3.08/1.60	WORST_CASE(NON_POLY, ?)
3.08/1.61	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.08/1.61	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.08/1.61	
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	(0) CpxTRS
3.08/1.61	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.08/1.61	(2) TRS for Loop Detection
3.08/1.61	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.08/1.61	(4) BEST
3.08/1.61	    (5) proven lower bound
3.08/1.61	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.08/1.61	        (7) BOUNDS(n^1, INF)
3.08/1.61	    (8) TRS for Loop Detection
3.08/1.61	        (9) DecreasingLoopProof [FINISHED, 0 ms]
3.08/1.61	        (10) BOUNDS(EXP, INF)
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(0)
3.08/1.61	Obligation:
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   f(X) -> cons(X, n__f(n__g(X)))
3.08/1.61	   g(0) -> s(0)
3.08/1.61	   g(s(X)) -> s(s(g(X)))
3.08/1.61	   sel(0, cons(X, Y)) -> X
3.08/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.08/1.61	   f(X) -> n__f(X)
3.08/1.61	   g(X) -> n__g(X)
3.08/1.61	   activate(n__f(X)) -> f(activate(X))
3.08/1.61	   activate(n__g(X)) -> g(activate(X))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.08/1.61	Transformed a relative TRS into a decreasing-loop problem.
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(2)
3.08/1.61	Obligation:
3.08/1.61	Analyzing the following TRS for decreasing loops:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   f(X) -> cons(X, n__f(n__g(X)))
3.08/1.61	   g(0) -> s(0)
3.08/1.61	   g(s(X)) -> s(s(g(X)))
3.08/1.61	   sel(0, cons(X, Y)) -> X
3.08/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.08/1.61	   f(X) -> n__f(X)
3.08/1.61	   g(X) -> n__g(X)
3.08/1.61	   activate(n__f(X)) -> f(activate(X))
3.08/1.61	   activate(n__g(X)) -> g(activate(X))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.08/1.61	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	g(s(X)) ->^+ s(s(g(X)))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / s(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(4)
3.08/1.61	Complex Obligation (BEST)
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(5)
3.08/1.61	Obligation:
3.08/1.61	Proved the lower bound n^1 for the following obligation:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   f(X) -> cons(X, n__f(n__g(X)))
3.08/1.61	   g(0) -> s(0)
3.08/1.61	   g(s(X)) -> s(s(g(X)))
3.08/1.61	   sel(0, cons(X, Y)) -> X
3.08/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.08/1.61	   f(X) -> n__f(X)
3.08/1.61	   g(X) -> n__g(X)
3.08/1.61	   activate(n__f(X)) -> f(activate(X))
3.08/1.61	   activate(n__g(X)) -> g(activate(X))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(6) LowerBoundPropagationProof (FINISHED)
3.08/1.61	Propagated lower bound.
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(7)
3.08/1.61	BOUNDS(n^1, INF)
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(8)
3.08/1.61	Obligation:
3.08/1.61	Analyzing the following TRS for decreasing loops:
3.08/1.61	
3.08/1.61	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(EXP, INF).
3.08/1.61	
3.08/1.61	
3.08/1.61	The TRS R consists of the following rules:
3.08/1.61	
3.08/1.61	   f(X) -> cons(X, n__f(n__g(X)))
3.08/1.61	   g(0) -> s(0)
3.08/1.61	   g(s(X)) -> s(s(g(X)))
3.08/1.61	   sel(0, cons(X, Y)) -> X
3.08/1.61	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
3.08/1.61	   f(X) -> n__f(X)
3.08/1.61	   g(X) -> n__g(X)
3.08/1.61	   activate(n__f(X)) -> f(activate(X))
3.08/1.61	   activate(n__g(X)) -> g(activate(X))
3.08/1.61	   activate(X) -> X
3.08/1.61	
3.08/1.61	S is empty.
3.08/1.61	Rewrite Strategy: FULL
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(9) DecreasingLoopProof (FINISHED)
3.08/1.61	The following loop(s) give(s) rise to the lower bound EXP:
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	activate(n__f(X)) ->^+ cons(activate(X), n__f(n__g(activate(X))))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / n__f(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	The rewrite sequence
3.08/1.61	
3.08/1.61	activate(n__f(X)) ->^+ cons(activate(X), n__f(n__g(activate(X))))
3.08/1.61	
3.08/1.61	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
3.08/1.61	
3.08/1.61	The pumping substitution is [X / n__f(X)].
3.08/1.61	
3.08/1.61	The result substitution is [ ].
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	
3.08/1.61	----------------------------------------
3.08/1.61	
3.08/1.61	(10)
3.08/1.61	BOUNDS(EXP, INF)
3.32/1.64	EOF