312.92/291.57	WORST_CASE(Omega(n^1), ?)
312.92/291.58	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
312.92/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
312.92/291.58	
312.92/291.58	
312.92/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
312.92/291.58	
312.92/291.58	(0) CpxTRS
312.92/291.58	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
312.92/291.58	(2) TRS for Loop Detection
312.92/291.58	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
312.92/291.58	(4) BEST
312.92/291.58	    (5) proven lower bound
312.92/291.58	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
312.92/291.58	        (7) BOUNDS(n^1, INF)
312.92/291.58	    (8) TRS for Loop Detection
312.92/291.58	
312.92/291.58	
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(0)
312.92/291.58	Obligation:
312.92/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
312.92/291.58	
312.92/291.58	
312.92/291.58	The TRS R consists of the following rules:
312.92/291.58	
312.92/291.58	   min(x, 0) -> 0
312.92/291.58	   min(0, y) -> 0
312.92/291.58	   min(s(x), s(y)) -> s(min(x, y))
312.92/291.58	   max(x, 0) -> x
312.92/291.58	   max(0, y) -> y
312.92/291.58	   max(s(x), s(y)) -> s(max(x, y))
312.92/291.58	   -(x, 0) -> x
312.92/291.58	   -(s(x), s(y)) -> -(x, y)
312.92/291.58	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
312.92/291.58	
312.92/291.58	S is empty.
312.92/291.58	Rewrite Strategy: FULL
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
312.92/291.58	Transformed a relative TRS into a decreasing-loop problem.
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(2)
312.92/291.58	Obligation:
312.92/291.58	Analyzing the following TRS for decreasing loops:
312.92/291.58	
312.92/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
312.92/291.58	
312.92/291.58	
312.92/291.58	The TRS R consists of the following rules:
312.92/291.58	
312.92/291.58	   min(x, 0) -> 0
312.92/291.58	   min(0, y) -> 0
312.92/291.58	   min(s(x), s(y)) -> s(min(x, y))
312.92/291.58	   max(x, 0) -> x
312.92/291.58	   max(0, y) -> y
312.92/291.58	   max(s(x), s(y)) -> s(max(x, y))
312.92/291.58	   -(x, 0) -> x
312.92/291.58	   -(s(x), s(y)) -> -(x, y)
312.92/291.58	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
312.92/291.58	
312.92/291.58	S is empty.
312.92/291.58	Rewrite Strategy: FULL
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
312.92/291.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
312.92/291.58	
312.92/291.58	The rewrite sequence
312.92/291.58	
312.92/291.58	-(s(x), s(y)) ->^+ -(x, y)
312.92/291.58	
312.92/291.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
312.92/291.58	
312.92/291.58	The pumping substitution is [x / s(x), y / s(y)].
312.92/291.58	
312.92/291.58	The result substitution is [ ].
312.92/291.58	
312.92/291.58	
312.92/291.58	
312.92/291.58	
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(4)
312.92/291.58	Complex Obligation (BEST)
312.92/291.58	
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(5)
312.92/291.58	Obligation:
312.92/291.58	Proved the lower bound n^1 for the following obligation:
312.92/291.58	
312.92/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
312.92/291.58	
312.92/291.58	
312.92/291.58	The TRS R consists of the following rules:
312.92/291.58	
312.92/291.58	   min(x, 0) -> 0
312.92/291.58	   min(0, y) -> 0
312.92/291.58	   min(s(x), s(y)) -> s(min(x, y))
312.92/291.58	   max(x, 0) -> x
312.92/291.58	   max(0, y) -> y
312.92/291.58	   max(s(x), s(y)) -> s(max(x, y))
312.92/291.58	   -(x, 0) -> x
312.92/291.58	   -(s(x), s(y)) -> -(x, y)
312.92/291.58	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
312.92/291.58	
312.92/291.58	S is empty.
312.92/291.58	Rewrite Strategy: FULL
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(6) LowerBoundPropagationProof (FINISHED)
312.92/291.58	Propagated lower bound.
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(7)
312.92/291.58	BOUNDS(n^1, INF)
312.92/291.58	
312.92/291.58	----------------------------------------
312.92/291.58	
312.92/291.58	(8)
312.92/291.58	Obligation:
312.92/291.58	Analyzing the following TRS for decreasing loops:
312.92/291.58	
312.92/291.58	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
312.92/291.58	
312.92/291.58	
312.92/291.58	The TRS R consists of the following rules:
312.92/291.58	
312.92/291.58	   min(x, 0) -> 0
312.92/291.58	   min(0, y) -> 0
312.92/291.58	   min(s(x), s(y)) -> s(min(x, y))
312.92/291.58	   max(x, 0) -> x
312.92/291.58	   max(0, y) -> y
312.92/291.58	   max(s(x), s(y)) -> s(max(x, y))
312.92/291.58	   -(x, 0) -> x
312.92/291.58	   -(s(x), s(y)) -> -(x, y)
312.92/291.58	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
312.92/291.58	
312.92/291.58	S is empty.
312.92/291.58	Rewrite Strategy: FULL
312.92/291.61	EOF