306.99/291.50	WORST_CASE(Omega(n^1), ?)
306.99/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
306.99/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
306.99/291.50	
306.99/291.50	
306.99/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.99/291.50	
306.99/291.50	(0) CpxTRS
306.99/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
306.99/291.50	(2) TRS for Loop Detection
306.99/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
306.99/291.50	(4) BEST
306.99/291.50	    (5) proven lower bound
306.99/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
306.99/291.50	        (7) BOUNDS(n^1, INF)
306.99/291.50	    (8) TRS for Loop Detection
306.99/291.50	
306.99/291.50	
306.99/291.50	----------------------------------------
306.99/291.50	
306.99/291.50	(0)
306.99/291.50	Obligation:
306.99/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.99/291.50	
306.99/291.50	
306.99/291.50	The TRS R consists of the following rules:
306.99/291.50	
306.99/291.50	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
306.99/291.50	   round(0) -> 0
306.99/291.50	   round(s(0)) -> s(s(0))
306.99/291.50	   round(s(s(x))) -> s(s(round(x)))
306.99/291.50	   gt(0, v) -> false
306.99/291.50	   gt(s(u), 0) -> true
306.99/291.50	   gt(s(u), s(v)) -> gt(u, v)
306.99/291.50	
306.99/291.50	S is empty.
306.99/291.50	Rewrite Strategy: FULL
306.99/291.50	----------------------------------------
306.99/291.50	
306.99/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
306.99/291.50	Transformed a relative TRS into a decreasing-loop problem.
306.99/291.50	----------------------------------------
306.99/291.50	
306.99/291.50	(2)
306.99/291.50	Obligation:
306.99/291.50	Analyzing the following TRS for decreasing loops:
306.99/291.50	
306.99/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.99/291.50	
306.99/291.50	
306.99/291.50	The TRS R consists of the following rules:
306.99/291.50	
306.99/291.50	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
306.99/291.50	   round(0) -> 0
306.99/291.50	   round(s(0)) -> s(s(0))
306.99/291.50	   round(s(s(x))) -> s(s(round(x)))
306.99/291.50	   gt(0, v) -> false
306.99/291.50	   gt(s(u), 0) -> true
306.99/291.50	   gt(s(u), s(v)) -> gt(u, v)
306.99/291.50	
306.99/291.50	S is empty.
306.99/291.50	Rewrite Strategy: FULL
306.99/291.50	----------------------------------------
306.99/291.50	
306.99/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
306.99/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
306.99/291.50	
306.99/291.50	The rewrite sequence
306.99/291.50	
306.99/291.50	gt(s(u), s(v)) ->^+ gt(u, v)
306.99/291.50	
306.99/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
306.99/291.50	
306.99/291.50	The pumping substitution is [u / s(u), v / s(v)].
306.99/291.50	
306.99/291.50	The result substitution is [ ].
306.99/291.50	
306.99/291.50	
306.99/291.50	
306.99/291.50	
306.99/291.50	----------------------------------------
306.99/291.50	
306.99/291.51	(4)
306.99/291.51	Complex Obligation (BEST)
306.99/291.51	
306.99/291.51	----------------------------------------
306.99/291.51	
306.99/291.51	(5)
306.99/291.51	Obligation:
306.99/291.51	Proved the lower bound n^1 for the following obligation:
306.99/291.51	
306.99/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.99/291.51	
306.99/291.51	
306.99/291.51	The TRS R consists of the following rules:
306.99/291.51	
306.99/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
306.99/291.51	   round(0) -> 0
306.99/291.51	   round(s(0)) -> s(s(0))
306.99/291.51	   round(s(s(x))) -> s(s(round(x)))
306.99/291.51	   gt(0, v) -> false
306.99/291.51	   gt(s(u), 0) -> true
306.99/291.51	   gt(s(u), s(v)) -> gt(u, v)
306.99/291.51	
306.99/291.51	S is empty.
306.99/291.51	Rewrite Strategy: FULL
306.99/291.51	----------------------------------------
306.99/291.51	
306.99/291.51	(6) LowerBoundPropagationProof (FINISHED)
306.99/291.51	Propagated lower bound.
306.99/291.51	----------------------------------------
306.99/291.51	
306.99/291.51	(7)
306.99/291.51	BOUNDS(n^1, INF)
306.99/291.51	
306.99/291.51	----------------------------------------
306.99/291.51	
306.99/291.51	(8)
306.99/291.51	Obligation:
306.99/291.51	Analyzing the following TRS for decreasing loops:
306.99/291.51	
306.99/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.99/291.51	
306.99/291.51	
306.99/291.51	The TRS R consists of the following rules:
306.99/291.51	
306.99/291.51	   f(true, x, y) -> f(gt(x, y), x, round(s(y)))
306.99/291.51	   round(0) -> 0
306.99/291.51	   round(s(0)) -> s(s(0))
306.99/291.51	   round(s(s(x))) -> s(s(round(x)))
306.99/291.51	   gt(0, v) -> false
306.99/291.51	   gt(s(u), 0) -> true
306.99/291.51	   gt(s(u), s(v)) -> gt(u, v)
306.99/291.51	
306.99/291.51	S is empty.
306.99/291.51	Rewrite Strategy: FULL
307.09/291.57	EOF