1114.97/291.50	WORST_CASE(Omega(n^1), O(n^2))
1115.21/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1115.21/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
1115.21/291.54	
1115.21/291.54	(0) CpxRelTRS
1115.21/291.54	(1) STerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 163 ms]
1115.21/291.54	(2) CpxRelTRS
1115.21/291.54	(3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	(4) CpxWeightedTrs
1115.21/291.54	    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	    (6) CpxTypedWeightedTrs
1115.21/291.54	    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (8) CpxTypedWeightedCompleteTrs
1115.21/291.54	    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	    (10) CpxTypedWeightedCompleteTrs
1115.21/291.54	    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 13 ms]
1115.21/291.54	    (12) CpxRNTS
1115.21/291.54	    (13) InliningProof [UPPER BOUND(ID), 731 ms]
1115.21/291.54	    (14) CpxRNTS
1115.21/291.54	    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	    (16) CpxRNTS
1115.21/291.54	    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms]
1115.21/291.54	    (18) CpxRNTS
1115.21/291.54	    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (20) CpxRNTS
1115.21/291.54	    (21) IntTrsBoundProof [UPPER BOUND(ID), 1111 ms]
1115.21/291.54	    (22) CpxRNTS
1115.21/291.54	    (23) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
1115.21/291.54	    (24) CpxRNTS
1115.21/291.54	    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (26) CpxRNTS
1115.21/291.54	    (27) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
1115.21/291.54	    (28) CpxRNTS
1115.21/291.54	    (29) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
1115.21/291.54	    (30) CpxRNTS
1115.21/291.54	    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (32) CpxRNTS
1115.21/291.54	    (33) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
1115.21/291.54	    (34) CpxRNTS
1115.21/291.54	    (35) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
1115.21/291.54	    (36) CpxRNTS
1115.21/291.54	    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (38) CpxRNTS
1115.21/291.54	    (39) IntTrsBoundProof [UPPER BOUND(ID), 704 ms]
1115.21/291.54	    (40) CpxRNTS
1115.21/291.54	    (41) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
1115.21/291.54	    (42) CpxRNTS
1115.21/291.54	    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (44) CpxRNTS
1115.21/291.54	    (45) IntTrsBoundProof [UPPER BOUND(ID), 1540 ms]
1115.21/291.54	    (46) CpxRNTS
1115.21/291.54	    (47) IntTrsBoundProof [UPPER BOUND(ID), 436 ms]
1115.21/291.54	    (48) CpxRNTS
1115.21/291.54	    (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (50) CpxRNTS
1115.21/291.54	    (51) IntTrsBoundProof [UPPER BOUND(ID), 1720 ms]
1115.21/291.54	    (52) CpxRNTS
1115.21/291.54	    (53) IntTrsBoundProof [UPPER BOUND(ID), 596 ms]
1115.21/291.54	    (54) CpxRNTS
1115.21/291.54	    (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (56) CpxRNTS
1115.21/291.54	    (57) IntTrsBoundProof [UPPER BOUND(ID), 401 ms]
1115.21/291.54	    (58) CpxRNTS
1115.21/291.54	    (59) IntTrsBoundProof [UPPER BOUND(ID), 34 ms]
1115.21/291.54	    (60) CpxRNTS
1115.21/291.54	    (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (62) CpxRNTS
1115.21/291.54	    (63) IntTrsBoundProof [UPPER BOUND(ID), 218 ms]
1115.21/291.54	    (64) CpxRNTS
1115.21/291.54	    (65) IntTrsBoundProof [UPPER BOUND(ID), 45 ms]
1115.21/291.54	    (66) CpxRNTS
1115.21/291.54	    (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (68) CpxRNTS
1115.21/291.54	    (69) IntTrsBoundProof [UPPER BOUND(ID), 257 ms]
1115.21/291.54	    (70) CpxRNTS
1115.21/291.54	    (71) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
1115.21/291.54	    (72) CpxRNTS
1115.21/291.54	    (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (74) CpxRNTS
1115.21/291.54	    (75) IntTrsBoundProof [UPPER BOUND(ID), 177 ms]
1115.21/291.54	    (76) CpxRNTS
1115.21/291.54	    (77) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
1115.21/291.54	    (78) CpxRNTS
1115.21/291.54	    (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (80) CpxRNTS
1115.21/291.54	    (81) IntTrsBoundProof [UPPER BOUND(ID), 205 ms]
1115.21/291.54	    (82) CpxRNTS
1115.21/291.54	    (83) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (84) CpxRNTS
1115.21/291.54	    (85) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (86) CpxRNTS
1115.21/291.54	    (87) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
1115.21/291.54	    (88) CpxRNTS
1115.21/291.54	    (89) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
1115.21/291.54	    (90) CpxRNTS
1115.21/291.54	    (91) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
1115.21/291.54	    (92) CpxRNTS
1115.21/291.54	    (93) IntTrsBoundProof [UPPER BOUND(ID), 279 ms]
1115.21/291.54	    (94) CpxRNTS
1115.21/291.54	    (95) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
1115.21/291.54	    (96) CpxRNTS
1115.21/291.54	    (97) FinalProof [FINISHED, 0 ms]
1115.21/291.54	    (98) BOUNDS(1, n^2)
1115.21/291.54	(99) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	(100) CpxRelTRS
1115.21/291.54	    (101) SlicingProof [LOWER BOUND(ID), 0 ms]
1115.21/291.54	    (102) CpxRelTRS
1115.21/291.54	    (103) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
1115.21/291.54	    (104) typed CpxTrs
1115.21/291.54	    (105) OrderProof [LOWER BOUND(ID), 0 ms]
1115.21/291.54	    (106) typed CpxTrs
1115.21/291.54	    (107) RewriteLemmaProof [LOWER BOUND(ID), 1987 ms]
1115.21/291.54	    (108) typed CpxTrs
1115.21/291.54	    (109) RewriteLemmaProof [LOWER BOUND(ID), 428 ms]
1115.21/291.54	    (110) BEST
1115.21/291.54	        (111) proven lower bound
1115.21/291.54	            (112) LowerBoundPropagationProof [FINISHED, 0 ms]
1115.21/291.54	            (113) BOUNDS(n^1, INF)
1115.21/291.54	        (114) typed CpxTrs
1115.21/291.54	            (115) RewriteLemmaProof [LOWER BOUND(ID), 494 ms]
1115.21/291.54	            (116) typed CpxTrs
1115.21/291.54	            (117) RewriteLemmaProof [LOWER BOUND(ID), 565 ms]
1115.21/291.54	            (118) typed CpxTrs
1115.21/291.54	            (119) RewriteLemmaProof [LOWER BOUND(ID), 595 ms]
1115.21/291.54	            (120) typed CpxTrs
1115.21/291.54	            (121) RewriteLemmaProof [LOWER BOUND(ID), 657 ms]
1115.21/291.54	            (122) typed CpxTrs
1115.21/291.54	            (123) RewriteLemmaProof [LOWER BOUND(ID), 621 ms]
1115.21/291.54	            (124) BOUNDS(1, INF)
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(0)
1115.21/291.54	Obligation:
1115.21/291.54	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x)
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x)
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x))
1115.21/291.54	   #less(@x, @y) -> #cklt(#compare(@x, @y))
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x)
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil)
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x)
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil)
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l)
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.21/291.54	   insertionsort#1(nil) -> nil
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l)
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.21/291.54	   insertionsortD#1(nil) -> nil
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(testList(#unit))
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(testList(#unit))
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.21/291.54	
1115.21/291.54	The (relative) TRS S consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #cklt(#EQ) -> #false
1115.21/291.54	   #cklt(#GT) -> #false
1115.21/291.54	   #cklt(#LT) -> #true
1115.21/291.54	   #compare(#0, #0) -> #EQ
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.21/291.54	   #compare(#s(@x), #0) -> #GT
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(1) STerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
1115.21/291.54	proved termination of relative rules
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(2)
1115.21/291.54	Obligation:
1115.21/291.54	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, n^2).
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x)
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x)
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x))
1115.21/291.54	   #less(@x, @y) -> #cklt(#compare(@x, @y))
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x)
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil)
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x)
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil)
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l)
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.21/291.54	   insertionsort#1(nil) -> nil
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l)
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.21/291.54	   insertionsortD#1(nil) -> nil
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(testList(#unit))
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(testList(#unit))
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.21/291.54	
1115.21/291.54	The (relative) TRS S consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #cklt(#EQ) -> #false
1115.21/291.54	   #cklt(#GT) -> #false
1115.21/291.54	   #cklt(#LT) -> #true
1115.21/291.54	   #compare(#0, #0) -> #EQ
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.21/291.54	   #compare(#s(@x), #0) -> #GT
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(3) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
1115.21/291.54	Transformed relative TRS to weighted TRS
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(4)
1115.21/291.54	Obligation:
1115.21/291.54	The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0 [1]
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
1115.21/291.54	   #less(@x, @y) -> #cklt(#compare(@x, @y)) [1]
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x) [1]
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1]
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x) [1]
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1]
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l) [1]
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1]
1115.21/291.54	   insertionsort#1(nil) -> nil [1]
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l) [1]
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1]
1115.21/291.54	   insertionsortD#1(nil) -> nil [1]
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(testList(#unit)) [1]
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1]
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
1115.21/291.54	   #cklt(#EQ) -> #false [0]
1115.21/291.54	   #cklt(#GT) -> #false [0]
1115.21/291.54	   #cklt(#LT) -> #true [0]
1115.21/291.54	   #compare(#0, #0) -> #EQ [0]
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #compare(#s(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(5) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1115.21/291.54	Infered types.
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(6)
1115.21/291.54	Obligation:
1115.21/291.54	Runtime Complexity Weighted TRS with Types.
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0 [1]
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
1115.21/291.54	   #less(@x, @y) -> #cklt(#compare(@x, @y)) [1]
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x) [1]
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1]
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x) [1]
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1]
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l) [1]
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1]
1115.21/291.54	   insertionsort#1(nil) -> nil [1]
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l) [1]
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1]
1115.21/291.54	   insertionsortD#1(nil) -> nil [1]
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(testList(#unit)) [1]
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1]
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
1115.21/291.54	   #cklt(#EQ) -> #false [0]
1115.21/291.54	   #cklt(#GT) -> #false [0]
1115.21/291.54	   #cklt(#LT) -> #true [0]
1115.21/291.54	   #compare(#0, #0) -> #EQ [0]
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #compare(#s(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	
1115.21/291.54	The TRS has the following type information:
1115.21/291.54	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#0 :: #0:#neg:#pos:#s
1115.21/291.54	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.21/291.54	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.21/291.54	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.21/291.54	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	nil :: :::nil
1115.21/291.54	#false :: #false:#true
1115.21/291.54	#true :: #false:#true
1115.21/291.54	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertionsort :: :::nil -> :::nil
1115.21/291.54	insertionsort#1 :: :::nil -> :::nil
1115.21/291.54	insertionsortD :: :::nil -> :::nil
1115.21/291.54	insertionsortD#1 :: :::nil -> :::nil
1115.21/291.54	testInsertionsort :: a -> :::nil
1115.21/291.54	testList :: #unit -> :::nil
1115.21/291.54	#unit :: #unit
1115.21/291.54	testInsertionsortD :: b -> :::nil
1115.21/291.54	#EQ :: #EQ:#GT:#LT
1115.21/291.54	#GT :: #EQ:#GT:#LT
1115.21/291.54	#LT :: #EQ:#GT:#LT
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(7) CompletionProof (UPPER BOUND(ID))
1115.21/291.54	The transformation into a RNTS is sound, since:
1115.21/291.54	
1115.21/291.54	(a) The obligation is a constructor system where every type has a constant constructor,
1115.21/291.54	
1115.21/291.54	(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
1115.21/291.54	
1115.21/291.54	   testInsertionsort_1
1115.21/291.54	   testInsertionsortD_1
1115.21/291.54	
1115.21/291.54	(c) The following functions are completely defined:
1115.21/291.54	
1115.21/291.54	   insertionsortD_1
1115.21/291.54	   insertionsort_1
1115.21/291.54	   #less_2
1115.21/291.54	   testList_1
1115.21/291.54	   #abs_1
1115.21/291.54	   insertionsortD#1_1
1115.21/291.54	   insertionsort#1_1
1115.21/291.54	   insertD_2
1115.21/291.54	   insert_2
1115.21/291.54	   insertD#1_2
1115.21/291.54	   insertD#2_4
1115.21/291.54	   insert#1_2
1115.21/291.54	   insert#2_4
1115.21/291.54	   #cklt_1
1115.21/291.54	   #compare_2
1115.21/291.54	
1115.21/291.54	Due to the following rules being added:
1115.21/291.54	
1115.21/291.54	   #cklt(v0) -> null_#cklt [0]
1115.21/291.54	   #compare(v0, v1) -> null_#compare [0]
1115.21/291.54	   insertD#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	   insert#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	
1115.21/291.54	And the following fresh constants:    null_#cklt, null_#compare, const, const1
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(8)
1115.21/291.54	Obligation:
1115.21/291.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1115.21/291.54	
1115.21/291.54	Runtime Complexity Weighted TRS with Types.
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0 [1]
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
1115.21/291.54	   #less(@x, @y) -> #cklt(#compare(@x, @y)) [1]
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x) [1]
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1]
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x) [1]
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys) [1]
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1]
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l) [1]
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs)) [1]
1115.21/291.54	   insertionsort#1(nil) -> nil [1]
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l) [1]
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs)) [1]
1115.21/291.54	   insertionsortD#1(nil) -> nil [1]
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(testList(#unit)) [1]
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(testList(#unit)) [1]
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
1115.21/291.54	   #cklt(#EQ) -> #false [0]
1115.21/291.54	   #cklt(#GT) -> #false [0]
1115.21/291.54	   #cklt(#LT) -> #true [0]
1115.21/291.54	   #compare(#0, #0) -> #EQ [0]
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #compare(#s(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #cklt(v0) -> null_#cklt [0]
1115.21/291.54	   #compare(v0, v1) -> null_#compare [0]
1115.21/291.54	   insertD#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	   insert#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	
1115.21/291.54	The TRS has the following type information:
1115.21/291.54	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#0 :: #0:#neg:#pos:#s
1115.21/291.54	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt
1115.21/291.54	#cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt
1115.21/291.54	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare
1115.21/291.54	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	nil :: :::nil
1115.21/291.54	#false :: #false:#true:null_#cklt
1115.21/291.54	#true :: #false:#true:null_#cklt
1115.21/291.54	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	insertD#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertionsort :: :::nil -> :::nil
1115.21/291.54	insertionsort#1 :: :::nil -> :::nil
1115.21/291.54	insertionsortD :: :::nil -> :::nil
1115.21/291.54	insertionsortD#1 :: :::nil -> :::nil
1115.21/291.54	testInsertionsort :: a -> :::nil
1115.21/291.54	testList :: #unit -> :::nil
1115.21/291.54	#unit :: #unit
1115.21/291.54	testInsertionsortD :: b -> :::nil
1115.21/291.54	#EQ :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	#GT :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	#LT :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	null_#cklt :: #false:#true:null_#cklt
1115.21/291.54	null_#compare :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	const :: a
1115.21/291.54	const1 :: b
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(9) NarrowingProof (BOTH BOUNDS(ID, ID))
1115.21/291.54	Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(10)
1115.21/291.54	Obligation:
1115.21/291.54	Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:
1115.21/291.54	
1115.21/291.54	Runtime Complexity Weighted TRS with Types.
1115.21/291.54	The TRS R consists of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(#0) -> #0 [1]
1115.21/291.54	   #abs(#neg(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#pos(@x)) -> #pos(@x) [1]
1115.21/291.54	   #abs(#s(@x)) -> #pos(#s(@x)) [1]
1115.21/291.54	   #less(#0, #0) -> #cklt(#EQ) [1]
1115.21/291.54	   #less(#0, #neg(@y')) -> #cklt(#GT) [1]
1115.21/291.54	   #less(#0, #pos(@y'')) -> #cklt(#LT) [1]
1115.21/291.54	   #less(#0, #s(@y1)) -> #cklt(#LT) [1]
1115.21/291.54	   #less(#neg(@x'), #0) -> #cklt(#LT) [1]
1115.21/291.54	   #less(#neg(@x''), #neg(@y2)) -> #cklt(#compare(@y2, @x'')) [1]
1115.21/291.54	   #less(#neg(@x1), #pos(@y3)) -> #cklt(#LT) [1]
1115.21/291.54	   #less(#pos(@x2), #0) -> #cklt(#GT) [1]
1115.21/291.54	   #less(#pos(@x3), #neg(@y4)) -> #cklt(#GT) [1]
1115.21/291.54	   #less(#pos(@x4), #pos(@y5)) -> #cklt(#compare(@x4, @y5)) [1]
1115.21/291.54	   #less(#s(@x5), #0) -> #cklt(#GT) [1]
1115.21/291.54	   #less(#s(@x6), #s(@y6)) -> #cklt(#compare(@x6, @y6)) [1]
1115.21/291.54	   #less(@x, @y) -> #cklt(null_#compare) [1]
1115.21/291.54	   insert(@x, @l) -> insert#1(@l, @x) [1]
1115.21/291.54	   insert#1(::(@y, @ys), @x) -> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) [2]
1115.21/291.54	   insert#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys)) [1]
1115.21/291.54	   insertD(@x, @l) -> insertD#1(@l, @x) [1]
1115.21/291.54	   insertD#1(::(@y, @ys), @x) -> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) [2]
1115.21/291.54	   insertD#1(nil, @x) -> ::(@x, nil) [1]
1115.21/291.54	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys)) [1]
1115.21/291.54	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys)) [1]
1115.21/291.54	   insertionsort(@l) -> insertionsort#1(@l) [1]
1115.21/291.54	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort#1(@xs)) [2]
1115.21/291.54	   insertionsort#1(nil) -> nil [1]
1115.21/291.54	   insertionsortD(@l) -> insertionsortD#1(@l) [1]
1115.21/291.54	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD#1(@xs)) [2]
1115.21/291.54	   insertionsortD#1(nil) -> nil [1]
1115.21/291.54	   testInsertionsort(@x) -> insertionsort(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2]
1115.21/291.54	   testInsertionsortD(@x) -> insertionsortD(::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))) [2]
1115.21/291.54	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil)))))))))) [1]
1115.21/291.54	   #cklt(#EQ) -> #false [0]
1115.21/291.54	   #cklt(#GT) -> #false [0]
1115.21/291.54	   #cklt(#LT) -> #true [0]
1115.21/291.54	   #compare(#0, #0) -> #EQ [0]
1115.21/291.54	   #compare(#0, #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#0, #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#0, #s(@y)) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #0) -> #LT [0]
1115.21/291.54	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) [0]
1115.21/291.54	   #compare(#neg(@x), #pos(@y)) -> #LT [0]
1115.21/291.54	   #compare(#pos(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #neg(@y)) -> #GT [0]
1115.21/291.54	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #compare(#s(@x), #0) -> #GT [0]
1115.21/291.54	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y) [0]
1115.21/291.54	   #cklt(v0) -> null_#cklt [0]
1115.21/291.54	   #compare(v0, v1) -> null_#compare [0]
1115.21/291.54	   insertD#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	   insert#2(v0, v1, v2, v3) -> nil [0]
1115.21/291.54	
1115.21/291.54	The TRS has the following type information:
1115.21/291.54	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#0 :: #0:#neg:#pos:#s
1115.21/291.54	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.21/291.54	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true:null_#cklt
1115.21/291.54	#cklt :: #EQ:#GT:#LT:null_#compare -> #false:#true:null_#cklt
1115.21/291.54	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT:null_#compare
1115.21/291.54	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insert#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	nil :: :::nil
1115.21/291.54	#false :: #false:#true:null_#cklt
1115.21/291.54	#true :: #false:#true:null_#cklt
1115.21/291.54	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.21/291.54	insertD#2 :: #false:#true:null_#cklt -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.21/291.54	insertionsort :: :::nil -> :::nil
1115.21/291.54	insertionsort#1 :: :::nil -> :::nil
1115.21/291.54	insertionsortD :: :::nil -> :::nil
1115.21/291.54	insertionsortD#1 :: :::nil -> :::nil
1115.21/291.54	testInsertionsort :: a -> :::nil
1115.21/291.54	testList :: #unit -> :::nil
1115.21/291.54	#unit :: #unit
1115.21/291.54	testInsertionsortD :: b -> :::nil
1115.21/291.54	#EQ :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	#GT :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	#LT :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	null_#cklt :: #false:#true:null_#cklt
1115.21/291.54	null_#compare :: #EQ:#GT:#LT:null_#compare
1115.21/291.54	const :: a
1115.21/291.54	const1 :: b
1115.21/291.54	
1115.21/291.54	Rewrite Strategy: INNERMOST
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(11) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
1115.21/291.54	Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
1115.21/291.54	The constant constructors are abstracted as follows:
1115.21/291.54	
1115.21/291.54	   #0 => 0
1115.21/291.54	   nil => 0
1115.21/291.54	   #false => 1
1115.21/291.54	   #true => 2
1115.21/291.54	   #unit => 0
1115.21/291.54	   #EQ => 1
1115.21/291.54	   #GT => 2
1115.21/291.54	   #LT => 3
1115.21/291.54	   null_#cklt => 0
1115.21/291.54	   null_#compare => 0
1115.21/291.54	   const => 0
1115.21/291.54	   const1 => 0
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(12)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(3) :|: z = 1 + @x', @x' >= 0, z' = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(2) :|: @y' >= 0, z' = 1 + @y', z = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(1) :|: z = 0, z' = 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insertD(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(@l) :|: z = @l, @l >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: @_ >= 0, z = @_
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(13) InliningProof (UPPER BOUND(ID))
1115.21/291.54	Inlined the following terminating rules on right-hand sides where appropriate:
1115.21/291.54	
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(14)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + @x :|: @x >= 0, z = 1 + @x
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + @x) :|: @x >= 0, z = 1 + @x
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insert(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, @x)), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + @x + 0 :|: @x >= 0, z = 0, z' = @x
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @x + (1 + @y + @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + @y + insertD(@x, @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(@l) :|: z = @l, @l >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(@l) :|: z = @l, @l >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z = @x, @x >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: @_ >= 0, z = @_
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(15) SimplificationProof (BOTH BOUNDS(ID, ID))
1115.21/291.54	Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(16)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.54	
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(17) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
1115.21/291.54	Found the following analysis order by SCC decomposition:
1115.21/291.54	
1115.21/291.54	   { #compare }
1115.21/291.54	   { #cklt }
1115.21/291.54	   { #abs }
1115.21/291.54	   { #less }
1115.21/291.54	   { insertD#2, insertD#1, insertD }
1115.21/291.54	   { insert#2, insert, insert#1 }
1115.21/291.54	   { testList }
1115.21/291.54	   { insertionsortD#1 }
1115.21/291.54	   { insertionsort#1 }
1115.21/291.54	   { insertionsortD }
1115.21/291.54	   { insertionsort }
1115.21/291.54	   { testInsertionsortD }
1115.21/291.54	   { testInsertionsort }
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(18)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.54	
1115.21/291.54	Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(19) ResultPropagationProof (UPPER BOUND(ID))
1115.21/291.54	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(20)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.54	
1115.21/291.54	Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(21) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.54	
1115.21/291.54	Computed SIZE bound using CoFloCo for: #compare
1115.21/291.54	after applying outer abstraction to obtain an ITS,
1115.21/291.54	resulting in: O(1) with polynomial bound: 3
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(22)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.54	
1115.21/291.54	Function symbols to be analyzed: {#compare}, {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.54	Previous analysis results are:
1115.21/291.54	  #compare: runtime: ?, size: O(1) [3]
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(23) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.54	
1115.21/291.54	Computed RUNTIME bound using CoFloCo for: #compare
1115.21/291.54	after applying outer abstraction to obtain an ITS,
1115.21/291.54	resulting in: O(1) with polynomial bound: 0
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(24)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(#compare(@y, z')), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.54	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.54	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.54	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.54	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.54	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.54	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.54	
1115.21/291.54	Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.54	Previous analysis results are:
1115.21/291.54	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.54	
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(25) ResultPropagationProof (UPPER BOUND(ID))
1115.21/291.54	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.21/291.54	----------------------------------------
1115.21/291.54	
1115.21/291.54	(26)
1115.21/291.54	Obligation:
1115.21/291.54	Complexity RNTS consisting of the following rules:
1115.21/291.54	
1115.21/291.54	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.54	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.54	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.54	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.54	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.54	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.54	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.54	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.55	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.55	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.55	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.55	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.55	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.55	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.55	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.55	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.55	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.55	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.55	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.55	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.55	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.55	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.55	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.55	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.55	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.55	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.55	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.55	
1115.21/291.55	Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.55	Previous analysis results are:
1115.21/291.55	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.55	
1115.21/291.55	----------------------------------------
1115.21/291.55	
1115.21/291.55	(27) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.55	
1115.21/291.55	Computed SIZE bound using CoFloCo for: #cklt
1115.21/291.55	after applying outer abstraction to obtain an ITS,
1115.21/291.55	resulting in: O(1) with polynomial bound: 2
1115.21/291.55	
1115.21/291.55	----------------------------------------
1115.21/291.55	
1115.21/291.55	(28)
1115.21/291.55	Obligation:
1115.21/291.55	Complexity RNTS consisting of the following rules:
1115.21/291.55	
1115.21/291.55	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.55	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.55	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.55	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.55	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.55	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.55	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.55	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.55	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.55	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.55	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.55	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.55	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.55	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.55	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.55	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.55	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.55	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.55	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.55	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#cklt}, {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: ?, size: O(1) [2]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(29) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.57	
1115.21/291.57	Computed RUNTIME bound using CoFloCo for: #cklt
1115.21/291.57	after applying outer abstraction to obtain an ITS,
1115.21/291.57	resulting in: O(1) with polynomial bound: 0
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(30)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(#cklt(s2), z', @y, @ys) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(#cklt(s3), z', @y, @ys) :|: s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(31) ResultPropagationProof (UPPER BOUND(ID))
1115.21/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(32)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(33) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.57	
1115.21/291.57	Computed SIZE bound using CoFloCo for: #abs
1115.21/291.57	after applying outer abstraction to obtain an ITS,
1115.21/291.57	resulting in: O(n^1) with polynomial bound: 1 + z
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(34)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#abs}, {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.21/291.57	  #abs: runtime: ?, size: O(n^1) [1 + z]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(35) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.57	
1115.21/291.57	Computed RUNTIME bound using CoFloCo for: #abs
1115.21/291.57	after applying outer abstraction to obtain an ITS,
1115.21/291.57	resulting in: O(1) with polynomial bound: 1
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(36)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 2 }-> insertionsort(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 2 }-> insertionsortD(1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0)))))))))) :|: z >= 0
1115.21/291.57	   testList(z) -{ 1 }-> 1 + #abs(0) + (1 + #abs(1 + (1 + (1 + (1 + (1 + 0))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + (1 + #abs(1 + (1 + 0)) + (1 + #abs(1 + (1 + (1 + 0))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + (1 + #abs(1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + (1 + #abs(1 + (1 + (1 + (1 + 0)))) + 0))))))))) :|: z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.21/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(37) ResultPropagationProof (UPPER BOUND(ID))
1115.21/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(38)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.21/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.21/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.21/291.57	
1115.21/291.57	Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.21/291.57	Previous analysis results are:
1115.21/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.21/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.21/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(39) IntTrsBoundProof (UPPER BOUND(ID))
1115.21/291.57	
1115.21/291.57	Computed SIZE bound using CoFloCo for: #less
1115.21/291.57	after applying outer abstraction to obtain an ITS,
1115.21/291.57	resulting in: O(1) with polynomial bound: 2
1115.21/291.57	
1115.21/291.57	----------------------------------------
1115.21/291.57	
1115.21/291.57	(40)
1115.21/291.57	Obligation:
1115.21/291.57	Complexity RNTS consisting of the following rules:
1115.21/291.57	
1115.21/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.21/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.21/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.21/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.21/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.21/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.21/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.21/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.21/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.21/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.21/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.21/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.21/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.21/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.21/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {#less}, {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: ?, size: O(1) [2]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(41) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: #less
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(1) with polynomial bound: 1
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(42)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(43) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(44)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(45) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insertD#2
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insertD#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + z + z'
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insertD
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + z + z'
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(46)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertD#2,insertD#1,insertD}, {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: ?, size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: ?, size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(47) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insertD#2
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 3 + 4*z1
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insertD#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + 4*z
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insertD
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 2 + 4*z'
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(48)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 1 }-> insertD#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 2 }-> insertD#2(s7, z', @y, @ys) :|: s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insertD(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(49) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(50)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(51) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insert#2
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 2 + z' + z'' + z1
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insert
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + z + z'
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insert#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + z + z'
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(52)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insert#2,insert,insert#1}, {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: ?, size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: ?, size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: ?, size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(53) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insert#2
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 3 + 4*z1
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insert
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 2 + 4*z'
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insert#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: 1 + 4*z
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(54)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 1 }-> insert#1(z', z) :|: z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 2 }-> insert#2(s6, z', @y, @ys) :|: s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z'' + insert(z', z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(55) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(56)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(57) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: testList
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(1) with polynomial bound: 74
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(58)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {testList}, {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: ?, size: O(1) [74]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(59) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: testList
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(1) with polynomial bound: 11
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(60)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(61) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(62)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(63) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insertionsortD#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: z
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(64)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertionsortD#1}, {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.57	  insertionsortD#1: runtime: ?, size: O(n^1) [z]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(65) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed RUNTIME bound using CoFloCo for: insertionsortD#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^2) with polynomial bound: 1 + 4*z^2
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(66)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 1 }-> insertionsortD#1(z) :|: z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 2 }-> insertD(@x, insertionsortD#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.57	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(67) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.57	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(68)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.57	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.57	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.57	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.57	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.57	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.57	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.57	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.57	
1115.38/291.57	Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.57	Previous analysis results are:
1115.38/291.57	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.57	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.57	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.57	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.57	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.57	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.57	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.57	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.57	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(69) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.57	
1115.38/291.57	Computed SIZE bound using CoFloCo for: insertionsort#1
1115.38/291.57	after applying outer abstraction to obtain an ITS,
1115.38/291.57	resulting in: O(n^1) with polynomial bound: z
1115.38/291.57	
1115.38/291.57	----------------------------------------
1115.38/291.57	
1115.38/291.57	(70)
1115.38/291.57	Obligation:
1115.38/291.57	Complexity RNTS consisting of the following rules:
1115.38/291.57	
1115.38/291.57	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.57	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.57	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.57	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.57	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.57	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.57	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsort#1}, {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: ?, size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(71) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed RUNTIME bound using CoFloCo for: insertionsort#1
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(n^2) with polynomial bound: 1 + 4*z^2
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(72)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 1 }-> insertionsort#1(z) :|: z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 2 }-> insert(@x, insertionsort#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(73) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(74)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(75) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed SIZE bound using CoFloCo for: insertionsortD
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(n^1) with polynomial bound: z
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(76)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsortD}, {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: ?, size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(77) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed RUNTIME bound using KoAT for: insertionsortD
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(n^2) with polynomial bound: 2 + 4*z^2
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(78)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 12 }-> insertionsortD(1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0)))))))))) :|: s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(79) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(80)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(81) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed SIZE bound using CoFloCo for: insertionsort
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(n^1) with polynomial bound: z
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(82)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {insertionsort}, {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: ?, size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(83) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed RUNTIME bound using KoAT for: insertionsort
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(n^2) with polynomial bound: 2 + 4*z^2
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(84)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 12 }-> insertionsort(1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0)))))))))) :|: s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(85) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(86)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(87) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed SIZE bound using CoFloCo for: testInsertionsortD
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(1) with polynomial bound: 74
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(88)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {testInsertionsortD}, {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  testInsertionsortD: runtime: ?, size: O(1) [74]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(89) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed RUNTIME bound using CoFloCo for: testInsertionsortD
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(1) with polynomial bound: 21918
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(90)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  testInsertionsortD: runtime: O(1) [21918], size: O(1) [74]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(91) ResultPropagationProof (UPPER BOUND(ID))
1115.38/291.58	Applied inner abstraction using the recently inferred runtime/size bounds where possible.
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(92)
1115.38/291.58	Obligation:
1115.38/291.58	Complexity RNTS consisting of the following rules:
1115.38/291.58	
1115.38/291.58	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.58	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.58	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.58	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.58	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.58	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.58	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.58	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.58	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.58	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.58	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.58	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.58	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.58	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.58	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.58	
1115.38/291.58	Function symbols to be analyzed: {testInsertionsort}
1115.38/291.58	Previous analysis results are:
1115.38/291.58	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.58	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.58	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.58	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.58	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.58	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.58	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.58	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.58	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.58	  testInsertionsortD: runtime: O(1) [21918], size: O(1) [74]
1115.38/291.58	
1115.38/291.58	----------------------------------------
1115.38/291.58	
1115.38/291.58	(93) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.58	
1115.38/291.58	Computed SIZE bound using CoFloCo for: testInsertionsort
1115.38/291.58	after applying outer abstraction to obtain an ITS,
1115.38/291.58	resulting in: O(1) with polynomial bound: 74
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(94)
1115.38/291.59	Obligation:
1115.38/291.59	Complexity RNTS consisting of the following rules:
1115.38/291.59	
1115.38/291.59	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.59	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.59	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.59	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.59	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.59	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.59	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.59	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.59	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.59	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.59	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.59	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.59	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.59	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.59	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	
1115.38/291.59	Function symbols to be analyzed: {testInsertionsort}
1115.38/291.59	Previous analysis results are:
1115.38/291.59	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.59	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.59	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.59	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.59	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.59	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.59	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.59	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.59	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.59	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.59	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.59	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  testInsertionsortD: runtime: O(1) [21918], size: O(1) [74]
1115.38/291.59	  testInsertionsort: runtime: ?, size: O(1) [74]
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(95) IntTrsBoundProof (UPPER BOUND(ID))
1115.38/291.59	
1115.38/291.59	Computed RUNTIME bound using CoFloCo for: testInsertionsort
1115.38/291.59	after applying outer abstraction to obtain an ITS,
1115.38/291.59	resulting in: O(1) with polynomial bound: 21918
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(96)
1115.38/291.59	Obligation:
1115.38/291.59	Complexity RNTS consisting of the following rules:
1115.38/291.59	
1115.38/291.59	   #abs(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   #abs(z) -{ 1 }-> 1 + (z - 1) :|: z - 1 >= 0
1115.38/291.59	   #abs(z) -{ 1 }-> 1 + (1 + (z - 1)) :|: z - 1 >= 0
1115.38/291.59	   #cklt(z) -{ 0 }-> 2 :|: z = 3
1115.38/291.59	   #cklt(z) -{ 0 }-> 1 :|: z = 1
1115.38/291.59	   #cklt(z) -{ 0 }-> 1 :|: z = 2
1115.38/291.59	   #cklt(z) -{ 0 }-> 0 :|: z >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z = 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 3 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z = 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 2 :|: z - 1 >= 0, z' - 1 >= 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 1 :|: z = 0, z' = 0
1115.38/291.59	   #compare(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> s4 :|: s4 >= 0, s4 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> s5 :|: s5 >= 0, s5 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z - 1 >= 0, z' = 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z = 0, z' = 0, 1 = 1
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
1115.38/291.59	   #less(z, z') -{ 1 }-> 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
1115.38/291.59	   insert(z, z') -{ 2 + 4*z' }-> s41 :|: s41 >= 0, s41 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.59	   insert#1(z, z') -{ 5 + 4*@ys }-> s43 :|: s43 >= 0, s43 <= z' + @y + @ys + 2, s6 >= 0, s6 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.59	   insert#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.59	   insert#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s42 :|: s42 >= 0, s42 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD(z, z') -{ 2 + 4*z' }-> s38 :|: s38 >= 0, s38 <= z' + z + 1, z' >= 0, z >= 0
1115.38/291.59	   insertD#1(z, z') -{ 5 + 4*@ys }-> s40 :|: s40 >= 0, s40 <= z' + @y + @ys + 2, s7 >= 0, s7 <= 2, s3 >= 0, s3 <= 3, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
1115.38/291.59	   insertD#1(z, z') -{ 1 }-> 1 + z' + 0 :|: z' >= 0, z = 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 1 }-> 1 + z' + (1 + z'' + z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
1115.38/291.59	   insertD#2(z, z', z'', z1) -{ 3 + 4*z1 }-> 1 + z'' + s39 :|: s39 >= 0, s39 <= z' + z1 + 1, z = 2, z' >= 0, z'' >= 0, z1 >= 0
1115.38/291.59	   insertionsort(z) -{ 2 + 4*z^2 }-> s47 :|: s47 >= 0, s47 <= z, z >= 0
1115.38/291.59	   insertionsort#1(z) -{ 5 + 4*@xs^2 + 4*s48 }-> s49 :|: s48 >= 0, s48 <= @xs, s49 >= 0, s49 <= @x + s48 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.59	   insertionsort#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   insertionsortD(z) -{ 2 + 4*z^2 }-> s44 :|: s44 >= 0, s44 <= z, z >= 0
1115.38/291.59	   insertionsortD#1(z) -{ 5 + 4*@xs^2 + 4*s45 }-> s46 :|: s45 >= 0, s45 <= @xs, s46 >= 0, s46 <= @x + s45 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
1115.38/291.59	   insertionsortD#1(z) -{ 1 }-> 0 :|: z = 0
1115.38/291.59	   testInsertionsort(z) -{ 414 + 80*s10 + 8*s10*s11 + 8*s10*s12 + 8*s10*s13 + 8*s10*s14 + 8*s10*s15 + 8*s10*s16 + 8*s10*s17 + 8*s10*s8 + 8*s10*s9 + 4*s10^2 + 80*s11 + 8*s11*s12 + 8*s11*s13 + 8*s11*s14 + 8*s11*s15 + 8*s11*s16 + 8*s11*s17 + 8*s11*s8 + 8*s11*s9 + 4*s11^2 + 80*s12 + 8*s12*s13 + 8*s12*s14 + 8*s12*s15 + 8*s12*s16 + 8*s12*s17 + 8*s12*s8 + 8*s12*s9 + 4*s12^2 + 80*s13 + 8*s13*s14 + 8*s13*s15 + 8*s13*s16 + 8*s13*s17 + 8*s13*s8 + 8*s13*s9 + 4*s13^2 + 80*s14 + 8*s14*s15 + 8*s14*s16 + 8*s14*s17 + 8*s14*s8 + 8*s14*s9 + 4*s14^2 + 80*s15 + 8*s15*s16 + 8*s15*s17 + 8*s15*s8 + 8*s15*s9 + 4*s15^2 + 80*s16 + 8*s16*s17 + 8*s16*s8 + 8*s16*s9 + 4*s16^2 + 80*s17 + 8*s17*s8 + 8*s17*s9 + 4*s17^2 + 80*s8 + 8*s8*s9 + 4*s8^2 + 80*s9 + 4*s9^2 }-> s51 :|: s51 >= 0, s51 <= 1 + s8 + (1 + s9 + (1 + s10 + (1 + s11 + (1 + s12 + (1 + s13 + (1 + s14 + (1 + s15 + (1 + s16 + (1 + s17 + 0))))))))), s8 >= 0, s8 <= 0 + 1, s9 >= 0, s9 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s10 >= 0, s10 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s11 >= 0, s11 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s12 >= 0, s12 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s13 >= 0, s13 <= 1 + (1 + 0) + 1, s14 >= 0, s14 <= 1 + (1 + (1 + 0)) + 1, s15 >= 0, s15 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s16 >= 0, s16 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s17 >= 0, s17 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	   testInsertionsortD(z) -{ 414 + 80*s18 + 8*s18*s19 + 8*s18*s20 + 8*s18*s21 + 8*s18*s22 + 8*s18*s23 + 8*s18*s24 + 8*s18*s25 + 8*s18*s26 + 8*s18*s27 + 4*s18^2 + 80*s19 + 8*s19*s20 + 8*s19*s21 + 8*s19*s22 + 8*s19*s23 + 8*s19*s24 + 8*s19*s25 + 8*s19*s26 + 8*s19*s27 + 4*s19^2 + 80*s20 + 8*s20*s21 + 8*s20*s22 + 8*s20*s23 + 8*s20*s24 + 8*s20*s25 + 8*s20*s26 + 8*s20*s27 + 4*s20^2 + 80*s21 + 8*s21*s22 + 8*s21*s23 + 8*s21*s24 + 8*s21*s25 + 8*s21*s26 + 8*s21*s27 + 4*s21^2 + 80*s22 + 8*s22*s23 + 8*s22*s24 + 8*s22*s25 + 8*s22*s26 + 8*s22*s27 + 4*s22^2 + 80*s23 + 8*s23*s24 + 8*s23*s25 + 8*s23*s26 + 8*s23*s27 + 4*s23^2 + 80*s24 + 8*s24*s25 + 8*s24*s26 + 8*s24*s27 + 4*s24^2 + 80*s25 + 8*s25*s26 + 8*s25*s27 + 4*s25^2 + 80*s26 + 8*s26*s27 + 4*s26^2 + 80*s27 + 4*s27^2 }-> s50 :|: s50 >= 0, s50 <= 1 + s18 + (1 + s19 + (1 + s20 + (1 + s21 + (1 + s22 + (1 + s23 + (1 + s24 + (1 + s25 + (1 + s26 + (1 + s27 + 0))))))))), s18 >= 0, s18 <= 0 + 1, s19 >= 0, s19 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s20 >= 0, s20 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s21 >= 0, s21 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s22 >= 0, s22 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s23 >= 0, s23 <= 1 + (1 + 0) + 1, s24 >= 0, s24 <= 1 + (1 + (1 + 0)) + 1, s25 >= 0, s25 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s26 >= 0, s26 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s27 >= 0, s27 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	   testList(z) -{ 11 }-> 1 + s28 + (1 + s29 + (1 + s30 + (1 + s31 + (1 + s32 + (1 + s33 + (1 + s34 + (1 + s35 + (1 + s36 + (1 + s37 + 0))))))))) :|: s28 >= 0, s28 <= 0 + 1, s29 >= 0, s29 <= 1 + (1 + (1 + (1 + (1 + 0)))) + 1, s30 >= 0, s30 <= 1 + (1 + (1 + (1 + (1 + (1 + 0))))) + 1, s31 >= 0, s31 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))))) + 1, s32 >= 0, s32 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0))))))) + 1, s33 >= 0, s33 <= 1 + (1 + 0) + 1, s34 >= 0, s34 <= 1 + (1 + (1 + 0)) + 1, s35 >= 0, s35 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))))) + 1, s36 >= 0, s36 <= 1 + (1 + (1 + (1 + (1 + (1 + (1 + 0)))))) + 1, s37 >= 0, s37 <= 1 + (1 + (1 + (1 + 0))) + 1, z >= 0
1115.38/291.59	
1115.38/291.59	Function symbols to be analyzed: 
1115.38/291.59	Previous analysis results are:
1115.38/291.59	  #compare: runtime: O(1) [0], size: O(1) [3]
1115.38/291.59	  #cklt: runtime: O(1) [0], size: O(1) [2]
1115.38/291.59	  #abs: runtime: O(1) [1], size: O(n^1) [1 + z]
1115.38/291.59	  #less: runtime: O(1) [1], size: O(1) [2]
1115.38/291.59	  insertD#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.59	  insertD#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.59	  insertD: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.59	  insert#2: runtime: O(n^1) [3 + 4*z1], size: O(n^1) [2 + z' + z'' + z1]
1115.38/291.59	  insert: runtime: O(n^1) [2 + 4*z'], size: O(n^1) [1 + z + z']
1115.38/291.59	  insert#1: runtime: O(n^1) [1 + 4*z], size: O(n^1) [1 + z + z']
1115.38/291.59	  testList: runtime: O(1) [11], size: O(1) [74]
1115.38/291.59	  insertionsortD#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsort#1: runtime: O(n^2) [1 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsortD: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  insertionsort: runtime: O(n^2) [2 + 4*z^2], size: O(n^1) [z]
1115.38/291.59	  testInsertionsortD: runtime: O(1) [21918], size: O(1) [74]
1115.38/291.59	  testInsertionsort: runtime: O(1) [21918], size: O(1) [74]
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(97) FinalProof (FINISHED)
1115.38/291.59	Computed overall runtime complexity
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(98)
1115.38/291.59	BOUNDS(1, n^2)
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(99) RenamingProof (BOTH BOUNDS(ID, ID))
1115.38/291.59	Renamed function symbols to avoid clashes with predefined symbol.
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(100)
1115.38/291.59	Obligation:
1115.38/291.59	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1115.38/291.59	
1115.38/291.59	
1115.38/291.59	The TRS R consists of the following rules:
1115.38/291.59	
1115.38/291.59	   #abs(#0) -> #0
1115.38/291.59	   #abs(#neg(@x)) -> #pos(@x)
1115.38/291.59	   #abs(#pos(@x)) -> #pos(@x)
1115.38/291.59	   #abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.59	   #less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.59	   insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.59	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	   insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.59	   insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.59	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	   insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.59	   insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.59	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.59	   insertionsort#1(nil) -> nil
1115.38/291.59	   insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.59	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.59	   insertionsortD#1(nil) -> nil
1115.38/291.59	   testInsertionsort(@x) -> insertionsort(testList(#unit))
1115.38/291.59	   testInsertionsortD(@x) -> insertionsortD(testList(#unit))
1115.38/291.59	   testList(@_) -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.59	
1115.38/291.59	The (relative) TRS S consists of the following rules:
1115.38/291.59	
1115.38/291.59	   #cklt(#EQ) -> #false
1115.38/291.59	   #cklt(#GT) -> #false
1115.38/291.59	   #cklt(#LT) -> #true
1115.38/291.59	   #compare(#0, #0) -> #EQ
1115.38/291.59	   #compare(#0, #neg(@y)) -> #GT
1115.38/291.59	   #compare(#0, #pos(@y)) -> #LT
1115.38/291.59	   #compare(#0, #s(@y)) -> #LT
1115.38/291.59	   #compare(#neg(@x), #0) -> #LT
1115.38/291.59	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.59	   #compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.59	   #compare(#pos(@x), #0) -> #GT
1115.38/291.59	   #compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.59	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.59	   #compare(#s(@x), #0) -> #GT
1115.38/291.59	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.59	
1115.38/291.59	Rewrite Strategy: INNERMOST
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(101) SlicingProof (LOWER BOUND(ID))
1115.38/291.59	Sliced the following arguments:
1115.38/291.59	testInsertionsort/0
1115.38/291.59	testList/0
1115.38/291.59	testInsertionsortD/0
1115.38/291.59	
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(102)
1115.38/291.59	Obligation:
1115.38/291.59	The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).
1115.38/291.59	
1115.38/291.59	
1115.38/291.59	The TRS R consists of the following rules:
1115.38/291.59	
1115.38/291.59	   #abs(#0) -> #0
1115.38/291.59	   #abs(#neg(@x)) -> #pos(@x)
1115.38/291.59	   #abs(#pos(@x)) -> #pos(@x)
1115.38/291.59	   #abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.59	   #less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.59	   insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.59	   insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	   insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	   insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	   insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.59	   insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.59	   insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	   insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	   insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	   insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.59	   insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.59	   insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.59	   insertionsort#1(nil) -> nil
1115.38/291.59	   insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.59	   insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.59	   insertionsortD#1(nil) -> nil
1115.38/291.59	   testInsertionsort -> insertionsort(testList)
1115.38/291.59	   testInsertionsortD -> insertionsortD(testList)
1115.38/291.59	   testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.59	
1115.38/291.59	The (relative) TRS S consists of the following rules:
1115.38/291.59	
1115.38/291.59	   #cklt(#EQ) -> #false
1115.38/291.59	   #cklt(#GT) -> #false
1115.38/291.59	   #cklt(#LT) -> #true
1115.38/291.59	   #compare(#0, #0) -> #EQ
1115.38/291.59	   #compare(#0, #neg(@y)) -> #GT
1115.38/291.59	   #compare(#0, #pos(@y)) -> #LT
1115.38/291.59	   #compare(#0, #s(@y)) -> #LT
1115.38/291.59	   #compare(#neg(@x), #0) -> #LT
1115.38/291.59	   #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.59	   #compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.59	   #compare(#pos(@x), #0) -> #GT
1115.38/291.59	   #compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.59	   #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.59	   #compare(#s(@x), #0) -> #GT
1115.38/291.59	   #compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.59	
1115.38/291.59	Rewrite Strategy: INNERMOST
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(103) TypeInferenceProof (BOTH BOUNDS(ID, ID))
1115.38/291.59	Infered types.
1115.38/291.59	----------------------------------------
1115.38/291.59	
1115.38/291.59	(104)
1115.38/291.59	Obligation:
1115.38/291.59	Innermost TRS:
1115.38/291.59	Rules:
1115.38/291.59	#abs(#0) -> #0
1115.38/291.59	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.59	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.59	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.59	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.59	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.59	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.59	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.59	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.59	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.59	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.59	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.59	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.59	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.59	insertionsort#1(nil) -> nil
1115.38/291.59	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.59	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.59	insertionsortD#1(nil) -> nil
1115.38/291.59	testInsertionsort -> insertionsort(testList)
1115.38/291.59	testInsertionsortD -> insertionsortD(testList)
1115.38/291.59	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.59	#cklt(#EQ) -> #false
1115.38/291.59	#cklt(#GT) -> #false
1115.38/291.59	#cklt(#LT) -> #true
1115.38/291.59	#compare(#0, #0) -> #EQ
1115.38/291.59	#compare(#0, #neg(@y)) -> #GT
1115.38/291.59	#compare(#0, #pos(@y)) -> #LT
1115.38/291.59	#compare(#0, #s(@y)) -> #LT
1115.38/291.59	#compare(#neg(@x), #0) -> #LT
1115.38/291.59	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.59	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.59	#compare(#pos(@x), #0) -> #GT
1115.38/291.59	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.59	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.59	#compare(#s(@x), #0) -> #GT
1115.38/291.59	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.59	
1115.38/291.59	Types:
1115.38/291.59	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.59	#0 :: #0:#neg:#pos:#s
1115.38/291.59	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.59	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.59	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(105) OrderProof (LOWER BOUND(ID))
1115.38/291.60	Heuristically decided to analyse the following defined symbols:
1115.38/291.60	#compare, insert, insert#1, insertD, insertD#1, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertD = insertD#1
1115.38/291.60	insertD < insertionsortD#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(106)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	#compare, insert, insert#1, insertD, insertD#1, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertD = insertD#1
1115.38/291.60	insertD < insertionsortD#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(107) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(0), gen_#0:#neg:#pos:#s5_3(0)) ->_R^Omega(0)
1115.38/291.60	#EQ
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(+(n8_3, 1)), gen_#0:#neg:#pos:#s5_3(+(n8_3, 1))) ->_R^Omega(0)
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) ->_IH
1115.38/291.60	#EQ
1115.38/291.60	
1115.38/291.60	We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(108)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertD#1, insert, insert#1, insertD, insertionsort, insertionsort#1, insertionsortD, insertionsortD#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertD = insertD#1
1115.38/291.60	insertD < insertionsortD#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(109) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(+(n319123_3, 1))) ->_R^Omega(1)
1115.38/291.60	insertD(#0, insertionsortD(gen_:::nil6_3(n319123_3))) ->_R^Omega(1)
1115.38/291.60	insertD(#0, insertionsortD#1(gen_:::nil6_3(n319123_3))) ->_IH
1115.38/291.60	insertD(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(110)
1115.38/291.60	Complex Obligation (BEST)
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(111)
1115.38/291.60	Obligation:
1115.38/291.60	Proved the lower bound n^1 for the following obligation:
1115.38/291.60	
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertionsortD#1, insert, insert#1, insertionsort, insertionsort#1, insertionsortD
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(112) LowerBoundPropagationProof (FINISHED)
1115.38/291.60	Propagated lower bound.
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(113)
1115.38/291.60	BOUNDS(n^1, INF)
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(114)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertionsortD, insert, insert#1, insertionsort, insertionsort#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(115) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsortD(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsortD(gen_:::nil6_3(+(n322764_3, 1))) ->_R^Omega(1)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(+(n322764_3, 1))) ->_R^Omega(1)
1115.38/291.60	insertD(#0, insertionsortD(gen_:::nil6_3(n322764_3))) ->_IH
1115.38/291.60	insertD(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(116)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n319123_3)) -> *7_3, rt in Omega(n319123_3)
1115.38/291.60	insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertionsortD#1, insert, insert#1, insertionsort, insertionsort#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	insertionsortD = insertionsortD#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(117) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(+(n331845_3, 1))) ->_R^Omega(1)
1115.38/291.60	insertD(#0, insertionsortD(gen_:::nil6_3(n331845_3))) ->_R^Omega(1)
1115.38/291.60	insertD(#0, insertionsortD#1(gen_:::nil6_3(n331845_3))) ->_IH
1115.38/291.60	insertD(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(118)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3)
1115.38/291.60	insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insert#1, insert, insertionsort, insertionsort#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insert = insert#1
1115.38/291.60	insert < insertionsort#1
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(119) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(+(n346798_3, 1))) ->_R^Omega(1)
1115.38/291.60	insert(#0, insertionsort(gen_:::nil6_3(n346798_3))) ->_R^Omega(1)
1115.38/291.60	insert(#0, insertionsort#1(gen_:::nil6_3(n346798_3))) ->_IH
1115.38/291.60	insert(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(120)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3)
1115.38/291.60	insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3)
1115.38/291.60	insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertionsort
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(121) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsort(gen_:::nil6_3(n361251_3)) -> *7_3, rt in Omega(n361251_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsort(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsort(gen_:::nil6_3(+(n361251_3, 1))) ->_R^Omega(1)
1115.38/291.60	insertionsort#1(gen_:::nil6_3(+(n361251_3, 1))) ->_R^Omega(1)
1115.38/291.60	insert(#0, insertionsort(gen_:::nil6_3(n361251_3))) ->_IH
1115.38/291.60	insert(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(122)
1115.38/291.60	Obligation:
1115.38/291.60	Innermost TRS:
1115.38/291.60	Rules:
1115.38/291.60	#abs(#0) -> #0
1115.38/291.60	#abs(#neg(@x)) -> #pos(@x)
1115.38/291.60	#abs(#pos(@x)) -> #pos(@x)
1115.38/291.60	#abs(#s(@x)) -> #pos(#s(@x))
1115.38/291.60	#less(@x, @y) -> #cklt(#compare(@x, @y))
1115.38/291.60	insert(@x, @l) -> insert#1(@l, @x)
1115.38/291.60	insert#1(::(@y, @ys), @x) -> insert#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insert#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insert#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insert#2(#true, @x, @y, @ys) -> ::(@y, insert(@x, @ys))
1115.38/291.60	insertD(@x, @l) -> insertD#1(@l, @x)
1115.38/291.60	insertD#1(::(@y, @ys), @x) -> insertD#2(#less(@y, @x), @x, @y, @ys)
1115.38/291.60	insertD#1(nil, @x) -> ::(@x, nil)
1115.38/291.60	insertD#2(#false, @x, @y, @ys) -> ::(@x, ::(@y, @ys))
1115.38/291.60	insertD#2(#true, @x, @y, @ys) -> ::(@y, insertD(@x, @ys))
1115.38/291.60	insertionsort(@l) -> insertionsort#1(@l)
1115.38/291.60	insertionsort#1(::(@x, @xs)) -> insert(@x, insertionsort(@xs))
1115.38/291.60	insertionsort#1(nil) -> nil
1115.38/291.60	insertionsortD(@l) -> insertionsortD#1(@l)
1115.38/291.60	insertionsortD#1(::(@x, @xs)) -> insertD(@x, insertionsortD(@xs))
1115.38/291.60	insertionsortD#1(nil) -> nil
1115.38/291.60	testInsertionsort -> insertionsort(testList)
1115.38/291.60	testInsertionsortD -> insertionsortD(testList)
1115.38/291.60	testList -> ::(#abs(#0), ::(#abs(#pos(#s(#s(#s(#s(#0)))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#0))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#s(#0))))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#0))))))))), ::(#abs(#pos(#s(#0))), ::(#abs(#pos(#s(#s(#0)))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#s(#s(#0)))))))))), ::(#abs(#pos(#s(#s(#s(#s(#s(#s(#0)))))))), ::(#abs(#pos(#s(#s(#s(#0))))), nil))))))))))
1115.38/291.60	#cklt(#EQ) -> #false
1115.38/291.60	#cklt(#GT) -> #false
1115.38/291.60	#cklt(#LT) -> #true
1115.38/291.60	#compare(#0, #0) -> #EQ
1115.38/291.60	#compare(#0, #neg(@y)) -> #GT
1115.38/291.60	#compare(#0, #pos(@y)) -> #LT
1115.38/291.60	#compare(#0, #s(@y)) -> #LT
1115.38/291.60	#compare(#neg(@x), #0) -> #LT
1115.38/291.60	#compare(#neg(@x), #neg(@y)) -> #compare(@y, @x)
1115.38/291.60	#compare(#neg(@x), #pos(@y)) -> #LT
1115.38/291.60	#compare(#pos(@x), #0) -> #GT
1115.38/291.60	#compare(#pos(@x), #neg(@y)) -> #GT
1115.38/291.60	#compare(#pos(@x), #pos(@y)) -> #compare(@x, @y)
1115.38/291.60	#compare(#s(@x), #0) -> #GT
1115.38/291.60	#compare(#s(@x), #s(@y)) -> #compare(@x, @y)
1115.38/291.60	
1115.38/291.60	Types:
1115.38/291.60	#abs :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#0 :: #0:#neg:#pos:#s
1115.38/291.60	#neg :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#pos :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#s :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s
1115.38/291.60	#less :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #false:#true
1115.38/291.60	#cklt :: #EQ:#GT:#LT -> #false:#true
1115.38/291.60	#compare :: #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> #EQ:#GT:#LT
1115.38/291.60	insert :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	:: :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insert#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	nil :: :::nil
1115.38/291.60	#false :: #false:#true
1115.38/291.60	#true :: #false:#true
1115.38/291.60	insertD :: #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertD#1 :: :::nil -> #0:#neg:#pos:#s -> :::nil
1115.38/291.60	insertD#2 :: #false:#true -> #0:#neg:#pos:#s -> #0:#neg:#pos:#s -> :::nil -> :::nil
1115.38/291.60	insertionsort :: :::nil -> :::nil
1115.38/291.60	insertionsort#1 :: :::nil -> :::nil
1115.38/291.60	insertionsortD :: :::nil -> :::nil
1115.38/291.60	insertionsortD#1 :: :::nil -> :::nil
1115.38/291.60	testInsertionsort :: :::nil
1115.38/291.60	testList :: :::nil
1115.38/291.60	testInsertionsortD :: :::nil
1115.38/291.60	#EQ :: #EQ:#GT:#LT
1115.38/291.60	#GT :: #EQ:#GT:#LT
1115.38/291.60	#LT :: #EQ:#GT:#LT
1115.38/291.60	hole_#0:#neg:#pos:#s1_3 :: #0:#neg:#pos:#s
1115.38/291.60	hole_#false:#true2_3 :: #false:#true
1115.38/291.60	hole_#EQ:#GT:#LT3_3 :: #EQ:#GT:#LT
1115.38/291.60	hole_:::nil4_3 :: :::nil
1115.38/291.60	gen_#0:#neg:#pos:#s5_3 :: Nat -> #0:#neg:#pos:#s
1115.38/291.60	gen_:::nil6_3 :: Nat -> :::nil
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Lemmas:
1115.38/291.60	#compare(gen_#0:#neg:#pos:#s5_3(n8_3), gen_#0:#neg:#pos:#s5_3(n8_3)) -> #EQ, rt in Omega(0)
1115.38/291.60	insertionsortD#1(gen_:::nil6_3(n331845_3)) -> *7_3, rt in Omega(n331845_3)
1115.38/291.60	insertionsortD(gen_:::nil6_3(n322764_3)) -> *7_3, rt in Omega(n322764_3)
1115.38/291.60	insertionsort#1(gen_:::nil6_3(n346798_3)) -> *7_3, rt in Omega(n346798_3)
1115.38/291.60	insertionsort(gen_:::nil6_3(n361251_3)) -> *7_3, rt in Omega(n361251_3)
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	Generator Equations:
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(0) <=> #0
1115.38/291.60	gen_#0:#neg:#pos:#s5_3(+(x, 1)) <=> #neg(gen_#0:#neg:#pos:#s5_3(x))
1115.38/291.60	gen_:::nil6_3(0) <=> nil
1115.38/291.60	gen_:::nil6_3(+(x, 1)) <=> ::(#0, gen_:::nil6_3(x))
1115.38/291.60	
1115.38/291.60	
1115.38/291.60	The following defined symbols remain to be analysed:
1115.38/291.60	insertionsort#1
1115.38/291.60	
1115.38/291.60	They will be analysed ascendingly in the following order:
1115.38/291.60	insertionsort = insertionsort#1
1115.38/291.60	
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(123) RewriteLemmaProof (LOWER BOUND(ID))
1115.38/291.60	Proved the following rewrite lemma:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(n380522_3)) -> *7_3, rt in Omega(n380522_3)
1115.38/291.60	
1115.38/291.60	Induction Base:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(0))
1115.38/291.60	
1115.38/291.60	Induction Step:
1115.38/291.60	insertionsort#1(gen_:::nil6_3(+(n380522_3, 1))) ->_R^Omega(1)
1115.38/291.60	insert(#0, insertionsort(gen_:::nil6_3(n380522_3))) ->_R^Omega(1)
1115.38/291.60	insert(#0, insertionsort#1(gen_:::nil6_3(n380522_3))) ->_IH
1115.38/291.60	insert(#0, *7_3)
1115.38/291.60	
1115.38/291.60	We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
1115.38/291.60	----------------------------------------
1115.38/291.60	
1115.38/291.60	(124)
1115.38/291.60	BOUNDS(1, INF)
1115.63/291.70	EOF