309.12/291.56	WORST_CASE(Omega(n^1), ?)
309.12/291.57	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
309.12/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.12/291.57	
309.12/291.57	
309.12/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.12/291.57	
309.12/291.57	(0) CpxTRS
309.12/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
309.12/291.57	(2) TRS for Loop Detection
309.12/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
309.12/291.57	(4) BEST
309.12/291.57	    (5) proven lower bound
309.12/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
309.12/291.57	        (7) BOUNDS(n^1, INF)
309.12/291.57	    (8) TRS for Loop Detection
309.12/291.57	
309.12/291.57	
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(0)
309.12/291.57	Obligation:
309.12/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.12/291.57	
309.12/291.57	
309.12/291.57	The TRS R consists of the following rules:
309.12/291.57	
309.12/291.57	   min(x, 0) -> 0
309.12/291.57	   min(0, y) -> 0
309.12/291.57	   min(s(x), s(y)) -> s(min(x, y))
309.12/291.57	   max(x, 0) -> x
309.12/291.57	   max(0, y) -> y
309.12/291.57	   max(s(x), s(y)) -> s(max(x, y))
309.12/291.57	   -(x, 0) -> x
309.12/291.57	   -(s(x), s(y)) -> -(x, y)
309.12/291.57	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
309.12/291.57	   gcd(s(x), 0) -> s(x)
309.12/291.57	   gcd(0, s(y)) -> s(y)
309.12/291.57	
309.12/291.57	S is empty.
309.12/291.57	Rewrite Strategy: FULL
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
309.12/291.57	Transformed a relative TRS into a decreasing-loop problem.
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(2)
309.12/291.57	Obligation:
309.12/291.57	Analyzing the following TRS for decreasing loops:
309.12/291.57	
309.12/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.12/291.57	
309.12/291.57	
309.12/291.57	The TRS R consists of the following rules:
309.12/291.57	
309.12/291.57	   min(x, 0) -> 0
309.12/291.57	   min(0, y) -> 0
309.12/291.57	   min(s(x), s(y)) -> s(min(x, y))
309.12/291.57	   max(x, 0) -> x
309.12/291.57	   max(0, y) -> y
309.12/291.57	   max(s(x), s(y)) -> s(max(x, y))
309.12/291.57	   -(x, 0) -> x
309.12/291.57	   -(s(x), s(y)) -> -(x, y)
309.12/291.57	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
309.12/291.57	   gcd(s(x), 0) -> s(x)
309.12/291.57	   gcd(0, s(y)) -> s(y)
309.12/291.57	
309.12/291.57	S is empty.
309.12/291.57	Rewrite Strategy: FULL
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
309.12/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
309.12/291.57	
309.12/291.57	The rewrite sequence
309.12/291.57	
309.12/291.57	-(s(x), s(y)) ->^+ -(x, y)
309.12/291.57	
309.12/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
309.12/291.57	
309.12/291.57	The pumping substitution is [x / s(x), y / s(y)].
309.12/291.57	
309.12/291.57	The result substitution is [ ].
309.12/291.57	
309.12/291.57	
309.12/291.57	
309.12/291.57	
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(4)
309.12/291.57	Complex Obligation (BEST)
309.12/291.57	
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(5)
309.12/291.57	Obligation:
309.12/291.57	Proved the lower bound n^1 for the following obligation:
309.12/291.57	
309.12/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.12/291.57	
309.12/291.57	
309.12/291.57	The TRS R consists of the following rules:
309.12/291.57	
309.12/291.57	   min(x, 0) -> 0
309.12/291.57	   min(0, y) -> 0
309.12/291.57	   min(s(x), s(y)) -> s(min(x, y))
309.12/291.57	   max(x, 0) -> x
309.12/291.57	   max(0, y) -> y
309.12/291.57	   max(s(x), s(y)) -> s(max(x, y))
309.12/291.57	   -(x, 0) -> x
309.12/291.57	   -(s(x), s(y)) -> -(x, y)
309.12/291.57	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
309.12/291.57	   gcd(s(x), 0) -> s(x)
309.12/291.57	   gcd(0, s(y)) -> s(y)
309.12/291.57	
309.12/291.57	S is empty.
309.12/291.57	Rewrite Strategy: FULL
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(6) LowerBoundPropagationProof (FINISHED)
309.12/291.57	Propagated lower bound.
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(7)
309.12/291.57	BOUNDS(n^1, INF)
309.12/291.57	
309.12/291.57	----------------------------------------
309.12/291.57	
309.12/291.57	(8)
309.12/291.57	Obligation:
309.12/291.57	Analyzing the following TRS for decreasing loops:
309.12/291.57	
309.12/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.12/291.57	
309.12/291.57	
309.12/291.57	The TRS R consists of the following rules:
309.12/291.57	
309.12/291.57	   min(x, 0) -> 0
309.12/291.57	   min(0, y) -> 0
309.12/291.57	   min(s(x), s(y)) -> s(min(x, y))
309.12/291.57	   max(x, 0) -> x
309.12/291.57	   max(0, y) -> y
309.12/291.57	   max(s(x), s(y)) -> s(max(x, y))
309.12/291.57	   -(x, 0) -> x
309.12/291.57	   -(s(x), s(y)) -> -(x, y)
309.12/291.57	   gcd(s(x), s(y)) -> gcd(-(s(max(x, y)), s(min(x, y))), s(min(x, y)))
309.12/291.57	   gcd(s(x), 0) -> s(x)
309.12/291.57	   gcd(0, s(y)) -> s(y)
309.12/291.57	
309.12/291.57	S is empty.
309.12/291.57	Rewrite Strategy: FULL
309.21/291.60	EOF