303.73/291.51	WORST_CASE(Omega(n^1), ?)
303.73/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
303.73/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.73/291.51	
303.73/291.51	
303.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.73/291.51	
303.73/291.51	(0) CpxTRS
303.73/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.73/291.51	(2) TRS for Loop Detection
303.73/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 34 ms]
303.73/291.51	(4) BEST
303.73/291.51	    (5) proven lower bound
303.73/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.73/291.51	        (7) BOUNDS(n^1, INF)
303.73/291.51	    (8) TRS for Loop Detection
303.73/291.51	
303.73/291.51	
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(0)
303.73/291.51	Obligation:
303.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.73/291.51	
303.73/291.51	
303.73/291.51	The TRS R consists of the following rules:
303.73/291.51	
303.73/291.51	   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
303.73/291.51	   fst(pair(XS, YS)) -> XS
303.73/291.51	   snd(pair(XS, YS)) -> YS
303.73/291.51	   splitAt(0, XS) -> pair(nil, XS)
303.73/291.51	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
303.73/291.51	   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
303.73/291.51	   head(cons(N, XS)) -> N
303.73/291.51	   tail(cons(N, XS)) -> activate(XS)
303.73/291.51	   sel(N, XS) -> head(afterNth(N, XS))
303.73/291.51	   take(N, XS) -> fst(splitAt(N, XS))
303.73/291.51	   afterNth(N, XS) -> snd(splitAt(N, XS))
303.73/291.51	   natsFrom(X) -> n__natsFrom(X)
303.73/291.51	   activate(n__natsFrom(X)) -> natsFrom(X)
303.73/291.51	   activate(X) -> X
303.73/291.51	
303.73/291.51	S is empty.
303.73/291.51	Rewrite Strategy: FULL
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.73/291.51	Transformed a relative TRS into a decreasing-loop problem.
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(2)
303.73/291.51	Obligation:
303.73/291.51	Analyzing the following TRS for decreasing loops:
303.73/291.51	
303.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.73/291.51	
303.73/291.51	
303.73/291.51	The TRS R consists of the following rules:
303.73/291.51	
303.73/291.51	   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
303.73/291.51	   fst(pair(XS, YS)) -> XS
303.73/291.51	   snd(pair(XS, YS)) -> YS
303.73/291.51	   splitAt(0, XS) -> pair(nil, XS)
303.73/291.51	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
303.73/291.51	   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
303.73/291.51	   head(cons(N, XS)) -> N
303.73/291.51	   tail(cons(N, XS)) -> activate(XS)
303.73/291.51	   sel(N, XS) -> head(afterNth(N, XS))
303.73/291.51	   take(N, XS) -> fst(splitAt(N, XS))
303.73/291.51	   afterNth(N, XS) -> snd(splitAt(N, XS))
303.73/291.51	   natsFrom(X) -> n__natsFrom(X)
303.73/291.51	   activate(n__natsFrom(X)) -> natsFrom(X)
303.73/291.51	   activate(X) -> X
303.73/291.51	
303.73/291.51	S is empty.
303.73/291.51	Rewrite Strategy: FULL
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.73/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.73/291.51	
303.73/291.51	The rewrite sequence
303.73/291.51	
303.73/291.51	splitAt(s(N), cons(X, XS)) ->^+ u(splitAt(N, XS), N, X, activate(XS))
303.73/291.51	
303.73/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
303.73/291.51	
303.73/291.51	The pumping substitution is [N / s(N), XS / cons(X, XS)].
303.73/291.51	
303.73/291.51	The result substitution is [ ].
303.73/291.51	
303.73/291.51	
303.73/291.51	
303.73/291.51	
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(4)
303.73/291.51	Complex Obligation (BEST)
303.73/291.51	
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(5)
303.73/291.51	Obligation:
303.73/291.51	Proved the lower bound n^1 for the following obligation:
303.73/291.51	
303.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.73/291.51	
303.73/291.51	
303.73/291.51	The TRS R consists of the following rules:
303.73/291.51	
303.73/291.51	   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
303.73/291.51	   fst(pair(XS, YS)) -> XS
303.73/291.51	   snd(pair(XS, YS)) -> YS
303.73/291.51	   splitAt(0, XS) -> pair(nil, XS)
303.73/291.51	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
303.73/291.51	   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
303.73/291.51	   head(cons(N, XS)) -> N
303.73/291.51	   tail(cons(N, XS)) -> activate(XS)
303.73/291.51	   sel(N, XS) -> head(afterNth(N, XS))
303.73/291.51	   take(N, XS) -> fst(splitAt(N, XS))
303.73/291.51	   afterNth(N, XS) -> snd(splitAt(N, XS))
303.73/291.51	   natsFrom(X) -> n__natsFrom(X)
303.73/291.51	   activate(n__natsFrom(X)) -> natsFrom(X)
303.73/291.51	   activate(X) -> X
303.73/291.51	
303.73/291.51	S is empty.
303.73/291.51	Rewrite Strategy: FULL
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(6) LowerBoundPropagationProof (FINISHED)
303.73/291.51	Propagated lower bound.
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(7)
303.73/291.51	BOUNDS(n^1, INF)
303.73/291.51	
303.73/291.51	----------------------------------------
303.73/291.51	
303.73/291.51	(8)
303.73/291.51	Obligation:
303.73/291.51	Analyzing the following TRS for decreasing loops:
303.73/291.51	
303.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.73/291.51	
303.73/291.51	
303.73/291.51	The TRS R consists of the following rules:
303.73/291.51	
303.73/291.51	   natsFrom(N) -> cons(N, n__natsFrom(s(N)))
303.73/291.51	   fst(pair(XS, YS)) -> XS
303.73/291.51	   snd(pair(XS, YS)) -> YS
303.73/291.51	   splitAt(0, XS) -> pair(nil, XS)
303.73/291.51	   splitAt(s(N), cons(X, XS)) -> u(splitAt(N, activate(XS)), N, X, activate(XS))
303.73/291.51	   u(pair(YS, ZS), N, X, XS) -> pair(cons(activate(X), YS), ZS)
303.73/291.51	   head(cons(N, XS)) -> N
303.73/291.51	   tail(cons(N, XS)) -> activate(XS)
303.73/291.51	   sel(N, XS) -> head(afterNth(N, XS))
303.73/291.51	   take(N, XS) -> fst(splitAt(N, XS))
303.73/291.51	   afterNth(N, XS) -> snd(splitAt(N, XS))
303.73/291.51	   natsFrom(X) -> n__natsFrom(X)
303.73/291.51	   activate(n__natsFrom(X)) -> natsFrom(X)
303.73/291.51	   activate(X) -> X
303.73/291.51	
303.73/291.51	S is empty.
303.73/291.51	Rewrite Strategy: FULL
303.73/291.54	EOF