319.88/291.54	WORST_CASE(Omega(n^1), ?)
319.88/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
319.88/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
319.88/291.54	
319.88/291.54	
319.88/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
319.88/291.54	
319.88/291.54	(0) CpxTRS
319.88/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
319.88/291.54	(2) TRS for Loop Detection
319.88/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
319.88/291.54	(4) BEST
319.88/291.54	    (5) proven lower bound
319.88/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
319.88/291.54	        (7) BOUNDS(n^1, INF)
319.88/291.54	    (8) TRS for Loop Detection
319.88/291.54	
319.88/291.54	
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(0)
319.88/291.54	Obligation:
319.88/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
319.88/291.54	
319.88/291.54	
319.88/291.54	The TRS R consists of the following rules:
319.88/291.54	
319.88/291.54	   from(X) -> cons(X, n__from(s(X)))
319.88/291.54	   first(0, Z) -> nil
319.88/291.54	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
319.88/291.54	   sel(0, cons(X, Z)) -> X
319.88/291.54	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
319.88/291.54	   from(X) -> n__from(X)
319.88/291.54	   first(X1, X2) -> n__first(X1, X2)
319.88/291.54	   activate(n__from(X)) -> from(X)
319.88/291.54	   activate(n__first(X1, X2)) -> first(X1, X2)
319.88/291.54	   activate(X) -> X
319.88/291.54	
319.88/291.54	S is empty.
319.88/291.54	Rewrite Strategy: FULL
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
319.88/291.54	Transformed a relative TRS into a decreasing-loop problem.
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(2)
319.88/291.54	Obligation:
319.88/291.54	Analyzing the following TRS for decreasing loops:
319.88/291.54	
319.88/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
319.88/291.54	
319.88/291.54	
319.88/291.54	The TRS R consists of the following rules:
319.88/291.54	
319.88/291.54	   from(X) -> cons(X, n__from(s(X)))
319.88/291.54	   first(0, Z) -> nil
319.88/291.54	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
319.88/291.54	   sel(0, cons(X, Z)) -> X
319.88/291.54	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
319.88/291.54	   from(X) -> n__from(X)
319.88/291.54	   first(X1, X2) -> n__first(X1, X2)
319.88/291.54	   activate(n__from(X)) -> from(X)
319.88/291.54	   activate(n__first(X1, X2)) -> first(X1, X2)
319.88/291.54	   activate(X) -> X
319.88/291.54	
319.88/291.54	S is empty.
319.88/291.54	Rewrite Strategy: FULL
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
319.88/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
319.88/291.54	
319.88/291.54	The rewrite sequence
319.88/291.54	
319.88/291.54	sel(s(X), cons(Y, Z)) ->^+ sel(X, Z)
319.88/291.54	
319.88/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
319.88/291.54	
319.88/291.54	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
319.88/291.54	
319.88/291.54	The result substitution is [ ].
319.88/291.54	
319.88/291.54	
319.88/291.54	
319.88/291.54	
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(4)
319.88/291.54	Complex Obligation (BEST)
319.88/291.54	
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(5)
319.88/291.54	Obligation:
319.88/291.54	Proved the lower bound n^1 for the following obligation:
319.88/291.54	
319.88/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
319.88/291.54	
319.88/291.54	
319.88/291.54	The TRS R consists of the following rules:
319.88/291.54	
319.88/291.54	   from(X) -> cons(X, n__from(s(X)))
319.88/291.54	   first(0, Z) -> nil
319.88/291.54	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
319.88/291.54	   sel(0, cons(X, Z)) -> X
319.88/291.54	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
319.88/291.54	   from(X) -> n__from(X)
319.88/291.54	   first(X1, X2) -> n__first(X1, X2)
319.88/291.54	   activate(n__from(X)) -> from(X)
319.88/291.54	   activate(n__first(X1, X2)) -> first(X1, X2)
319.88/291.54	   activate(X) -> X
319.88/291.54	
319.88/291.54	S is empty.
319.88/291.54	Rewrite Strategy: FULL
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(6) LowerBoundPropagationProof (FINISHED)
319.88/291.54	Propagated lower bound.
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(7)
319.88/291.54	BOUNDS(n^1, INF)
319.88/291.54	
319.88/291.54	----------------------------------------
319.88/291.54	
319.88/291.54	(8)
319.88/291.54	Obligation:
319.88/291.54	Analyzing the following TRS for decreasing loops:
319.88/291.54	
319.88/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
319.88/291.54	
319.88/291.54	
319.88/291.54	The TRS R consists of the following rules:
319.88/291.54	
319.88/291.54	   from(X) -> cons(X, n__from(s(X)))
319.88/291.54	   first(0, Z) -> nil
319.88/291.54	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
319.88/291.54	   sel(0, cons(X, Z)) -> X
319.88/291.54	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
319.88/291.54	   from(X) -> n__from(X)
319.88/291.54	   first(X1, X2) -> n__first(X1, X2)
319.88/291.54	   activate(n__from(X)) -> from(X)
319.88/291.54	   activate(n__first(X1, X2)) -> first(X1, X2)
319.88/291.54	   activate(X) -> X
319.88/291.54	
319.88/291.54	S is empty.
319.88/291.54	Rewrite Strategy: FULL
319.93/291.58	EOF