317.92/291.50	WORST_CASE(Omega(n^1), ?)
317.92/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
317.92/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
317.92/291.51	
317.92/291.51	
317.92/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.92/291.51	
317.92/291.51	(0) CpxTRS
317.92/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
317.92/291.51	(2) TRS for Loop Detection
317.92/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
317.92/291.51	(4) BEST
317.92/291.51	    (5) proven lower bound
317.92/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
317.92/291.51	        (7) BOUNDS(n^1, INF)
317.92/291.51	    (8) TRS for Loop Detection
317.92/291.51	
317.92/291.51	
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(0)
317.92/291.51	Obligation:
317.92/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.92/291.51	
317.92/291.51	
317.92/291.51	The TRS R consists of the following rules:
317.92/291.51	
317.92/291.51	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
317.92/291.51	   g(b) -> c
317.92/291.51	   b -> c
317.92/291.51	   g(X) -> n__g(X)
317.92/291.51	   activate(n__g(X)) -> g(activate(X))
317.92/291.51	   activate(X) -> X
317.92/291.51	
317.92/291.51	S is empty.
317.92/291.51	Rewrite Strategy: FULL
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
317.92/291.51	Transformed a relative TRS into a decreasing-loop problem.
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(2)
317.92/291.51	Obligation:
317.92/291.51	Analyzing the following TRS for decreasing loops:
317.92/291.51	
317.92/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.92/291.51	
317.92/291.51	
317.92/291.51	The TRS R consists of the following rules:
317.92/291.51	
317.92/291.51	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
317.92/291.51	   g(b) -> c
317.92/291.51	   b -> c
317.92/291.51	   g(X) -> n__g(X)
317.92/291.51	   activate(n__g(X)) -> g(activate(X))
317.92/291.51	   activate(X) -> X
317.92/291.51	
317.92/291.51	S is empty.
317.92/291.51	Rewrite Strategy: FULL
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
317.92/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
317.92/291.51	
317.92/291.51	The rewrite sequence
317.92/291.51	
317.92/291.51	activate(n__g(X)) ->^+ g(activate(X))
317.92/291.51	
317.92/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
317.92/291.51	
317.92/291.51	The pumping substitution is [X / n__g(X)].
317.92/291.51	
317.92/291.51	The result substitution is [ ].
317.92/291.51	
317.92/291.51	
317.92/291.51	
317.92/291.51	
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(4)
317.92/291.51	Complex Obligation (BEST)
317.92/291.51	
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(5)
317.92/291.51	Obligation:
317.92/291.51	Proved the lower bound n^1 for the following obligation:
317.92/291.51	
317.92/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.92/291.51	
317.92/291.51	
317.92/291.51	The TRS R consists of the following rules:
317.92/291.51	
317.92/291.51	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
317.92/291.51	   g(b) -> c
317.92/291.51	   b -> c
317.92/291.51	   g(X) -> n__g(X)
317.92/291.51	   activate(n__g(X)) -> g(activate(X))
317.92/291.51	   activate(X) -> X
317.92/291.51	
317.92/291.51	S is empty.
317.92/291.51	Rewrite Strategy: FULL
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(6) LowerBoundPropagationProof (FINISHED)
317.92/291.51	Propagated lower bound.
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(7)
317.92/291.51	BOUNDS(n^1, INF)
317.92/291.51	
317.92/291.51	----------------------------------------
317.92/291.51	
317.92/291.51	(8)
317.92/291.51	Obligation:
317.92/291.51	Analyzing the following TRS for decreasing loops:
317.92/291.51	
317.92/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.92/291.51	
317.92/291.51	
317.92/291.51	The TRS R consists of the following rules:
317.92/291.51	
317.92/291.51	   f(X, n__g(X), Y) -> f(activate(Y), activate(Y), activate(Y))
317.92/291.51	   g(b) -> c
317.92/291.51	   b -> c
317.92/291.51	   g(X) -> n__g(X)
317.92/291.51	   activate(n__g(X)) -> g(activate(X))
317.92/291.51	   activate(X) -> X
317.92/291.51	
317.92/291.51	S is empty.
317.92/291.51	Rewrite Strategy: FULL
318.00/291.54	EOF