3.33/1.66	WORST_CASE(NON_POLY, ?)
3.33/1.66	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.33/1.66	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.33/1.66	
3.33/1.66	
3.33/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.33/1.66	
3.33/1.66	(0) CpxTRS
3.33/1.66	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
3.33/1.66	(2) TRS for Loop Detection
3.33/1.66	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
3.33/1.66	(4) BEST
3.33/1.66	    (5) proven lower bound
3.33/1.66	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
3.33/1.66	        (7) BOUNDS(n^1, INF)
3.33/1.66	    (8) TRS for Loop Detection
3.33/1.66	        (9) InfiniteLowerBoundProof [FINISHED, 0 ms]
3.33/1.66	        (10) BOUNDS(INF, INF)
3.33/1.66	
3.33/1.66	
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(0)
3.33/1.66	Obligation:
3.33/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.33/1.66	
3.33/1.66	
3.33/1.66	The TRS R consists of the following rules:
3.33/1.66	
3.33/1.66	   f(s(x), x) -> f(s(x), round(x))
3.33/1.66	   round(0) -> 0
3.33/1.66	   round(0) -> s(0)
3.33/1.66	   round(s(0)) -> s(0)
3.33/1.66	   round(s(s(x))) -> s(s(round(x)))
3.33/1.66	
3.33/1.66	S is empty.
3.33/1.66	Rewrite Strategy: FULL
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
3.33/1.66	Transformed a relative TRS into a decreasing-loop problem.
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(2)
3.33/1.66	Obligation:
3.33/1.66	Analyzing the following TRS for decreasing loops:
3.33/1.66	
3.33/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.33/1.66	
3.33/1.66	
3.33/1.66	The TRS R consists of the following rules:
3.33/1.66	
3.33/1.66	   f(s(x), x) -> f(s(x), round(x))
3.33/1.66	   round(0) -> 0
3.33/1.66	   round(0) -> s(0)
3.33/1.66	   round(s(0)) -> s(0)
3.33/1.66	   round(s(s(x))) -> s(s(round(x)))
3.33/1.66	
3.33/1.66	S is empty.
3.33/1.66	Rewrite Strategy: FULL
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(3) DecreasingLoopProof (LOWER BOUND(ID))
3.33/1.66	The following loop(s) give(s) rise to the lower bound Omega(n^1):
3.33/1.66	
3.33/1.66	The rewrite sequence
3.33/1.66	
3.33/1.66	round(s(s(x))) ->^+ s(s(round(x)))
3.33/1.66	
3.33/1.66	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
3.33/1.66	
3.33/1.66	The pumping substitution is [x / s(s(x))].
3.33/1.66	
3.33/1.66	The result substitution is [ ].
3.33/1.66	
3.33/1.66	
3.33/1.66	
3.33/1.66	
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(4)
3.33/1.66	Complex Obligation (BEST)
3.33/1.66	
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(5)
3.33/1.66	Obligation:
3.33/1.66	Proved the lower bound n^1 for the following obligation:
3.33/1.66	
3.33/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.33/1.66	
3.33/1.66	
3.33/1.66	The TRS R consists of the following rules:
3.33/1.66	
3.33/1.66	   f(s(x), x) -> f(s(x), round(x))
3.33/1.66	   round(0) -> 0
3.33/1.66	   round(0) -> s(0)
3.33/1.66	   round(s(0)) -> s(0)
3.33/1.66	   round(s(s(x))) -> s(s(round(x)))
3.33/1.66	
3.33/1.66	S is empty.
3.33/1.66	Rewrite Strategy: FULL
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(6) LowerBoundPropagationProof (FINISHED)
3.33/1.66	Propagated lower bound.
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(7)
3.33/1.66	BOUNDS(n^1, INF)
3.33/1.66	
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(8)
3.33/1.66	Obligation:
3.33/1.66	Analyzing the following TRS for decreasing loops:
3.33/1.66	
3.33/1.66	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(INF, INF).
3.33/1.66	
3.33/1.66	
3.33/1.66	The TRS R consists of the following rules:
3.33/1.66	
3.33/1.66	   f(s(x), x) -> f(s(x), round(x))
3.33/1.66	   round(0) -> 0
3.33/1.66	   round(0) -> s(0)
3.33/1.66	   round(s(0)) -> s(0)
3.33/1.66	   round(s(s(x))) -> s(s(round(x)))
3.33/1.66	
3.33/1.66	S is empty.
3.33/1.66	Rewrite Strategy: FULL
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(9) InfiniteLowerBoundProof (FINISHED)
3.33/1.66	The following loop proves infinite runtime complexity:
3.33/1.66	
3.33/1.66	The rewrite sequence
3.33/1.66	
3.33/1.66	f(s(0), 0) ->^+ f(s(0), 0)
3.33/1.66	
3.33/1.66	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
3.33/1.66	
3.33/1.66	The pumping substitution is [ ].
3.33/1.66	
3.33/1.66	The result substitution is [ ].
3.33/1.66	
3.33/1.66	
3.33/1.66	
3.33/1.66	
3.33/1.66	----------------------------------------
3.33/1.66	
3.33/1.66	(10)
3.33/1.66	BOUNDS(INF, INF)
3.33/1.69	EOF