1060.58/291.51	WORST_CASE(Omega(n^1), ?)
1060.82/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1060.82/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1060.82/291.53	
1060.82/291.53	(0) CpxTRS
1060.82/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1060.82/291.53	(2) TRS for Loop Detection
1060.82/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1060.82/291.53	(4) BEST
1060.82/291.53	    (5) proven lower bound
1060.82/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1060.82/291.53	        (7) BOUNDS(n^1, INF)
1060.82/291.53	    (8) TRS for Loop Detection
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(0)
1060.82/291.53	Obligation:
1060.82/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	The TRS R consists of the following rules:
1060.82/291.53	
1060.82/291.53	   f(X) -> cons(X, n__f(n__g(X)))
1060.82/291.53	   g(0) -> s(0)
1060.82/291.53	   g(s(X)) -> s(s(g(X)))
1060.82/291.53	   sel(0, cons(X, Y)) -> X
1060.82/291.53	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
1060.82/291.53	   f(X) -> n__f(X)
1060.82/291.53	   g(X) -> n__g(X)
1060.82/291.53	   activate(n__f(X)) -> f(activate(X))
1060.82/291.53	   activate(n__g(X)) -> g(activate(X))
1060.82/291.53	   activate(X) -> X
1060.82/291.53	
1060.82/291.53	S is empty.
1060.82/291.53	Rewrite Strategy: INNERMOST
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1060.82/291.53	Transformed a relative TRS into a decreasing-loop problem.
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(2)
1060.82/291.53	Obligation:
1060.82/291.53	Analyzing the following TRS for decreasing loops:
1060.82/291.53	
1060.82/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	The TRS R consists of the following rules:
1060.82/291.53	
1060.82/291.53	   f(X) -> cons(X, n__f(n__g(X)))
1060.82/291.53	   g(0) -> s(0)
1060.82/291.53	   g(s(X)) -> s(s(g(X)))
1060.82/291.53	   sel(0, cons(X, Y)) -> X
1060.82/291.53	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
1060.82/291.53	   f(X) -> n__f(X)
1060.82/291.53	   g(X) -> n__g(X)
1060.82/291.53	   activate(n__f(X)) -> f(activate(X))
1060.82/291.53	   activate(n__g(X)) -> g(activate(X))
1060.82/291.53	   activate(X) -> X
1060.82/291.53	
1060.82/291.53	S is empty.
1060.82/291.53	Rewrite Strategy: INNERMOST
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
1060.82/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1060.82/291.53	
1060.82/291.53	The rewrite sequence
1060.82/291.53	
1060.82/291.53	g(s(X)) ->^+ s(s(g(X)))
1060.82/291.53	
1060.82/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
1060.82/291.53	
1060.82/291.53	The pumping substitution is [X / s(X)].
1060.82/291.53	
1060.82/291.53	The result substitution is [ ].
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(4)
1060.82/291.53	Complex Obligation (BEST)
1060.82/291.53	
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(5)
1060.82/291.53	Obligation:
1060.82/291.53	Proved the lower bound n^1 for the following obligation:
1060.82/291.53	
1060.82/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	The TRS R consists of the following rules:
1060.82/291.53	
1060.82/291.53	   f(X) -> cons(X, n__f(n__g(X)))
1060.82/291.53	   g(0) -> s(0)
1060.82/291.53	   g(s(X)) -> s(s(g(X)))
1060.82/291.53	   sel(0, cons(X, Y)) -> X
1060.82/291.53	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
1060.82/291.53	   f(X) -> n__f(X)
1060.82/291.53	   g(X) -> n__g(X)
1060.82/291.53	   activate(n__f(X)) -> f(activate(X))
1060.82/291.53	   activate(n__g(X)) -> g(activate(X))
1060.82/291.53	   activate(X) -> X
1060.82/291.53	
1060.82/291.53	S is empty.
1060.82/291.53	Rewrite Strategy: INNERMOST
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(6) LowerBoundPropagationProof (FINISHED)
1060.82/291.53	Propagated lower bound.
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(7)
1060.82/291.53	BOUNDS(n^1, INF)
1060.82/291.53	
1060.82/291.53	----------------------------------------
1060.82/291.53	
1060.82/291.53	(8)
1060.82/291.53	Obligation:
1060.82/291.53	Analyzing the following TRS for decreasing loops:
1060.82/291.53	
1060.82/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1060.82/291.53	
1060.82/291.53	
1060.82/291.53	The TRS R consists of the following rules:
1060.82/291.53	
1060.82/291.53	   f(X) -> cons(X, n__f(n__g(X)))
1060.82/291.53	   g(0) -> s(0)
1060.82/291.53	   g(s(X)) -> s(s(g(X)))
1060.82/291.53	   sel(0, cons(X, Y)) -> X
1060.82/291.53	   sel(s(X), cons(Y, Z)) -> sel(X, activate(Z))
1060.82/291.53	   f(X) -> n__f(X)
1060.82/291.53	   g(X) -> n__g(X)
1060.82/291.53	   activate(n__f(X)) -> f(activate(X))
1060.82/291.53	   activate(n__g(X)) -> g(activate(X))
1060.82/291.53	   activate(X) -> X
1060.82/291.53	
1060.82/291.53	S is empty.
1060.82/291.53	Rewrite Strategy: INNERMOST
1060.90/291.59	EOF