3.35/1.71	WORST_CASE(NON_POLY, ?)
3.35/1.72	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.35/1.72	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.35/1.72	
3.35/1.72	
3.35/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.35/1.72	
3.35/1.72	(0) CpxTRS
3.35/1.72	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.35/1.72	(2) TRS for Loop Detection
3.35/1.72	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.35/1.72	(4) BEST
3.35/1.72	    (5) proven lower bound
3.35/1.72	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.35/1.72	        (7) BOUNDS(n^1, INF)
3.35/1.72	    (8) TRS for Loop Detection
3.35/1.72	        (9) DecreasingLoopProof [FINISHED, 16 ms]
3.35/1.72	        (10) BOUNDS(EXP, INF)
3.35/1.72	
3.35/1.72	
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(0)
3.35/1.72	Obligation:
3.35/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.35/1.72	
3.35/1.72	
3.35/1.72	The TRS R consists of the following rules:
3.35/1.72	
3.35/1.72	   U11(tt, N, X, XS) -> U12(splitAt(activate(N), activate(XS)), activate(X))
3.35/1.72	   U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.35/1.72	   afterNth(N, XS) -> snd(splitAt(N, XS))
3.35/1.72	   and(tt, X) -> activate(X)
3.35/1.72	   fst(pair(X, Y)) -> X
3.35/1.72	   head(cons(N, XS)) -> N
3.35/1.72	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.35/1.72	   sel(N, XS) -> head(afterNth(N, XS))
3.35/1.72	   snd(pair(X, Y)) -> Y
3.35/1.72	   splitAt(0, XS) -> pair(nil, XS)
3.35/1.72	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, activate(XS))
3.35/1.72	   tail(cons(N, XS)) -> activate(XS)
3.35/1.72	   take(N, XS) -> fst(splitAt(N, XS))
3.35/1.72	   natsFrom(X) -> n__natsFrom(X)
3.35/1.72	   s(X) -> n__s(X)
3.35/1.72	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.35/1.72	   activate(n__s(X)) -> s(activate(X))
3.35/1.72	   activate(X) -> X
3.35/1.72	
3.35/1.72	S is empty.
3.35/1.72	Rewrite Strategy: FULL
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.35/1.72	Transformed a relative TRS into a decreasing-loop problem.
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(2)
3.35/1.72	Obligation:
3.35/1.72	Analyzing the following TRS for decreasing loops:
3.35/1.72	
3.35/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.35/1.72	
3.35/1.72	
3.35/1.72	The TRS R consists of the following rules:
3.35/1.72	
3.35/1.72	   U11(tt, N, X, XS) -> U12(splitAt(activate(N), activate(XS)), activate(X))
3.35/1.72	   U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.35/1.72	   afterNth(N, XS) -> snd(splitAt(N, XS))
3.35/1.72	   and(tt, X) -> activate(X)
3.35/1.72	   fst(pair(X, Y)) -> X
3.35/1.72	   head(cons(N, XS)) -> N
3.35/1.72	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.35/1.72	   sel(N, XS) -> head(afterNth(N, XS))
3.35/1.72	   snd(pair(X, Y)) -> Y
3.35/1.72	   splitAt(0, XS) -> pair(nil, XS)
3.35/1.72	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, activate(XS))
3.35/1.72	   tail(cons(N, XS)) -> activate(XS)
3.35/1.72	   take(N, XS) -> fst(splitAt(N, XS))
3.35/1.72	   natsFrom(X) -> n__natsFrom(X)
3.35/1.72	   s(X) -> n__s(X)
3.35/1.72	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.35/1.72	   activate(n__s(X)) -> s(activate(X))
3.35/1.72	   activate(X) -> X
3.35/1.72	
3.35/1.72	S is empty.
3.35/1.72	Rewrite Strategy: FULL
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.35/1.72	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.35/1.72	
3.35/1.72	The rewrite sequence
3.35/1.72	
3.35/1.72	activate(n__s(X)) ->^+ s(activate(X))
3.35/1.72	
3.35/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.35/1.72	
3.35/1.72	The pumping substitution is [X / n__s(X)].
3.35/1.72	
3.35/1.72	The result substitution is [ ].
3.35/1.72	
3.35/1.72	
3.35/1.72	
3.35/1.72	
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(4)
3.35/1.72	Complex Obligation (BEST)
3.35/1.72	
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(5)
3.35/1.72	Obligation:
3.35/1.72	Proved the lower bound n^1 for the following obligation:
3.35/1.72	
3.35/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.35/1.72	
3.35/1.72	
3.35/1.72	The TRS R consists of the following rules:
3.35/1.72	
3.35/1.72	   U11(tt, N, X, XS) -> U12(splitAt(activate(N), activate(XS)), activate(X))
3.35/1.72	   U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.35/1.72	   afterNth(N, XS) -> snd(splitAt(N, XS))
3.35/1.72	   and(tt, X) -> activate(X)
3.35/1.72	   fst(pair(X, Y)) -> X
3.35/1.72	   head(cons(N, XS)) -> N
3.35/1.72	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.35/1.72	   sel(N, XS) -> head(afterNth(N, XS))
3.35/1.72	   snd(pair(X, Y)) -> Y
3.35/1.72	   splitAt(0, XS) -> pair(nil, XS)
3.35/1.72	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, activate(XS))
3.35/1.72	   tail(cons(N, XS)) -> activate(XS)
3.35/1.72	   take(N, XS) -> fst(splitAt(N, XS))
3.35/1.72	   natsFrom(X) -> n__natsFrom(X)
3.35/1.72	   s(X) -> n__s(X)
3.35/1.72	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.35/1.72	   activate(n__s(X)) -> s(activate(X))
3.35/1.72	   activate(X) -> X
3.35/1.72	
3.35/1.72	S is empty.
3.35/1.72	Rewrite Strategy: FULL
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(6) LowerBoundPropagationProof (FINISHED)
3.35/1.72	Propagated lower bound.
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(7)
3.35/1.72	BOUNDS(n^1, INF)
3.35/1.72	
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(8)
3.35/1.72	Obligation:
3.35/1.72	Analyzing the following TRS for decreasing loops:
3.35/1.72	
3.35/1.72	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.35/1.72	
3.35/1.72	
3.35/1.72	The TRS R consists of the following rules:
3.35/1.72	
3.35/1.72	   U11(tt, N, X, XS) -> U12(splitAt(activate(N), activate(XS)), activate(X))
3.35/1.72	   U12(pair(YS, ZS), X) -> pair(cons(activate(X), YS), ZS)
3.35/1.72	   afterNth(N, XS) -> snd(splitAt(N, XS))
3.35/1.72	   and(tt, X) -> activate(X)
3.35/1.72	   fst(pair(X, Y)) -> X
3.35/1.72	   head(cons(N, XS)) -> N
3.35/1.72	   natsFrom(N) -> cons(N, n__natsFrom(n__s(N)))
3.35/1.72	   sel(N, XS) -> head(afterNth(N, XS))
3.35/1.72	   snd(pair(X, Y)) -> Y
3.35/1.72	   splitAt(0, XS) -> pair(nil, XS)
3.35/1.72	   splitAt(s(N), cons(X, XS)) -> U11(tt, N, X, activate(XS))
3.35/1.72	   tail(cons(N, XS)) -> activate(XS)
3.35/1.72	   take(N, XS) -> fst(splitAt(N, XS))
3.35/1.72	   natsFrom(X) -> n__natsFrom(X)
3.35/1.72	   s(X) -> n__s(X)
3.35/1.72	   activate(n__natsFrom(X)) -> natsFrom(activate(X))
3.35/1.72	   activate(n__s(X)) -> s(activate(X))
3.35/1.72	   activate(X) -> X
3.35/1.72	
3.35/1.72	S is empty.
3.35/1.72	Rewrite Strategy: FULL
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(9) DecreasingLoopProof (FINISHED)
3.35/1.72	The following loop(s) give(s) rise to the lower bound EXP:
3.35/1.72	
3.35/1.72	The rewrite sequence
3.35/1.72	
3.35/1.72	activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X))))
3.35/1.72	
3.35/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.35/1.72	
3.35/1.72	The pumping substitution is [X / n__natsFrom(X)].
3.35/1.72	
3.35/1.72	The result substitution is [ ].
3.35/1.72	
3.35/1.72	
3.35/1.72	
3.35/1.72	The rewrite sequence
3.35/1.72	
3.35/1.72	activate(n__natsFrom(X)) ->^+ cons(activate(X), n__natsFrom(n__s(activate(X))))
3.35/1.72	
3.35/1.72	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.35/1.72	
3.35/1.72	The pumping substitution is [X / n__natsFrom(X)].
3.35/1.72	
3.35/1.72	The result substitution is [ ].
3.35/1.72	
3.35/1.72	
3.35/1.72	
3.35/1.72	
3.35/1.72	----------------------------------------
3.35/1.72	
3.35/1.72	(10)
3.35/1.72	BOUNDS(EXP, INF)
3.35/1.76	EOF