1135.63/291.55	WORST_CASE(Omega(n^1), ?)
1140.84/292.85	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1140.84/292.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1140.84/292.85	
1140.84/292.85	(0) CpxTRS
1140.84/292.85	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1140.84/292.85	(2) TRS for Loop Detection
1140.84/292.85	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1140.84/292.85	(4) BEST
1140.84/292.85	    (5) proven lower bound
1140.84/292.85	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1140.84/292.85	        (7) BOUNDS(n^1, INF)
1140.84/292.85	    (8) TRS for Loop Detection
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(0)
1140.84/292.85	Obligation:
1140.84/292.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	The TRS R consists of the following rules:
1140.84/292.85	
1140.84/292.85	   qsort(xs) -> qs(half(length(xs)), xs)
1140.84/292.85	   qs(n, nil) -> nil
1140.84/292.85	   qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))
1140.84/292.85	   filterlow(n, nil) -> nil
1140.84/292.85	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1140.84/292.85	   if1(true, n, x, xs) -> filterlow(n, xs)
1140.84/292.85	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1140.84/292.85	   filterhigh(n, nil) -> nil
1140.84/292.85	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1140.84/292.85	   if2(true, n, x, xs) -> filterhigh(n, xs)
1140.84/292.85	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1140.84/292.85	   ge(x, 0) -> true
1140.84/292.85	   ge(0, s(x)) -> false
1140.84/292.85	   ge(s(x), s(y)) -> ge(x, y)
1140.84/292.85	   append(nil, ys) -> ys
1140.84/292.85	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1140.84/292.85	   length(nil) -> 0
1140.84/292.85	   length(cons(x, xs)) -> s(length(xs))
1140.84/292.85	   half(0) -> 0
1140.84/292.85	   half(s(0)) -> 0
1140.84/292.85	   half(s(s(x))) -> s(half(x))
1140.84/292.85	   get(n, nil) -> 0
1140.84/292.85	   get(n, cons(x, nil)) -> x
1140.84/292.85	   get(0, cons(x, cons(y, xs))) -> x
1140.84/292.85	   get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs))
1140.84/292.85	
1140.84/292.85	S is empty.
1140.84/292.85	Rewrite Strategy: INNERMOST
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1140.84/292.85	Transformed a relative TRS into a decreasing-loop problem.
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(2)
1140.84/292.85	Obligation:
1140.84/292.85	Analyzing the following TRS for decreasing loops:
1140.84/292.85	
1140.84/292.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	The TRS R consists of the following rules:
1140.84/292.85	
1140.84/292.85	   qsort(xs) -> qs(half(length(xs)), xs)
1140.84/292.85	   qs(n, nil) -> nil
1140.84/292.85	   qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))
1140.84/292.85	   filterlow(n, nil) -> nil
1140.84/292.85	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1140.84/292.85	   if1(true, n, x, xs) -> filterlow(n, xs)
1140.84/292.85	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1140.84/292.85	   filterhigh(n, nil) -> nil
1140.84/292.85	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1140.84/292.85	   if2(true, n, x, xs) -> filterhigh(n, xs)
1140.84/292.85	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1140.84/292.85	   ge(x, 0) -> true
1140.84/292.85	   ge(0, s(x)) -> false
1140.84/292.85	   ge(s(x), s(y)) -> ge(x, y)
1140.84/292.85	   append(nil, ys) -> ys
1140.84/292.85	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1140.84/292.85	   length(nil) -> 0
1140.84/292.85	   length(cons(x, xs)) -> s(length(xs))
1140.84/292.85	   half(0) -> 0
1140.84/292.85	   half(s(0)) -> 0
1140.84/292.85	   half(s(s(x))) -> s(half(x))
1140.84/292.85	   get(n, nil) -> 0
1140.84/292.85	   get(n, cons(x, nil)) -> x
1140.84/292.85	   get(0, cons(x, cons(y, xs))) -> x
1140.84/292.85	   get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs))
1140.84/292.85	
1140.84/292.85	S is empty.
1140.84/292.85	Rewrite Strategy: INNERMOST
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(3) DecreasingLoopProof (LOWER BOUND(ID))
1140.84/292.85	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1140.84/292.85	
1140.84/292.85	The rewrite sequence
1140.84/292.85	
1140.84/292.85	append(cons(x, xs), ys) ->^+ cons(x, append(xs, ys))
1140.84/292.85	
1140.84/292.85	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
1140.84/292.85	
1140.84/292.85	The pumping substitution is [xs / cons(x, xs)].
1140.84/292.85	
1140.84/292.85	The result substitution is [ ].
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(4)
1140.84/292.85	Complex Obligation (BEST)
1140.84/292.85	
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(5)
1140.84/292.85	Obligation:
1140.84/292.85	Proved the lower bound n^1 for the following obligation:
1140.84/292.85	
1140.84/292.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	The TRS R consists of the following rules:
1140.84/292.85	
1140.84/292.85	   qsort(xs) -> qs(half(length(xs)), xs)
1140.84/292.85	   qs(n, nil) -> nil
1140.84/292.85	   qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))
1140.84/292.85	   filterlow(n, nil) -> nil
1140.84/292.85	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1140.84/292.85	   if1(true, n, x, xs) -> filterlow(n, xs)
1140.84/292.85	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1140.84/292.85	   filterhigh(n, nil) -> nil
1140.84/292.85	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1140.84/292.85	   if2(true, n, x, xs) -> filterhigh(n, xs)
1140.84/292.85	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1140.84/292.85	   ge(x, 0) -> true
1140.84/292.85	   ge(0, s(x)) -> false
1140.84/292.85	   ge(s(x), s(y)) -> ge(x, y)
1140.84/292.85	   append(nil, ys) -> ys
1140.84/292.85	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1140.84/292.85	   length(nil) -> 0
1140.84/292.85	   length(cons(x, xs)) -> s(length(xs))
1140.84/292.85	   half(0) -> 0
1140.84/292.85	   half(s(0)) -> 0
1140.84/292.85	   half(s(s(x))) -> s(half(x))
1140.84/292.85	   get(n, nil) -> 0
1140.84/292.85	   get(n, cons(x, nil)) -> x
1140.84/292.85	   get(0, cons(x, cons(y, xs))) -> x
1140.84/292.85	   get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs))
1140.84/292.85	
1140.84/292.85	S is empty.
1140.84/292.85	Rewrite Strategy: INNERMOST
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(6) LowerBoundPropagationProof (FINISHED)
1140.84/292.85	Propagated lower bound.
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(7)
1140.84/292.85	BOUNDS(n^1, INF)
1140.84/292.85	
1140.84/292.85	----------------------------------------
1140.84/292.85	
1140.84/292.85	(8)
1140.84/292.85	Obligation:
1140.84/292.85	Analyzing the following TRS for decreasing loops:
1140.84/292.85	
1140.84/292.85	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1140.84/292.85	
1140.84/292.85	
1140.84/292.85	The TRS R consists of the following rules:
1140.84/292.85	
1140.84/292.85	   qsort(xs) -> qs(half(length(xs)), xs)
1140.84/292.85	   qs(n, nil) -> nil
1140.84/292.85	   qs(n, cons(x, xs)) -> append(qs(half(n), filterlow(get(n, cons(x, xs)), cons(x, xs))), cons(get(n, cons(x, xs)), qs(half(n), filterhigh(get(n, cons(x, xs)), cons(x, xs)))))
1140.84/292.85	   filterlow(n, nil) -> nil
1140.84/292.85	   filterlow(n, cons(x, xs)) -> if1(ge(n, x), n, x, xs)
1140.84/292.85	   if1(true, n, x, xs) -> filterlow(n, xs)
1140.84/292.85	   if1(false, n, x, xs) -> cons(x, filterlow(n, xs))
1140.84/292.85	   filterhigh(n, nil) -> nil
1140.84/292.85	   filterhigh(n, cons(x, xs)) -> if2(ge(x, n), n, x, xs)
1140.84/292.85	   if2(true, n, x, xs) -> filterhigh(n, xs)
1140.84/292.85	   if2(false, n, x, xs) -> cons(x, filterhigh(n, xs))
1140.84/292.85	   ge(x, 0) -> true
1140.84/292.85	   ge(0, s(x)) -> false
1140.84/292.86	   ge(s(x), s(y)) -> ge(x, y)
1140.84/292.86	   append(nil, ys) -> ys
1140.84/292.86	   append(cons(x, xs), ys) -> cons(x, append(xs, ys))
1140.84/292.86	   length(nil) -> 0
1140.84/292.86	   length(cons(x, xs)) -> s(length(xs))
1140.84/292.86	   half(0) -> 0
1140.84/292.86	   half(s(0)) -> 0
1140.84/292.86	   half(s(s(x))) -> s(half(x))
1140.84/292.86	   get(n, nil) -> 0
1140.84/292.86	   get(n, cons(x, nil)) -> x
1140.84/292.86	   get(0, cons(x, cons(y, xs))) -> x
1140.84/292.86	   get(s(n), cons(x, cons(y, xs))) -> get(n, cons(y, xs))
1140.84/292.86	
1140.84/292.86	S is empty.
1140.84/292.86	Rewrite Strategy: INNERMOST
1140.93/292.92	EOF