352.44/291.53	WORST_CASE(Omega(n^1), ?)
352.44/291.54	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
352.44/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
352.44/291.54	
352.44/291.54	
352.44/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.44/291.54	
352.44/291.54	(0) CpxTRS
352.44/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
352.44/291.54	(2) TRS for Loop Detection
352.44/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
352.44/291.54	(4) BEST
352.44/291.54	    (5) proven lower bound
352.44/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
352.44/291.54	        (7) BOUNDS(n^1, INF)
352.44/291.54	    (8) TRS for Loop Detection
352.44/291.54	
352.44/291.54	
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(0)
352.44/291.54	Obligation:
352.44/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.44/291.54	
352.44/291.54	
352.44/291.54	The TRS R consists of the following rules:
352.44/291.54	
352.44/291.54	   f(s(x), y) -> f(x, s(x))
352.44/291.54	   f(x, s(y)) -> f(y, x)
352.44/291.54	
352.44/291.54	S is empty.
352.44/291.54	Rewrite Strategy: FULL
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
352.44/291.54	Transformed a relative TRS into a decreasing-loop problem.
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(2)
352.44/291.54	Obligation:
352.44/291.54	Analyzing the following TRS for decreasing loops:
352.44/291.54	
352.44/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.44/291.54	
352.44/291.54	
352.44/291.54	The TRS R consists of the following rules:
352.44/291.54	
352.44/291.54	   f(s(x), y) -> f(x, s(x))
352.44/291.54	   f(x, s(y)) -> f(y, x)
352.44/291.54	
352.44/291.54	S is empty.
352.44/291.54	Rewrite Strategy: FULL
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
352.44/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
352.44/291.54	
352.44/291.54	The rewrite sequence
352.44/291.54	
352.44/291.54	f(s(x), y) ->^+ f(x, s(x))
352.44/291.54	
352.44/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
352.44/291.54	
352.44/291.54	The pumping substitution is [x / s(x)].
352.44/291.54	
352.44/291.54	The result substitution is [y / s(x)].
352.44/291.54	
352.44/291.54	
352.44/291.54	
352.44/291.54	
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(4)
352.44/291.54	Complex Obligation (BEST)
352.44/291.54	
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(5)
352.44/291.54	Obligation:
352.44/291.54	Proved the lower bound n^1 for the following obligation:
352.44/291.54	
352.44/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.44/291.54	
352.44/291.54	
352.44/291.54	The TRS R consists of the following rules:
352.44/291.54	
352.44/291.54	   f(s(x), y) -> f(x, s(x))
352.44/291.54	   f(x, s(y)) -> f(y, x)
352.44/291.54	
352.44/291.54	S is empty.
352.44/291.54	Rewrite Strategy: FULL
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(6) LowerBoundPropagationProof (FINISHED)
352.44/291.54	Propagated lower bound.
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(7)
352.44/291.54	BOUNDS(n^1, INF)
352.44/291.54	
352.44/291.54	----------------------------------------
352.44/291.54	
352.44/291.54	(8)
352.44/291.54	Obligation:
352.44/291.54	Analyzing the following TRS for decreasing loops:
352.44/291.54	
352.44/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.44/291.54	
352.44/291.54	
352.44/291.54	The TRS R consists of the following rules:
352.44/291.54	
352.44/291.54	   f(s(x), y) -> f(x, s(x))
352.44/291.54	   f(x, s(y)) -> f(y, x)
352.44/291.54	
352.44/291.54	S is empty.
352.44/291.54	Rewrite Strategy: FULL
352.50/291.58	EOF