3.24/1.56	WORST_CASE(NON_POLY, ?)
3.24/1.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.24/1.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.24/1.57	
3.24/1.57	
3.24/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.57	
3.24/1.57	(0) CpxTRS
3.24/1.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.24/1.57	(2) TRS for Loop Detection
3.24/1.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.24/1.57	(4) BEST
3.24/1.57	    (5) proven lower bound
3.24/1.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.24/1.57	        (7) BOUNDS(n^1, INF)
3.24/1.57	    (8) TRS for Loop Detection
3.24/1.57	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.24/1.57	        (10) BOUNDS(EXP, INF)
3.24/1.57	
3.24/1.57	
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(0)
3.24/1.57	Obligation:
3.24/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.57	
3.24/1.57	
3.24/1.57	The TRS R consists of the following rules:
3.24/1.57	
3.24/1.57	   first(0, X) -> nil
3.24/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.24/1.57	   from(X) -> cons(X, n__from(n__s(X)))
3.24/1.57	   first(X1, X2) -> n__first(X1, X2)
3.24/1.57	   from(X) -> n__from(X)
3.24/1.57	   s(X) -> n__s(X)
3.24/1.57	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.24/1.57	   activate(n__from(X)) -> from(activate(X))
3.24/1.57	   activate(n__s(X)) -> s(activate(X))
3.24/1.57	   activate(X) -> X
3.24/1.57	
3.24/1.57	S is empty.
3.24/1.57	Rewrite Strategy: FULL
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.24/1.57	Transformed a relative TRS into a decreasing-loop problem.
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(2)
3.24/1.57	Obligation:
3.24/1.57	Analyzing the following TRS for decreasing loops:
3.24/1.57	
3.24/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.57	
3.24/1.57	
3.24/1.57	The TRS R consists of the following rules:
3.24/1.57	
3.24/1.57	   first(0, X) -> nil
3.24/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.24/1.57	   from(X) -> cons(X, n__from(n__s(X)))
3.24/1.57	   first(X1, X2) -> n__first(X1, X2)
3.24/1.57	   from(X) -> n__from(X)
3.24/1.57	   s(X) -> n__s(X)
3.24/1.57	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.24/1.57	   activate(n__from(X)) -> from(activate(X))
3.24/1.57	   activate(n__s(X)) -> s(activate(X))
3.24/1.57	   activate(X) -> X
3.24/1.57	
3.24/1.57	S is empty.
3.24/1.57	Rewrite Strategy: FULL
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.24/1.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.24/1.57	
3.24/1.57	The rewrite sequence
3.24/1.57	
3.24/1.57	activate(n__first(X1, X2)) ->^+ first(activate(X1), activate(X2))
3.24/1.57	
3.24/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.24/1.57	
3.24/1.57	The pumping substitution is [X1 / n__first(X1, X2)].
3.24/1.57	
3.24/1.57	The result substitution is [ ].
3.24/1.57	
3.24/1.57	
3.24/1.57	
3.24/1.57	
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(4)
3.24/1.57	Complex Obligation (BEST)
3.24/1.57	
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(5)
3.24/1.57	Obligation:
3.24/1.57	Proved the lower bound n^1 for the following obligation:
3.24/1.57	
3.24/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.57	
3.24/1.57	
3.24/1.57	The TRS R consists of the following rules:
3.24/1.57	
3.24/1.57	   first(0, X) -> nil
3.24/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.24/1.57	   from(X) -> cons(X, n__from(n__s(X)))
3.24/1.57	   first(X1, X2) -> n__first(X1, X2)
3.24/1.57	   from(X) -> n__from(X)
3.24/1.57	   s(X) -> n__s(X)
3.24/1.57	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.24/1.57	   activate(n__from(X)) -> from(activate(X))
3.24/1.57	   activate(n__s(X)) -> s(activate(X))
3.24/1.57	   activate(X) -> X
3.24/1.57	
3.24/1.57	S is empty.
3.24/1.57	Rewrite Strategy: FULL
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(6) LowerBoundPropagationProof (FINISHED)
3.24/1.57	Propagated lower bound.
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(7)
3.24/1.57	BOUNDS(n^1, INF)
3.24/1.57	
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(8)
3.24/1.57	Obligation:
3.24/1.57	Analyzing the following TRS for decreasing loops:
3.24/1.57	
3.24/1.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.24/1.57	
3.24/1.57	
3.24/1.57	The TRS R consists of the following rules:
3.24/1.57	
3.24/1.57	   first(0, X) -> nil
3.24/1.57	   first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
3.24/1.57	   from(X) -> cons(X, n__from(n__s(X)))
3.24/1.57	   first(X1, X2) -> n__first(X1, X2)
3.24/1.57	   from(X) -> n__from(X)
3.24/1.57	   s(X) -> n__s(X)
3.24/1.57	   activate(n__first(X1, X2)) -> first(activate(X1), activate(X2))
3.24/1.57	   activate(n__from(X)) -> from(activate(X))
3.24/1.57	   activate(n__s(X)) -> s(activate(X))
3.24/1.57	   activate(X) -> X
3.24/1.57	
3.24/1.57	S is empty.
3.24/1.57	Rewrite Strategy: FULL
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(9) DecreasingLoopProof (FINISHED)
3.24/1.57	The following loop(s) give(s) rise to the lower bound EXP:
3.24/1.57	
3.24/1.57	The rewrite sequence
3.24/1.57	
3.24/1.57	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.24/1.57	
3.24/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.24/1.57	
3.24/1.57	The pumping substitution is [X / n__from(X)].
3.24/1.57	
3.24/1.57	The result substitution is [ ].
3.24/1.57	
3.24/1.57	
3.24/1.57	
3.24/1.57	The rewrite sequence
3.24/1.57	
3.24/1.57	activate(n__from(X)) ->^+ cons(activate(X), n__from(n__s(activate(X))))
3.24/1.57	
3.24/1.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.24/1.57	
3.24/1.57	The pumping substitution is [X / n__from(X)].
3.24/1.57	
3.24/1.57	The result substitution is [ ].
3.24/1.57	
3.24/1.57	
3.24/1.57	
3.24/1.57	
3.24/1.57	----------------------------------------
3.24/1.57	
3.24/1.57	(10)
3.24/1.57	BOUNDS(EXP, INF)
3.39/1.99	EOF