1133.79/291.56	WORST_CASE(Omega(n^1), ?)
1142.05/293.67	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1142.05/293.67	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1142.05/293.67	
1142.05/293.67	(0) CpxTRS
1142.05/293.67	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1142.05/293.67	(2) TRS for Loop Detection
1142.05/293.67	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1142.05/293.67	(4) BEST
1142.05/293.67	    (5) proven lower bound
1142.05/293.67	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1142.05/293.67	        (7) BOUNDS(n^1, INF)
1142.05/293.67	    (8) TRS for Loop Detection
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(0)
1142.05/293.67	Obligation:
1142.05/293.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	The TRS R consists of the following rules:
1142.05/293.67	
1142.05/293.67	   a__pairNs -> cons(0, incr(oddNs))
1142.05/293.67	   a__oddNs -> a__incr(a__pairNs)
1142.05/293.67	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1142.05/293.67	   a__take(0, XS) -> nil
1142.05/293.67	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
1142.05/293.67	   a__zip(nil, XS) -> nil
1142.05/293.67	   a__zip(X, nil) -> nil
1142.05/293.67	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
1142.05/293.67	   a__tail(cons(X, XS)) -> mark(XS)
1142.05/293.67	   a__repItems(nil) -> nil
1142.05/293.67	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
1142.05/293.67	   mark(pairNs) -> a__pairNs
1142.05/293.67	   mark(incr(X)) -> a__incr(mark(X))
1142.05/293.67	   mark(oddNs) -> a__oddNs
1142.05/293.67	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
1142.05/293.67	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
1142.05/293.67	   mark(tail(X)) -> a__tail(mark(X))
1142.05/293.67	   mark(repItems(X)) -> a__repItems(mark(X))
1142.05/293.67	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1142.05/293.67	   mark(0) -> 0
1142.05/293.67	   mark(s(X)) -> s(mark(X))
1142.05/293.67	   mark(nil) -> nil
1142.05/293.67	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
1142.05/293.67	   a__pairNs -> pairNs
1142.05/293.67	   a__incr(X) -> incr(X)
1142.05/293.67	   a__oddNs -> oddNs
1142.05/293.67	   a__take(X1, X2) -> take(X1, X2)
1142.05/293.67	   a__zip(X1, X2) -> zip(X1, X2)
1142.05/293.67	   a__tail(X) -> tail(X)
1142.05/293.67	   a__repItems(X) -> repItems(X)
1142.05/293.67	
1142.05/293.67	S is empty.
1142.05/293.67	Rewrite Strategy: INNERMOST
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1142.05/293.67	Transformed a relative TRS into a decreasing-loop problem.
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(2)
1142.05/293.67	Obligation:
1142.05/293.67	Analyzing the following TRS for decreasing loops:
1142.05/293.67	
1142.05/293.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	The TRS R consists of the following rules:
1142.05/293.67	
1142.05/293.67	   a__pairNs -> cons(0, incr(oddNs))
1142.05/293.67	   a__oddNs -> a__incr(a__pairNs)
1142.05/293.67	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1142.05/293.67	   a__take(0, XS) -> nil
1142.05/293.67	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
1142.05/293.67	   a__zip(nil, XS) -> nil
1142.05/293.67	   a__zip(X, nil) -> nil
1142.05/293.67	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
1142.05/293.67	   a__tail(cons(X, XS)) -> mark(XS)
1142.05/293.67	   a__repItems(nil) -> nil
1142.05/293.67	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
1142.05/293.67	   mark(pairNs) -> a__pairNs
1142.05/293.67	   mark(incr(X)) -> a__incr(mark(X))
1142.05/293.67	   mark(oddNs) -> a__oddNs
1142.05/293.67	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
1142.05/293.67	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
1142.05/293.67	   mark(tail(X)) -> a__tail(mark(X))
1142.05/293.67	   mark(repItems(X)) -> a__repItems(mark(X))
1142.05/293.67	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1142.05/293.67	   mark(0) -> 0
1142.05/293.67	   mark(s(X)) -> s(mark(X))
1142.05/293.67	   mark(nil) -> nil
1142.05/293.67	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
1142.05/293.67	   a__pairNs -> pairNs
1142.05/293.67	   a__incr(X) -> incr(X)
1142.05/293.67	   a__oddNs -> oddNs
1142.05/293.67	   a__take(X1, X2) -> take(X1, X2)
1142.05/293.67	   a__zip(X1, X2) -> zip(X1, X2)
1142.05/293.67	   a__tail(X) -> tail(X)
1142.05/293.67	   a__repItems(X) -> repItems(X)
1142.05/293.67	
1142.05/293.67	S is empty.
1142.05/293.67	Rewrite Strategy: INNERMOST
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(3) DecreasingLoopProof (LOWER BOUND(ID))
1142.05/293.67	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1142.05/293.67	
1142.05/293.67	The rewrite sequence
1142.05/293.67	
1142.05/293.67	mark(repItems(X)) ->^+ a__repItems(mark(X))
1142.05/293.67	
1142.05/293.67	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
1142.05/293.67	
1142.05/293.67	The pumping substitution is [X / repItems(X)].
1142.05/293.67	
1142.05/293.67	The result substitution is [ ].
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(4)
1142.05/293.67	Complex Obligation (BEST)
1142.05/293.67	
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(5)
1142.05/293.67	Obligation:
1142.05/293.67	Proved the lower bound n^1 for the following obligation:
1142.05/293.67	
1142.05/293.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	The TRS R consists of the following rules:
1142.05/293.67	
1142.05/293.67	   a__pairNs -> cons(0, incr(oddNs))
1142.05/293.67	   a__oddNs -> a__incr(a__pairNs)
1142.05/293.67	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1142.05/293.67	   a__take(0, XS) -> nil
1142.05/293.67	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
1142.05/293.67	   a__zip(nil, XS) -> nil
1142.05/293.67	   a__zip(X, nil) -> nil
1142.05/293.67	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
1142.05/293.67	   a__tail(cons(X, XS)) -> mark(XS)
1142.05/293.67	   a__repItems(nil) -> nil
1142.05/293.67	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
1142.05/293.67	   mark(pairNs) -> a__pairNs
1142.05/293.67	   mark(incr(X)) -> a__incr(mark(X))
1142.05/293.67	   mark(oddNs) -> a__oddNs
1142.05/293.67	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
1142.05/293.67	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
1142.05/293.67	   mark(tail(X)) -> a__tail(mark(X))
1142.05/293.67	   mark(repItems(X)) -> a__repItems(mark(X))
1142.05/293.67	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1142.05/293.67	   mark(0) -> 0
1142.05/293.67	   mark(s(X)) -> s(mark(X))
1142.05/293.67	   mark(nil) -> nil
1142.05/293.67	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
1142.05/293.67	   a__pairNs -> pairNs
1142.05/293.67	   a__incr(X) -> incr(X)
1142.05/293.67	   a__oddNs -> oddNs
1142.05/293.67	   a__take(X1, X2) -> take(X1, X2)
1142.05/293.67	   a__zip(X1, X2) -> zip(X1, X2)
1142.05/293.67	   a__tail(X) -> tail(X)
1142.05/293.67	   a__repItems(X) -> repItems(X)
1142.05/293.67	
1142.05/293.67	S is empty.
1142.05/293.67	Rewrite Strategy: INNERMOST
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(6) LowerBoundPropagationProof (FINISHED)
1142.05/293.67	Propagated lower bound.
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(7)
1142.05/293.67	BOUNDS(n^1, INF)
1142.05/293.67	
1142.05/293.67	----------------------------------------
1142.05/293.67	
1142.05/293.67	(8)
1142.05/293.67	Obligation:
1142.05/293.67	Analyzing the following TRS for decreasing loops:
1142.05/293.67	
1142.05/293.67	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1142.05/293.67	
1142.05/293.67	
1142.05/293.67	The TRS R consists of the following rules:
1142.05/293.67	
1142.05/293.67	   a__pairNs -> cons(0, incr(oddNs))
1142.05/293.67	   a__oddNs -> a__incr(a__pairNs)
1142.05/293.67	   a__incr(cons(X, XS)) -> cons(s(mark(X)), incr(XS))
1142.05/293.67	   a__take(0, XS) -> nil
1142.05/293.67	   a__take(s(N), cons(X, XS)) -> cons(mark(X), take(N, XS))
1142.05/293.67	   a__zip(nil, XS) -> nil
1142.05/293.67	   a__zip(X, nil) -> nil
1142.05/293.67	   a__zip(cons(X, XS), cons(Y, YS)) -> cons(pair(mark(X), mark(Y)), zip(XS, YS))
1142.05/293.67	   a__tail(cons(X, XS)) -> mark(XS)
1142.05/293.67	   a__repItems(nil) -> nil
1142.05/293.67	   a__repItems(cons(X, XS)) -> cons(mark(X), cons(X, repItems(XS)))
1142.05/293.67	   mark(pairNs) -> a__pairNs
1142.05/293.67	   mark(incr(X)) -> a__incr(mark(X))
1142.05/293.67	   mark(oddNs) -> a__oddNs
1142.05/293.67	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
1142.05/293.67	   mark(zip(X1, X2)) -> a__zip(mark(X1), mark(X2))
1142.05/293.67	   mark(tail(X)) -> a__tail(mark(X))
1142.05/293.67	   mark(repItems(X)) -> a__repItems(mark(X))
1142.05/293.67	   mark(cons(X1, X2)) -> cons(mark(X1), X2)
1142.05/293.67	   mark(0) -> 0
1142.05/293.67	   mark(s(X)) -> s(mark(X))
1142.05/293.67	   mark(nil) -> nil
1142.05/293.67	   mark(pair(X1, X2)) -> pair(mark(X1), mark(X2))
1142.05/293.67	   a__pairNs -> pairNs
1142.05/293.67	   a__incr(X) -> incr(X)
1142.05/293.67	   a__oddNs -> oddNs
1142.05/293.67	   a__take(X1, X2) -> take(X1, X2)
1142.05/293.67	   a__zip(X1, X2) -> zip(X1, X2)
1142.05/293.67	   a__tail(X) -> tail(X)
1142.05/293.67	   a__repItems(X) -> repItems(X)
1142.05/293.67	
1142.05/293.67	S is empty.
1142.05/293.67	Rewrite Strategy: INNERMOST
1142.26/293.74	EOF