364.56/291.48	WORST_CASE(Omega(n^1), ?)
364.56/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
364.56/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
364.56/291.49	
364.56/291.49	
364.56/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.56/291.49	
364.56/291.49	(0) CpxTRS
364.56/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
364.56/291.49	(2) TRS for Loop Detection
364.56/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
364.56/291.49	(4) BEST
364.56/291.49	    (5) proven lower bound
364.56/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
364.56/291.49	        (7) BOUNDS(n^1, INF)
364.56/291.49	    (8) TRS for Loop Detection
364.56/291.49	
364.56/291.49	
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(0)
364.56/291.49	Obligation:
364.56/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.56/291.49	
364.56/291.49	
364.56/291.49	The TRS R consists of the following rules:
364.56/291.49	
364.56/291.49	   f(s(x), y) -> f(x, s(s(x)))
364.56/291.49	   f(x, s(s(y))) -> f(y, x)
364.56/291.49	
364.56/291.49	S is empty.
364.56/291.49	Rewrite Strategy: INNERMOST
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
364.56/291.49	Transformed a relative TRS into a decreasing-loop problem.
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(2)
364.56/291.49	Obligation:
364.56/291.49	Analyzing the following TRS for decreasing loops:
364.56/291.49	
364.56/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.56/291.49	
364.56/291.49	
364.56/291.49	The TRS R consists of the following rules:
364.56/291.49	
364.56/291.49	   f(s(x), y) -> f(x, s(s(x)))
364.56/291.49	   f(x, s(s(y))) -> f(y, x)
364.56/291.49	
364.56/291.49	S is empty.
364.56/291.49	Rewrite Strategy: INNERMOST
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
364.56/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
364.56/291.49	
364.56/291.49	The rewrite sequence
364.56/291.49	
364.56/291.49	f(s(x), y) ->^+ f(x, s(s(x)))
364.56/291.49	
364.56/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
364.56/291.49	
364.56/291.49	The pumping substitution is [x / s(x)].
364.56/291.49	
364.56/291.49	The result substitution is [y / s(s(x))].
364.56/291.49	
364.56/291.49	
364.56/291.49	
364.56/291.49	
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(4)
364.56/291.49	Complex Obligation (BEST)
364.56/291.49	
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(5)
364.56/291.49	Obligation:
364.56/291.49	Proved the lower bound n^1 for the following obligation:
364.56/291.49	
364.56/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.56/291.49	
364.56/291.49	
364.56/291.49	The TRS R consists of the following rules:
364.56/291.49	
364.56/291.49	   f(s(x), y) -> f(x, s(s(x)))
364.56/291.49	   f(x, s(s(y))) -> f(y, x)
364.56/291.49	
364.56/291.49	S is empty.
364.56/291.49	Rewrite Strategy: INNERMOST
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(6) LowerBoundPropagationProof (FINISHED)
364.56/291.49	Propagated lower bound.
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(7)
364.56/291.49	BOUNDS(n^1, INF)
364.56/291.49	
364.56/291.49	----------------------------------------
364.56/291.49	
364.56/291.49	(8)
364.56/291.49	Obligation:
364.56/291.49	Analyzing the following TRS for decreasing loops:
364.56/291.49	
364.56/291.49	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.56/291.49	
364.56/291.49	
364.56/291.49	The TRS R consists of the following rules:
364.56/291.49	
364.56/291.49	   f(s(x), y) -> f(x, s(s(x)))
364.56/291.49	   f(x, s(s(y))) -> f(y, x)
364.56/291.49	
364.56/291.49	S is empty.
364.56/291.49	Rewrite Strategy: INNERMOST
364.56/291.53	EOF