1114.85/291.51	WORST_CASE(Omega(n^1), ?)
1119.21/292.64	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1119.21/292.64	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.21/292.64	
1119.21/292.64	(0) CpxTRS
1119.21/292.64	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1119.21/292.64	(2) TRS for Loop Detection
1119.21/292.64	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1119.21/292.64	(4) BEST
1119.21/292.64	    (5) proven lower bound
1119.21/292.64	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1119.21/292.64	        (7) BOUNDS(n^1, INF)
1119.21/292.64	    (8) TRS for Loop Detection
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(0)
1119.21/292.64	Obligation:
1119.21/292.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	The TRS R consists of the following rules:
1119.21/292.64	
1119.21/292.64	   qsort(nil) -> nil
1119.21/292.64	   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
1119.21/292.64	   filterlow(n, nil) -> nil
1119.21/292.64	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1119.21/292.64	   if1(true, n, x, xs) -> filterlow(n, xs)
1119.21/292.64	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1119.21/292.64	   filterhigh(n, nil) -> nil
1119.21/292.64	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1119.21/292.64	   if2(true, n, x, xs) -> filterhigh(n, xs)
1119.21/292.64	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1119.21/292.64	   ge(x, 0) -> true
1119.21/292.64	   ge(0, s(x)) -> false
1119.21/292.64	   ge(s(x), s(y)) -> ge(x, y)
1119.21/292.64	   append(nil, ys) -> ys
1119.21/292.64	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1119.21/292.64	   last(nil) -> 0
1119.21/292.64	   last(cons(x, nil)) -> x
1119.21/292.64	   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
1119.21/292.64	
1119.21/292.64	S is empty.
1119.21/292.64	Rewrite Strategy: INNERMOST
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1119.21/292.64	Transformed a relative TRS into a decreasing-loop problem.
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(2)
1119.21/292.64	Obligation:
1119.21/292.64	Analyzing the following TRS for decreasing loops:
1119.21/292.64	
1119.21/292.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	The TRS R consists of the following rules:
1119.21/292.64	
1119.21/292.64	   qsort(nil) -> nil
1119.21/292.64	   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
1119.21/292.64	   filterlow(n, nil) -> nil
1119.21/292.64	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1119.21/292.64	   if1(true, n, x, xs) -> filterlow(n, xs)
1119.21/292.64	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1119.21/292.64	   filterhigh(n, nil) -> nil
1119.21/292.64	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1119.21/292.64	   if2(true, n, x, xs) -> filterhigh(n, xs)
1119.21/292.64	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1119.21/292.64	   ge(x, 0) -> true
1119.21/292.64	   ge(0, s(x)) -> false
1119.21/292.64	   ge(s(x), s(y)) -> ge(x, y)
1119.21/292.64	   append(nil, ys) -> ys
1119.21/292.64	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1119.21/292.64	   last(nil) -> 0
1119.21/292.64	   last(cons(x, nil)) -> x
1119.21/292.64	   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
1119.21/292.64	
1119.21/292.64	S is empty.
1119.21/292.64	Rewrite Strategy: INNERMOST
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(3) DecreasingLoopProof (LOWER BOUND(ID))
1119.21/292.64	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1119.21/292.64	
1119.21/292.64	The rewrite sequence
1119.21/292.64	
1119.21/292.64	append(cons(x, xs), ys) ->^+ cons(x, append(xs, ys))
1119.21/292.64	
1119.21/292.64	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
1119.21/292.64	
1119.21/292.64	The pumping substitution is [xs / cons(x, xs)].
1119.21/292.64	
1119.21/292.64	The result substitution is [ ].
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(4)
1119.21/292.64	Complex Obligation (BEST)
1119.21/292.64	
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(5)
1119.21/292.64	Obligation:
1119.21/292.64	Proved the lower bound n^1 for the following obligation:
1119.21/292.64	
1119.21/292.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	The TRS R consists of the following rules:
1119.21/292.64	
1119.21/292.64	   qsort(nil) -> nil
1119.21/292.64	   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
1119.21/292.64	   filterlow(n, nil) -> nil
1119.21/292.64	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1119.21/292.64	   if1(true, n, x, xs) -> filterlow(n, xs)
1119.21/292.64	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1119.21/292.64	   filterhigh(n, nil) -> nil
1119.21/292.64	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1119.21/292.64	   if2(true, n, x, xs) -> filterhigh(n, xs)
1119.21/292.64	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1119.21/292.64	   ge(x, 0) -> true
1119.21/292.64	   ge(0, s(x)) -> false
1119.21/292.64	   ge(s(x), s(y)) -> ge(x, y)
1119.21/292.64	   append(nil, ys) -> ys
1119.21/292.64	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1119.21/292.64	   last(nil) -> 0
1119.21/292.64	   last(cons(x, nil)) -> x
1119.21/292.64	   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
1119.21/292.64	
1119.21/292.64	S is empty.
1119.21/292.64	Rewrite Strategy: INNERMOST
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(6) LowerBoundPropagationProof (FINISHED)
1119.21/292.64	Propagated lower bound.
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(7)
1119.21/292.64	BOUNDS(n^1, INF)
1119.21/292.64	
1119.21/292.64	----------------------------------------
1119.21/292.64	
1119.21/292.64	(8)
1119.21/292.64	Obligation:
1119.21/292.64	Analyzing the following TRS for decreasing loops:
1119.21/292.64	
1119.21/292.64	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1119.21/292.64	
1119.21/292.64	
1119.21/292.64	The TRS R consists of the following rules:
1119.21/292.64	
1119.21/292.64	   qsort(nil) -> nil
1119.21/292.64	   qsort(cons(x, xs)) -> append(qsort(filterlow(last(cons(x, xs)), cons(x, xs))), cons(last(cons(x, xs)), qsort(filterhigh(last(cons(x, xs)), cons(x, xs)))))
1119.21/292.64	   filterlow(n, nil) -> nil
1119.21/292.64	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1119.21/292.64	   if1(true, n, x, xs) -> filterlow(n, xs)
1119.21/292.64	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1119.21/292.64	   filterhigh(n, nil) -> nil
1119.21/292.64	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1119.21/292.64	   if2(true, n, x, xs) -> filterhigh(n, xs)
1119.21/292.64	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1119.21/292.64	   ge(x, 0) -> true
1119.21/292.64	   ge(0, s(x)) -> false
1119.21/292.64	   ge(s(x), s(y)) -> ge(x, y)
1119.21/292.64	   append(nil, ys) -> ys
1119.21/292.64	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1119.21/292.64	   last(nil) -> 0
1119.21/292.64	   last(cons(x, nil)) -> x
1119.21/292.64	   last(cons(x, cons(y, xs))) -> last(cons(y, xs))
1119.21/292.64	
1119.21/292.64	S is empty.
1119.21/292.64	Rewrite Strategy: INNERMOST
1119.55/292.70	EOF