354.59/291.52	WORST_CASE(Omega(n^1), ?)
354.59/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
354.59/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
354.59/291.53	
354.59/291.53	
354.59/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
354.59/291.53	
354.59/291.53	(0) CpxTRS
354.59/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
354.59/291.53	(2) TRS for Loop Detection
354.59/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
354.59/291.53	(4) BEST
354.59/291.53	    (5) proven lower bound
354.59/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
354.59/291.53	        (7) BOUNDS(n^1, INF)
354.59/291.53	    (8) TRS for Loop Detection
354.59/291.53	
354.59/291.53	
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(0)
354.59/291.53	Obligation:
354.59/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
354.59/291.53	
354.59/291.53	
354.59/291.53	The TRS R consists of the following rules:
354.59/291.53	
354.59/291.53	   f(s(x), y) -> f(x, s(x))
354.59/291.53	   f(x, s(y)) -> f(y, x)
354.59/291.53	
354.59/291.53	S is empty.
354.59/291.53	Rewrite Strategy: INNERMOST
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
354.59/291.53	Transformed a relative TRS into a decreasing-loop problem.
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(2)
354.59/291.53	Obligation:
354.59/291.53	Analyzing the following TRS for decreasing loops:
354.59/291.53	
354.59/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
354.59/291.53	
354.59/291.53	
354.59/291.53	The TRS R consists of the following rules:
354.59/291.53	
354.59/291.53	   f(s(x), y) -> f(x, s(x))
354.59/291.53	   f(x, s(y)) -> f(y, x)
354.59/291.53	
354.59/291.53	S is empty.
354.59/291.53	Rewrite Strategy: INNERMOST
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
354.59/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
354.59/291.53	
354.59/291.53	The rewrite sequence
354.59/291.53	
354.59/291.53	f(s(x), y) ->^+ f(x, s(x))
354.59/291.53	
354.59/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
354.59/291.53	
354.59/291.53	The pumping substitution is [x / s(x)].
354.59/291.53	
354.59/291.53	The result substitution is [y / s(x)].
354.59/291.53	
354.59/291.53	
354.59/291.53	
354.59/291.53	
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(4)
354.59/291.53	Complex Obligation (BEST)
354.59/291.53	
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(5)
354.59/291.53	Obligation:
354.59/291.53	Proved the lower bound n^1 for the following obligation:
354.59/291.53	
354.59/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
354.59/291.53	
354.59/291.53	
354.59/291.53	The TRS R consists of the following rules:
354.59/291.53	
354.59/291.53	   f(s(x), y) -> f(x, s(x))
354.59/291.53	   f(x, s(y)) -> f(y, x)
354.59/291.53	
354.59/291.53	S is empty.
354.59/291.53	Rewrite Strategy: INNERMOST
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(6) LowerBoundPropagationProof (FINISHED)
354.59/291.53	Propagated lower bound.
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(7)
354.59/291.53	BOUNDS(n^1, INF)
354.59/291.53	
354.59/291.53	----------------------------------------
354.59/291.53	
354.59/291.53	(8)
354.59/291.53	Obligation:
354.59/291.53	Analyzing the following TRS for decreasing loops:
354.59/291.53	
354.59/291.53	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
354.59/291.53	
354.59/291.53	
354.59/291.53	The TRS R consists of the following rules:
354.59/291.53	
354.59/291.53	   f(s(x), y) -> f(x, s(x))
354.59/291.53	   f(x, s(y)) -> f(y, x)
354.59/291.53	
354.59/291.53	S is empty.
354.59/291.53	Rewrite Strategy: INNERMOST
354.59/291.57	EOF