309.44/291.46	WORST_CASE(Omega(n^1), ?)
309.50/291.46	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
309.50/291.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.50/291.46	
309.50/291.46	
309.50/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.50/291.46	
309.50/291.46	(0) CpxTRS
309.50/291.46	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
309.50/291.46	(2) TRS for Loop Detection
309.50/291.46	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
309.50/291.46	(4) BEST
309.50/291.46	    (5) proven lower bound
309.50/291.46	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
309.50/291.46	        (7) BOUNDS(n^1, INF)
309.50/291.46	    (8) TRS for Loop Detection
309.50/291.46	
309.50/291.46	
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(0)
309.50/291.46	Obligation:
309.50/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.50/291.46	
309.50/291.46	
309.50/291.46	The TRS R consists of the following rules:
309.50/291.46	
309.50/291.46	   qsort(nil) -> nil
309.50/291.46	   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
309.50/291.46	   filterlow(n, nil) -> nil
309.50/291.46	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
309.50/291.46	   if1(true, n, x, xs) -> filterlow(n, xs)
309.50/291.46	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
309.50/291.46	   filterhigh(n, nil) -> nil
309.50/291.46	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
309.50/291.46	   if2(true, n, x, xs) -> filterhigh(n, xs)
309.50/291.46	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
309.50/291.46	   ge(x, 0) -> true
309.50/291.46	   ge(0, s(x)) -> false
309.50/291.46	   ge(s(x), s(y)) -> ge(x, y)
309.50/291.46	   append(nil, ys) -> ys
309.50/291.46	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
309.50/291.46	
309.50/291.46	S is empty.
309.50/291.46	Rewrite Strategy: FULL
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
309.50/291.46	Transformed a relative TRS into a decreasing-loop problem.
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(2)
309.50/291.46	Obligation:
309.50/291.46	Analyzing the following TRS for decreasing loops:
309.50/291.46	
309.50/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.50/291.46	
309.50/291.46	
309.50/291.46	The TRS R consists of the following rules:
309.50/291.46	
309.50/291.46	   qsort(nil) -> nil
309.50/291.46	   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
309.50/291.46	   filterlow(n, nil) -> nil
309.50/291.46	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
309.50/291.46	   if1(true, n, x, xs) -> filterlow(n, xs)
309.50/291.46	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
309.50/291.46	   filterhigh(n, nil) -> nil
309.50/291.46	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
309.50/291.46	   if2(true, n, x, xs) -> filterhigh(n, xs)
309.50/291.46	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
309.50/291.46	   ge(x, 0) -> true
309.50/291.46	   ge(0, s(x)) -> false
309.50/291.46	   ge(s(x), s(y)) -> ge(x, y)
309.50/291.46	   append(nil, ys) -> ys
309.50/291.46	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
309.50/291.46	
309.50/291.46	S is empty.
309.50/291.46	Rewrite Strategy: FULL
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(3) DecreasingLoopProof (LOWER BOUND(ID))
309.50/291.46	The following loop(s) give(s) rise to the lower bound Omega(n^1):
309.50/291.46	
309.50/291.46	The rewrite sequence
309.50/291.46	
309.50/291.46	append(cons(x, xs), ys) ->^+ cons(x, append(xs, ys))
309.50/291.46	
309.50/291.46	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
309.50/291.46	
309.50/291.46	The pumping substitution is [xs / cons(x, xs)].
309.50/291.46	
309.50/291.46	The result substitution is [ ].
309.50/291.46	
309.50/291.46	
309.50/291.46	
309.50/291.46	
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(4)
309.50/291.46	Complex Obligation (BEST)
309.50/291.46	
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(5)
309.50/291.46	Obligation:
309.50/291.46	Proved the lower bound n^1 for the following obligation:
309.50/291.46	
309.50/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.50/291.46	
309.50/291.46	
309.50/291.46	The TRS R consists of the following rules:
309.50/291.46	
309.50/291.46	   qsort(nil) -> nil
309.50/291.46	   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
309.50/291.46	   filterlow(n, nil) -> nil
309.50/291.46	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
309.50/291.46	   if1(true, n, x, xs) -> filterlow(n, xs)
309.50/291.46	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
309.50/291.46	   filterhigh(n, nil) -> nil
309.50/291.46	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
309.50/291.46	   if2(true, n, x, xs) -> filterhigh(n, xs)
309.50/291.46	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
309.50/291.46	   ge(x, 0) -> true
309.50/291.46	   ge(0, s(x)) -> false
309.50/291.46	   ge(s(x), s(y)) -> ge(x, y)
309.50/291.46	   append(nil, ys) -> ys
309.50/291.46	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
309.50/291.46	
309.50/291.46	S is empty.
309.50/291.46	Rewrite Strategy: FULL
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(6) LowerBoundPropagationProof (FINISHED)
309.50/291.46	Propagated lower bound.
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(7)
309.50/291.46	BOUNDS(n^1, INF)
309.50/291.46	
309.50/291.46	----------------------------------------
309.50/291.46	
309.50/291.46	(8)
309.50/291.46	Obligation:
309.50/291.46	Analyzing the following TRS for decreasing loops:
309.50/291.46	
309.50/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.50/291.46	
309.50/291.46	
309.50/291.46	The TRS R consists of the following rules:
309.50/291.46	
309.50/291.46	   qsort(nil) -> nil
309.50/291.46	   qsort(cons(x, xs)) -> append(qsort(filterlow(x, cons(x, xs))), cons(x, qsort(filterhigh(x, cons(x, xs)))))
309.50/291.46	   filterlow(n, nil) -> nil
309.50/291.46	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
309.50/291.46	   if1(true, n, x, xs) -> filterlow(n, xs)
309.50/291.46	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
309.50/291.46	   filterhigh(n, nil) -> nil
309.50/291.46	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
309.50/291.46	   if2(true, n, x, xs) -> filterhigh(n, xs)
309.50/291.46	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
309.50/291.46	   ge(x, 0) -> true
309.50/291.46	   ge(0, s(x)) -> false
309.50/291.46	   ge(s(x), s(y)) -> ge(x, y)
309.50/291.46	   append(nil, ys) -> ys
309.50/291.46	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
309.50/291.46	
309.50/291.46	S is empty.
309.50/291.46	Rewrite Strategy: FULL
309.50/291.49	EOF