316.50/291.50	WORST_CASE(Omega(n^1), ?)
316.50/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
316.50/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
316.50/291.50	
316.50/291.50	
316.50/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.50/291.50	
316.50/291.50	(0) CpxTRS
316.50/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
316.50/291.50	(2) TRS for Loop Detection
316.50/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
316.50/291.50	(4) BEST
316.50/291.50	    (5) proven lower bound
316.50/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
316.50/291.50	        (7) BOUNDS(n^1, INF)
316.50/291.50	    (8) TRS for Loop Detection
316.50/291.50	
316.50/291.50	
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(0)
316.50/291.50	Obligation:
316.50/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.50/291.50	
316.50/291.50	
316.50/291.50	The TRS R consists of the following rules:
316.50/291.50	
316.50/291.50	   from(X) -> cons(X, n__from(s(X)))
316.50/291.50	   sel(0, cons(X, Y)) -> X
316.50/291.50	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
316.50/291.50	   from(X) -> n__from(X)
316.50/291.50	   activate(n__from(X)) -> from(X)
316.50/291.50	   activate(X) -> X
316.50/291.50	
316.50/291.50	S is empty.
316.50/291.50	Rewrite Strategy: FULL
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
316.50/291.50	Transformed a relative TRS into a decreasing-loop problem.
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(2)
316.50/291.50	Obligation:
316.50/291.50	Analyzing the following TRS for decreasing loops:
316.50/291.50	
316.50/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.50/291.50	
316.50/291.50	
316.50/291.50	The TRS R consists of the following rules:
316.50/291.50	
316.50/291.50	   from(X) -> cons(X, n__from(s(X)))
316.50/291.50	   sel(0, cons(X, Y)) -> X
316.50/291.50	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
316.50/291.50	   from(X) -> n__from(X)
316.50/291.50	   activate(n__from(X)) -> from(X)
316.50/291.50	   activate(X) -> X
316.50/291.50	
316.50/291.50	S is empty.
316.50/291.50	Rewrite Strategy: FULL
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
316.50/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
316.50/291.50	
316.50/291.50	The rewrite sequence
316.50/291.50	
316.50/291.50	sel(s(X), cons(Y, Z)) ->^+ sel(X, Z)
316.50/291.50	
316.50/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
316.50/291.50	
316.50/291.50	The pumping substitution is [X / s(X), Z / cons(Y, Z)].
316.50/291.50	
316.50/291.50	The result substitution is [ ].
316.50/291.50	
316.50/291.50	
316.50/291.50	
316.50/291.50	
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(4)
316.50/291.50	Complex Obligation (BEST)
316.50/291.50	
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(5)
316.50/291.50	Obligation:
316.50/291.50	Proved the lower bound n^1 for the following obligation:
316.50/291.50	
316.50/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.50/291.50	
316.50/291.50	
316.50/291.50	The TRS R consists of the following rules:
316.50/291.50	
316.50/291.50	   from(X) -> cons(X, n__from(s(X)))
316.50/291.50	   sel(0, cons(X, Y)) -> X
316.50/291.50	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
316.50/291.50	   from(X) -> n__from(X)
316.50/291.50	   activate(n__from(X)) -> from(X)
316.50/291.50	   activate(X) -> X
316.50/291.50	
316.50/291.50	S is empty.
316.50/291.50	Rewrite Strategy: FULL
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(6) LowerBoundPropagationProof (FINISHED)
316.50/291.50	Propagated lower bound.
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(7)
316.50/291.50	BOUNDS(n^1, INF)
316.50/291.50	
316.50/291.50	----------------------------------------
316.50/291.50	
316.50/291.50	(8)
316.50/291.50	Obligation:
316.50/291.50	Analyzing the following TRS for decreasing loops:
316.50/291.50	
316.50/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.50/291.50	
316.50/291.50	
316.50/291.50	The TRS R consists of the following rules:
316.50/291.50	
316.50/291.50	   from(X) -> cons(X, n__from(s(X)))
316.50/291.50	   sel(0, cons(X, Y)) -> X
316.50/291.50	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
316.50/291.51	   from(X) -> n__from(X)
316.50/291.51	   activate(n__from(X)) -> from(X)
316.50/291.51	   activate(X) -> X
316.50/291.51	
316.50/291.51	S is empty.
316.50/291.51	Rewrite Strategy: FULL
316.50/291.53	EOF