314.77/291.55	WORST_CASE(Omega(n^1), ?)
314.77/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
314.77/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
314.77/291.56	
314.77/291.56	
314.77/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.77/291.56	
314.77/291.56	(0) CpxTRS
314.77/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
314.77/291.56	(2) TRS for Loop Detection
314.77/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
314.77/291.56	(4) BEST
314.77/291.56	    (5) proven lower bound
314.77/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
314.77/291.56	        (7) BOUNDS(n^1, INF)
314.77/291.56	    (8) TRS for Loop Detection
314.77/291.56	
314.77/291.56	
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(0)
314.77/291.56	Obligation:
314.77/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.77/291.56	
314.77/291.56	
314.77/291.56	The TRS R consists of the following rules:
314.77/291.56	
314.77/291.56	   -(x, 0) -> x
314.77/291.56	   -(s(x), s(y)) -> -(x, y)
314.77/291.56	   min(x, 0) -> 0
314.77/291.56	   min(0, y) -> 0
314.77/291.56	   min(s(x), s(y)) -> s(min(x, y))
314.77/291.56	   twice(0) -> 0
314.77/291.56	   twice(s(x)) -> s(s(twice(x)))
314.77/291.56	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
314.77/291.56	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
314.77/291.56	
314.77/291.56	S is empty.
314.77/291.56	Rewrite Strategy: FULL
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
314.77/291.56	Transformed a relative TRS into a decreasing-loop problem.
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(2)
314.77/291.56	Obligation:
314.77/291.56	Analyzing the following TRS for decreasing loops:
314.77/291.56	
314.77/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.77/291.56	
314.77/291.56	
314.77/291.56	The TRS R consists of the following rules:
314.77/291.56	
314.77/291.56	   -(x, 0) -> x
314.77/291.56	   -(s(x), s(y)) -> -(x, y)
314.77/291.56	   min(x, 0) -> 0
314.77/291.56	   min(0, y) -> 0
314.77/291.56	   min(s(x), s(y)) -> s(min(x, y))
314.77/291.56	   twice(0) -> 0
314.77/291.56	   twice(s(x)) -> s(s(twice(x)))
314.77/291.56	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
314.77/291.56	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
314.77/291.56	
314.77/291.56	S is empty.
314.77/291.56	Rewrite Strategy: FULL
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
314.77/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
314.77/291.56	
314.77/291.56	The rewrite sequence
314.77/291.56	
314.77/291.56	-(s(x), s(y)) ->^+ -(x, y)
314.77/291.56	
314.77/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
314.77/291.56	
314.77/291.56	The pumping substitution is [x / s(x), y / s(y)].
314.77/291.56	
314.77/291.56	The result substitution is [ ].
314.77/291.56	
314.77/291.56	
314.77/291.56	
314.77/291.56	
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(4)
314.77/291.56	Complex Obligation (BEST)
314.77/291.56	
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(5)
314.77/291.56	Obligation:
314.77/291.56	Proved the lower bound n^1 for the following obligation:
314.77/291.56	
314.77/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.77/291.56	
314.77/291.56	
314.77/291.56	The TRS R consists of the following rules:
314.77/291.56	
314.77/291.56	   -(x, 0) -> x
314.77/291.56	   -(s(x), s(y)) -> -(x, y)
314.77/291.56	   min(x, 0) -> 0
314.77/291.56	   min(0, y) -> 0
314.77/291.56	   min(s(x), s(y)) -> s(min(x, y))
314.77/291.56	   twice(0) -> 0
314.77/291.56	   twice(s(x)) -> s(s(twice(x)))
314.77/291.56	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
314.77/291.56	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
314.77/291.56	
314.77/291.56	S is empty.
314.77/291.56	Rewrite Strategy: FULL
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(6) LowerBoundPropagationProof (FINISHED)
314.77/291.56	Propagated lower bound.
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(7)
314.77/291.56	BOUNDS(n^1, INF)
314.77/291.56	
314.77/291.56	----------------------------------------
314.77/291.56	
314.77/291.56	(8)
314.77/291.56	Obligation:
314.77/291.56	Analyzing the following TRS for decreasing loops:
314.77/291.56	
314.77/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
314.77/291.56	
314.77/291.56	
314.77/291.56	The TRS R consists of the following rules:
314.77/291.56	
314.77/291.56	   -(x, 0) -> x
314.77/291.56	   -(s(x), s(y)) -> -(x, y)
314.77/291.56	   min(x, 0) -> 0
314.77/291.56	   min(0, y) -> 0
314.77/291.56	   min(s(x), s(y)) -> s(min(x, y))
314.77/291.56	   twice(0) -> 0
314.77/291.56	   twice(s(x)) -> s(s(twice(x)))
314.77/291.56	   f(s(x), s(y)) -> f(-(y, min(x, y)), s(twice(min(x, y))))
314.77/291.56	   f(s(x), s(y)) -> f(-(x, min(x, y)), s(twice(min(x, y))))
314.77/291.56	
314.77/291.56	S is empty.
314.77/291.56	Rewrite Strategy: FULL
314.77/291.60	EOF