337.12/291.55	WORST_CASE(Omega(n^1), ?)
337.20/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
337.20/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
337.20/291.56	
337.20/291.56	
337.20/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
337.20/291.56	
337.20/291.56	(0) CpxTRS
337.20/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
337.20/291.56	(2) TRS for Loop Detection
337.20/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 22 ms]
337.20/291.56	(4) BEST
337.20/291.56	    (5) proven lower bound
337.20/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
337.20/291.56	        (7) BOUNDS(n^1, INF)
337.20/291.56	    (8) TRS for Loop Detection
337.20/291.56	
337.20/291.56	
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(0)
337.20/291.56	Obligation:
337.20/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
337.20/291.56	
337.20/291.56	
337.20/291.56	The TRS R consists of the following rules:
337.20/291.56	
337.20/291.56	   from(X) -> cons(X, n__from(s(X)))
337.20/291.56	   head(cons(X, XS)) -> X
337.20/291.56	   2nd(cons(X, XS)) -> head(activate(XS))
337.20/291.56	   take(0, XS) -> nil
337.20/291.56	   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
337.20/291.56	   sel(0, cons(X, XS)) -> X
337.20/291.56	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
337.20/291.56	   from(X) -> n__from(X)
337.20/291.56	   take(X1, X2) -> n__take(X1, X2)
337.20/291.56	   activate(n__from(X)) -> from(X)
337.20/291.56	   activate(n__take(X1, X2)) -> take(X1, X2)
337.20/291.56	   activate(X) -> X
337.20/291.56	
337.20/291.56	S is empty.
337.20/291.56	Rewrite Strategy: FULL
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
337.20/291.56	Transformed a relative TRS into a decreasing-loop problem.
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(2)
337.20/291.56	Obligation:
337.20/291.56	Analyzing the following TRS for decreasing loops:
337.20/291.56	
337.20/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
337.20/291.56	
337.20/291.56	
337.20/291.56	The TRS R consists of the following rules:
337.20/291.56	
337.20/291.56	   from(X) -> cons(X, n__from(s(X)))
337.20/291.56	   head(cons(X, XS)) -> X
337.20/291.56	   2nd(cons(X, XS)) -> head(activate(XS))
337.20/291.56	   take(0, XS) -> nil
337.20/291.56	   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
337.20/291.56	   sel(0, cons(X, XS)) -> X
337.20/291.56	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
337.20/291.56	   from(X) -> n__from(X)
337.20/291.56	   take(X1, X2) -> n__take(X1, X2)
337.20/291.56	   activate(n__from(X)) -> from(X)
337.20/291.56	   activate(n__take(X1, X2)) -> take(X1, X2)
337.20/291.56	   activate(X) -> X
337.20/291.56	
337.20/291.56	S is empty.
337.20/291.56	Rewrite Strategy: FULL
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
337.20/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
337.20/291.56	
337.20/291.56	The rewrite sequence
337.20/291.56	
337.20/291.56	sel(s(N), cons(X, XS)) ->^+ sel(N, XS)
337.20/291.56	
337.20/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
337.20/291.56	
337.20/291.56	The pumping substitution is [N / s(N), XS / cons(X, XS)].
337.20/291.56	
337.20/291.56	The result substitution is [ ].
337.20/291.56	
337.20/291.56	
337.20/291.56	
337.20/291.56	
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(4)
337.20/291.56	Complex Obligation (BEST)
337.20/291.56	
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(5)
337.20/291.56	Obligation:
337.20/291.56	Proved the lower bound n^1 for the following obligation:
337.20/291.56	
337.20/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
337.20/291.56	
337.20/291.56	
337.20/291.56	The TRS R consists of the following rules:
337.20/291.56	
337.20/291.56	   from(X) -> cons(X, n__from(s(X)))
337.20/291.56	   head(cons(X, XS)) -> X
337.20/291.56	   2nd(cons(X, XS)) -> head(activate(XS))
337.20/291.56	   take(0, XS) -> nil
337.20/291.56	   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
337.20/291.56	   sel(0, cons(X, XS)) -> X
337.20/291.56	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
337.20/291.56	   from(X) -> n__from(X)
337.20/291.56	   take(X1, X2) -> n__take(X1, X2)
337.20/291.56	   activate(n__from(X)) -> from(X)
337.20/291.56	   activate(n__take(X1, X2)) -> take(X1, X2)
337.20/291.56	   activate(X) -> X
337.20/291.56	
337.20/291.56	S is empty.
337.20/291.56	Rewrite Strategy: FULL
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(6) LowerBoundPropagationProof (FINISHED)
337.20/291.56	Propagated lower bound.
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(7)
337.20/291.56	BOUNDS(n^1, INF)
337.20/291.56	
337.20/291.56	----------------------------------------
337.20/291.56	
337.20/291.56	(8)
337.20/291.56	Obligation:
337.20/291.56	Analyzing the following TRS for decreasing loops:
337.20/291.56	
337.20/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
337.20/291.56	
337.20/291.56	
337.20/291.56	The TRS R consists of the following rules:
337.20/291.56	
337.20/291.56	   from(X) -> cons(X, n__from(s(X)))
337.20/291.56	   head(cons(X, XS)) -> X
337.20/291.56	   2nd(cons(X, XS)) -> head(activate(XS))
337.20/291.56	   take(0, XS) -> nil
337.20/291.56	   take(s(N), cons(X, XS)) -> cons(X, n__take(N, activate(XS)))
337.20/291.56	   sel(0, cons(X, XS)) -> X
337.20/291.56	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
337.20/291.56	   from(X) -> n__from(X)
337.20/291.56	   take(X1, X2) -> n__take(X1, X2)
337.20/291.56	   activate(n__from(X)) -> from(X)
337.20/291.56	   activate(n__take(X1, X2)) -> take(X1, X2)
337.20/291.56	   activate(X) -> X
337.20/291.56	
337.20/291.56	S is empty.
337.20/291.56	Rewrite Strategy: FULL
337.21/291.60	EOF