315.51/291.48	WORST_CASE(Omega(n^1), ?)
315.51/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
315.51/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
315.51/291.49	
315.51/291.49	
315.51/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.51/291.49	
315.51/291.49	(0) CpxTRS
315.51/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
315.51/291.49	(2) TRS for Loop Detection
315.51/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
315.51/291.49	(4) BEST
315.51/291.49	    (5) proven lower bound
315.51/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
315.51/291.49	        (7) BOUNDS(n^1, INF)
315.51/291.49	    (8) TRS for Loop Detection
315.51/291.49	
315.51/291.49	
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(0)
315.51/291.49	Obligation:
315.51/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.51/291.49	
315.51/291.49	
315.51/291.49	The TRS R consists of the following rules:
315.51/291.49	
315.51/291.49	   minus(x, y) -> cond(min(x, y), x, y)
315.51/291.49	   cond(y, x, y) -> s(minus(x, s(y)))
315.51/291.49	   min(0, v) -> 0
315.51/291.49	   min(u, 0) -> 0
315.51/291.49	   min(s(u), s(v)) -> s(min(u, v))
315.51/291.49	
315.51/291.49	S is empty.
315.51/291.49	Rewrite Strategy: FULL
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
315.51/291.49	Transformed a relative TRS into a decreasing-loop problem.
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(2)
315.51/291.49	Obligation:
315.51/291.49	Analyzing the following TRS for decreasing loops:
315.51/291.49	
315.51/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.51/291.49	
315.51/291.49	
315.51/291.49	The TRS R consists of the following rules:
315.51/291.49	
315.51/291.49	   minus(x, y) -> cond(min(x, y), x, y)
315.51/291.49	   cond(y, x, y) -> s(minus(x, s(y)))
315.51/291.49	   min(0, v) -> 0
315.51/291.49	   min(u, 0) -> 0
315.51/291.49	   min(s(u), s(v)) -> s(min(u, v))
315.51/291.49	
315.51/291.49	S is empty.
315.51/291.49	Rewrite Strategy: FULL
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
315.51/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
315.51/291.49	
315.51/291.49	The rewrite sequence
315.51/291.49	
315.51/291.49	min(s(u), s(v)) ->^+ s(min(u, v))
315.51/291.49	
315.51/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
315.51/291.49	
315.51/291.49	The pumping substitution is [u / s(u), v / s(v)].
315.51/291.49	
315.51/291.49	The result substitution is [ ].
315.51/291.49	
315.51/291.49	
315.51/291.49	
315.51/291.49	
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(4)
315.51/291.49	Complex Obligation (BEST)
315.51/291.49	
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(5)
315.51/291.49	Obligation:
315.51/291.49	Proved the lower bound n^1 for the following obligation:
315.51/291.49	
315.51/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.51/291.49	
315.51/291.49	
315.51/291.49	The TRS R consists of the following rules:
315.51/291.49	
315.51/291.49	   minus(x, y) -> cond(min(x, y), x, y)
315.51/291.49	   cond(y, x, y) -> s(minus(x, s(y)))
315.51/291.49	   min(0, v) -> 0
315.51/291.49	   min(u, 0) -> 0
315.51/291.49	   min(s(u), s(v)) -> s(min(u, v))
315.51/291.49	
315.51/291.49	S is empty.
315.51/291.49	Rewrite Strategy: FULL
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(6) LowerBoundPropagationProof (FINISHED)
315.51/291.49	Propagated lower bound.
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(7)
315.51/291.49	BOUNDS(n^1, INF)
315.51/291.49	
315.51/291.49	----------------------------------------
315.51/291.49	
315.51/291.49	(8)
315.51/291.49	Obligation:
315.51/291.49	Analyzing the following TRS for decreasing loops:
315.51/291.49	
315.51/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
315.51/291.49	
315.51/291.49	
315.51/291.49	The TRS R consists of the following rules:
315.51/291.49	
315.51/291.49	   minus(x, y) -> cond(min(x, y), x, y)
315.51/291.49	   cond(y, x, y) -> s(minus(x, s(y)))
315.51/291.49	   min(0, v) -> 0
315.51/291.49	   min(u, 0) -> 0
315.51/291.49	   min(s(u), s(v)) -> s(min(u, v))
315.51/291.49	
315.51/291.49	S is empty.
315.51/291.49	Rewrite Strategy: FULL
315.55/291.51	EOF