1115.95/291.53	WORST_CASE(Omega(n^1), ?)
1116.27/291.58	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1116.27/291.58	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1116.27/291.58	
1116.27/291.58	(0) CpxTRS
1116.27/291.58	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1116.27/291.58	(2) TRS for Loop Detection
1116.27/291.58	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1116.27/291.58	(4) BEST
1116.27/291.58	    (5) proven lower bound
1116.27/291.58	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1116.27/291.58	        (7) BOUNDS(n^1, INF)
1116.27/291.58	    (8) TRS for Loop Detection
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(0)
1116.27/291.58	Obligation:
1116.27/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	The TRS R consists of the following rules:
1116.27/291.58	
1116.27/291.58	   p(0) -> 0
1116.27/291.58	   p(s(x)) -> x
1116.27/291.58	   minus(x, 0) -> x
1116.27/291.58	   minus(s(x), s(y)) -> minus(x, y)
1116.27/291.58	   minus(x, s(y)) -> p(minus(x, y))
1116.27/291.58	   div(0, s(y)) -> 0
1116.27/291.58	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1116.27/291.58	   log(s(0), s(s(y))) -> 0
1116.27/291.58	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1116.27/291.58	
1116.27/291.58	S is empty.
1116.27/291.58	Rewrite Strategy: INNERMOST
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1116.27/291.58	Transformed a relative TRS into a decreasing-loop problem.
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(2)
1116.27/291.58	Obligation:
1116.27/291.58	Analyzing the following TRS for decreasing loops:
1116.27/291.58	
1116.27/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	The TRS R consists of the following rules:
1116.27/291.58	
1116.27/291.58	   p(0) -> 0
1116.27/291.58	   p(s(x)) -> x
1116.27/291.58	   minus(x, 0) -> x
1116.27/291.58	   minus(s(x), s(y)) -> minus(x, y)
1116.27/291.58	   minus(x, s(y)) -> p(minus(x, y))
1116.27/291.58	   div(0, s(y)) -> 0
1116.27/291.58	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1116.27/291.58	   log(s(0), s(s(y))) -> 0
1116.27/291.58	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1116.27/291.58	
1116.27/291.58	S is empty.
1116.27/291.58	Rewrite Strategy: INNERMOST
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(3) DecreasingLoopProof (LOWER BOUND(ID))
1116.27/291.58	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1116.27/291.58	
1116.27/291.58	The rewrite sequence
1116.27/291.58	
1116.27/291.58	minus(s(x), s(y)) ->^+ minus(x, y)
1116.27/291.58	
1116.27/291.58	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1116.27/291.58	
1116.27/291.58	The pumping substitution is [x / s(x), y / s(y)].
1116.27/291.58	
1116.27/291.58	The result substitution is [ ].
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(4)
1116.27/291.58	Complex Obligation (BEST)
1116.27/291.58	
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(5)
1116.27/291.58	Obligation:
1116.27/291.58	Proved the lower bound n^1 for the following obligation:
1116.27/291.58	
1116.27/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	The TRS R consists of the following rules:
1116.27/291.58	
1116.27/291.58	   p(0) -> 0
1116.27/291.58	   p(s(x)) -> x
1116.27/291.58	   minus(x, 0) -> x
1116.27/291.58	   minus(s(x), s(y)) -> minus(x, y)
1116.27/291.58	   minus(x, s(y)) -> p(minus(x, y))
1116.27/291.58	   div(0, s(y)) -> 0
1116.27/291.58	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1116.27/291.58	   log(s(0), s(s(y))) -> 0
1116.27/291.58	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1116.27/291.58	
1116.27/291.58	S is empty.
1116.27/291.58	Rewrite Strategy: INNERMOST
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(6) LowerBoundPropagationProof (FINISHED)
1116.27/291.58	Propagated lower bound.
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(7)
1116.27/291.58	BOUNDS(n^1, INF)
1116.27/291.58	
1116.27/291.58	----------------------------------------
1116.27/291.58	
1116.27/291.58	(8)
1116.27/291.58	Obligation:
1116.27/291.58	Analyzing the following TRS for decreasing loops:
1116.27/291.58	
1116.27/291.58	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1116.27/291.58	
1116.27/291.58	
1116.27/291.58	The TRS R consists of the following rules:
1116.27/291.58	
1116.27/291.58	   p(0) -> 0
1116.27/291.58	   p(s(x)) -> x
1116.27/291.58	   minus(x, 0) -> x
1116.27/291.58	   minus(s(x), s(y)) -> minus(x, y)
1116.27/291.58	   minus(x, s(y)) -> p(minus(x, y))
1116.27/291.58	   div(0, s(y)) -> 0
1116.27/291.58	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1116.27/291.58	   log(s(0), s(s(y))) -> 0
1116.27/291.58	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1116.27/291.58	
1116.27/291.58	S is empty.
1116.27/291.58	Rewrite Strategy: INNERMOST
1116.46/291.65	EOF