306.66/291.51	WORST_CASE(Omega(n^1), ?)
306.66/291.52	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
306.66/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
306.66/291.52	
306.66/291.52	
306.66/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.66/291.52	
306.66/291.52	(0) CpxTRS
306.66/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
306.66/291.52	(2) TRS for Loop Detection
306.66/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
306.66/291.52	(4) BEST
306.66/291.52	    (5) proven lower bound
306.66/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
306.66/291.52	        (7) BOUNDS(n^1, INF)
306.66/291.52	    (8) TRS for Loop Detection
306.66/291.52	
306.66/291.52	
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(0)
306.66/291.52	Obligation:
306.66/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.66/291.52	
306.66/291.52	
306.66/291.52	The TRS R consists of the following rules:
306.66/291.52	
306.66/291.52	   from(X) -> cons(X, n__from(s(X)))
306.66/291.52	   after(0, XS) -> XS
306.66/291.52	   after(s(N), cons(X, XS)) -> after(N, activate(XS))
306.66/291.52	   from(X) -> n__from(X)
306.66/291.52	   activate(n__from(X)) -> from(X)
306.66/291.52	   activate(X) -> X
306.66/291.52	
306.66/291.52	S is empty.
306.66/291.52	Rewrite Strategy: FULL
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
306.66/291.52	Transformed a relative TRS into a decreasing-loop problem.
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(2)
306.66/291.52	Obligation:
306.66/291.52	Analyzing the following TRS for decreasing loops:
306.66/291.52	
306.66/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.66/291.52	
306.66/291.52	
306.66/291.52	The TRS R consists of the following rules:
306.66/291.52	
306.66/291.52	   from(X) -> cons(X, n__from(s(X)))
306.66/291.52	   after(0, XS) -> XS
306.66/291.52	   after(s(N), cons(X, XS)) -> after(N, activate(XS))
306.66/291.52	   from(X) -> n__from(X)
306.66/291.52	   activate(n__from(X)) -> from(X)
306.66/291.52	   activate(X) -> X
306.66/291.52	
306.66/291.52	S is empty.
306.66/291.52	Rewrite Strategy: FULL
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
306.66/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
306.66/291.52	
306.66/291.52	The rewrite sequence
306.66/291.52	
306.66/291.52	after(s(N), cons(X, XS)) ->^+ after(N, XS)
306.66/291.52	
306.66/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
306.66/291.52	
306.66/291.52	The pumping substitution is [N / s(N), XS / cons(X, XS)].
306.66/291.52	
306.66/291.52	The result substitution is [ ].
306.66/291.52	
306.66/291.52	
306.66/291.52	
306.66/291.52	
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(4)
306.66/291.52	Complex Obligation (BEST)
306.66/291.52	
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(5)
306.66/291.52	Obligation:
306.66/291.52	Proved the lower bound n^1 for the following obligation:
306.66/291.52	
306.66/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.66/291.52	
306.66/291.52	
306.66/291.52	The TRS R consists of the following rules:
306.66/291.52	
306.66/291.52	   from(X) -> cons(X, n__from(s(X)))
306.66/291.52	   after(0, XS) -> XS
306.66/291.52	   after(s(N), cons(X, XS)) -> after(N, activate(XS))
306.66/291.52	   from(X) -> n__from(X)
306.66/291.52	   activate(n__from(X)) -> from(X)
306.66/291.52	   activate(X) -> X
306.66/291.52	
306.66/291.52	S is empty.
306.66/291.52	Rewrite Strategy: FULL
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(6) LowerBoundPropagationProof (FINISHED)
306.66/291.52	Propagated lower bound.
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(7)
306.66/291.52	BOUNDS(n^1, INF)
306.66/291.52	
306.66/291.52	----------------------------------------
306.66/291.52	
306.66/291.52	(8)
306.66/291.52	Obligation:
306.66/291.52	Analyzing the following TRS for decreasing loops:
306.66/291.52	
306.66/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.66/291.52	
306.66/291.52	
306.66/291.52	The TRS R consists of the following rules:
306.66/291.52	
306.66/291.52	   from(X) -> cons(X, n__from(s(X)))
306.66/291.52	   after(0, XS) -> XS
306.66/291.52	   after(s(N), cons(X, XS)) -> after(N, activate(XS))
306.66/291.52	   from(X) -> n__from(X)
306.66/291.52	   activate(n__from(X)) -> from(X)
306.66/291.52	   activate(X) -> X
306.66/291.52	
306.66/291.52	S is empty.
306.66/291.52	Rewrite Strategy: FULL
306.66/291.55	EOF