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)
1174.14/299.11	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls))
1174.14/299.11	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls)
1174.14/299.11	   quicksort(@l) -> quicksort#1(@l)
1174.14/299.11	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z)
1174.14/299.11	   quicksort#1(nil) -> nil
1174.14/299.11	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
1174.14/299.11	   sortAll(@l) -> sortAll#1(@l)
1174.14/299.11	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs)
1174.14/299.11	   sortAll#1(nil) -> nil
1174.14/299.11	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs))
1174.14/299.11	   split(@l) -> split#1(@l)
1174.14/299.11	   split#1(::(@x, @xs)) -> insert(@x, split(@xs))
1174.14/299.11	   split#1(nil) -> nil
1174.14/299.11	   splitAndSort(@l) -> sortAll(split(@l))
1174.14/299.11	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot)
1174.14/299.11	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x)
1174.14/299.11	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil)
1174.14/299.11	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)
1174.14/299.11	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
1174.14/299.11	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
1174.14/299.11	
1174.14/299.11	The (relative) TRS S consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(#false, #false) -> #false
1174.14/299.11	   #and(#false, #true) -> #false
1174.14/299.11	   #and(#true, #false) -> #false
1174.14/299.11	   #and(#true, #true) -> #true
1174.14/299.11	   #ckgt(#EQ) -> #false
1174.14/299.11	   #ckgt(#GT) -> #true
1174.14/299.11	   #ckgt(#LT) -> #false
1174.14/299.11	   #compare(#0, #0) -> #EQ
1174.14/299.11	   #compare(#0, #neg(@y)) -> #GT
1174.14/299.11	   #compare(#0, #pos(@y)) -> #LT
1174.14/299.11	   #compare(#0, #s(@y)) -> #LT
1174.14/299.11	   #compare(#neg(@x), #0) -> #LT
1174.14/299.11	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1174.14/299.11	   #compare(#neg(@x), #pos(@y)) -> #LT
1174.14/299.11	   #compare(#pos(@x), #0) -> #GT
1174.14/299.11	   #compare(#pos(@x), #neg(@y)) -> #GT
1174.14/299.11	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1174.14/299.11	   #compare(#s(@x), #0) -> #GT
1174.14/299.11	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1174.14/299.11	   #eq(#0, #0) -> #true
1174.14/299.11	   #eq(#0, #neg(@y)) -> #false
1174.14/299.11	   #eq(#0, #pos(@y)) -> #false
1174.14/299.11	   #eq(#0, #s(@y)) -> #false
1174.14/299.11	   #eq(#neg(@x), #0) -> #false
1174.14/299.11	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#neg(@x), #pos(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #0) -> #false
1174.14/299.11	   #eq(#pos(@x), #neg(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#s(@x), #0) -> #false
1174.14/299.11	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	   #eq(::(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, nil) -> #true
1174.14/299.11	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	
1174.14/299.11	Rewrite Strategy: INNERMOST
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(134) LowerBoundPropagationProof (FINISHED)
1174.14/299.11	Propagated lower bound.
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(135)
1174.14/299.11	BOUNDS(n^1, INF)
1174.14/299.11	
1174.14/299.11	----------------------------------------
1174.14/299.11	
1174.14/299.11	(136)
1174.14/299.11	Obligation:
1174.14/299.11	Analyzing the following TRS for decreasing loops:
1174.14/299.11	
1174.14/299.11	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^3).
1174.14/299.11	
1174.14/299.11	
1174.14/299.11	The TRS R consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #equal(@x, @y) -> #eq(@x, @y)
1174.14/299.11	   #greater(@x, @y) -> #ckgt(#compare(@x, @y))
1174.14/299.11	   append(@l, @ys) -> append#1(@l, @ys)
1174.14/299.11	   append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys))
1174.14/299.11	   append#1(nil, @ys) -> @ys
1174.14/299.11	   insert(@x, @l) -> insert#1(@x, @l, @x)
1174.14/299.11	   insert#1(tuple#2(@valX, @keyX), @l, @x) -> insert#2(@l, @keyX, @valX, @x)
1174.14/299.11	   insert#2(::(@l1, @ls), @keyX, @valX, @x) -> insert#3(@l1, @keyX, @ls, @valX, @x)
1174.14/299.11	   insert#2(nil, @keyX, @valX, @x) -> ::(tuple#2(::(@valX, nil), @keyX), nil)
1174.14/299.11	   insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)
1174.14/299.11	   insert#4(#false, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(@vals1, @key1), insert(@x, @ls))
1174.14/299.11	   insert#4(#true, @key1, @ls, @valX, @vals1, @x) -> ::(tuple#2(::(@valX, @vals1), @key1), @ls)
1174.14/299.11	   quicksort(@l) -> quicksort#1(@l)
1174.14/299.11	   quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z)
1174.14/299.11	   quicksort#1(nil) -> nil
1174.14/299.11	   quicksort#2(tuple#2(@xs, @ys), @z) -> append(quicksort(@xs), ::(@z, quicksort(@ys)))
1174.14/299.11	   sortAll(@l) -> sortAll#1(@l)
1174.14/299.11	   sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs)
1174.14/299.11	   sortAll#1(nil) -> nil
1174.14/299.11	   sortAll#2(tuple#2(@vals, @key), @xs) -> ::(tuple#2(quicksort(@vals), @key), sortAll(@xs))
1174.14/299.11	   split(@l) -> split#1(@l)
1174.14/299.11	   split#1(::(@x, @xs)) -> insert(@x, split(@xs))
1174.14/299.11	   split#1(nil) -> nil
1174.14/299.11	   splitAndSort(@l) -> sortAll(split(@l))
1174.14/299.11	   splitqs(@pivot, @l) -> splitqs#1(@l, @pivot)
1174.14/299.11	   splitqs#1(::(@x, @xs), @pivot) -> splitqs#2(splitqs(@pivot, @xs), @pivot, @x)
1174.14/299.11	   splitqs#1(nil, @pivot) -> tuple#2(nil, nil)
1174.14/299.11	   splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)
1174.14/299.11	   splitqs#3(#false, @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs)
1174.14/299.11	   splitqs#3(#true, @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs))
1174.14/299.11	
1174.14/299.11	The (relative) TRS S consists of the following rules:
1174.14/299.11	
1174.14/299.11	   #and(#false, #false) -> #false
1174.14/299.11	   #and(#false, #true) -> #false
1174.14/299.11	   #and(#true, #false) -> #false
1174.14/299.11	   #and(#true, #true) -> #true
1174.14/299.11	   #ckgt(#EQ) -> #false
1174.14/299.11	   #ckgt(#GT) -> #true
1174.14/299.11	   #ckgt(#LT) -> #false
1174.14/299.11	   #compare(#0, #0) -> #EQ
1174.14/299.11	   #compare(#0, #neg(@y)) -> #GT
1174.14/299.11	   #compare(#0, #pos(@y)) -> #LT
1174.14/299.11	   #compare(#0, #s(@y)) -> #LT
1174.14/299.11	   #compare(#neg(@x), #0) -> #LT
1174.14/299.11	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1174.14/299.11	   #compare(#neg(@x), #pos(@y)) -> #LT
1174.14/299.11	   #compare(#pos(@x), #0) -> #GT
1174.14/299.11	   #compare(#pos(@x), #neg(@y)) -> #GT
1174.14/299.11	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1174.14/299.11	   #compare(#s(@x), #0) -> #GT
1174.14/299.11	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1174.14/299.11	   #eq(#0, #0) -> #true
1174.14/299.11	   #eq(#0, #neg(@y)) -> #false
1174.14/299.11	   #eq(#0, #pos(@y)) -> #false
1174.14/299.11	   #eq(#0, #s(@y)) -> #false
1174.14/299.11	   #eq(#neg(@x), #0) -> #false
1174.14/299.11	   #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#neg(@x), #pos(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #0) -> #false
1174.14/299.11	   #eq(#pos(@x), #neg(@y)) -> #false
1174.14/299.11	   #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(#s(@x), #0) -> #false
1174.14/299.11	   #eq(#s(@x), #s(@y)) -> #eq(@x, @y)
1174.14/299.11	   #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	   #eq(::(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(nil, nil) -> #true
1174.14/299.11	   #eq(nil, tuple#2(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), nil) -> #false
1174.14/299.11	   #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
1174.14/299.11	
1174.14/299.11	Rewrite Strategy: INNERMOST
1174.34/299.21	EOF