1114.67/291.52	WORST_CASE(Omega(n^1), ?)
1115.00/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1115.00/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.00/291.57	
1115.00/291.57	(0) CpxTRS
1115.00/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1115.00/291.57	(2) TRS for Loop Detection
1115.00/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1115.00/291.57	(4) BEST
1115.00/291.57	    (5) proven lower bound
1115.00/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1115.00/291.57	        (7) BOUNDS(n^1, INF)
1115.00/291.57	    (8) TRS for Loop Detection
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(0)
1115.00/291.57	Obligation:
1115.00/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	The TRS R consists of the following rules:
1115.00/291.57	
1115.00/291.57	   p(0) -> 0
1115.00/291.57	   p(s(x)) -> x
1115.00/291.57	   minus(x, 0) -> x
1115.00/291.57	   minus(s(x), s(y)) -> minus(x, y)
1115.00/291.57	   minus(x, s(y)) -> p(minus(x, y))
1115.00/291.57	   div(0, s(y)) -> 0
1115.00/291.57	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1115.00/291.57	   log(s(0), s(s(y))) -> 0
1115.00/291.57	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1115.00/291.57	
1115.00/291.57	S is empty.
1115.00/291.57	Rewrite Strategy: FULL
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1115.00/291.57	Transformed a relative TRS into a decreasing-loop problem.
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(2)
1115.00/291.57	Obligation:
1115.00/291.57	Analyzing the following TRS for decreasing loops:
1115.00/291.57	
1115.00/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	The TRS R consists of the following rules:
1115.00/291.57	
1115.00/291.57	   p(0) -> 0
1115.00/291.57	   p(s(x)) -> x
1115.00/291.57	   minus(x, 0) -> x
1115.00/291.57	   minus(s(x), s(y)) -> minus(x, y)
1115.00/291.57	   minus(x, s(y)) -> p(minus(x, y))
1115.00/291.57	   div(0, s(y)) -> 0
1115.00/291.57	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1115.00/291.57	   log(s(0), s(s(y))) -> 0
1115.00/291.57	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1115.00/291.57	
1115.00/291.57	S is empty.
1115.00/291.57	Rewrite Strategy: FULL
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
1115.00/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1115.00/291.57	
1115.00/291.57	The rewrite sequence
1115.00/291.57	
1115.00/291.57	minus(s(x), s(y)) ->^+ minus(x, y)
1115.00/291.57	
1115.00/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1115.00/291.57	
1115.00/291.57	The pumping substitution is [x / s(x), y / s(y)].
1115.00/291.57	
1115.00/291.57	The result substitution is [ ].
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(4)
1115.00/291.57	Complex Obligation (BEST)
1115.00/291.57	
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(5)
1115.00/291.57	Obligation:
1115.00/291.57	Proved the lower bound n^1 for the following obligation:
1115.00/291.57	
1115.00/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	The TRS R consists of the following rules:
1115.00/291.57	
1115.00/291.57	   p(0) -> 0
1115.00/291.57	   p(s(x)) -> x
1115.00/291.57	   minus(x, 0) -> x
1115.00/291.57	   minus(s(x), s(y)) -> minus(x, y)
1115.00/291.57	   minus(x, s(y)) -> p(minus(x, y))
1115.00/291.57	   div(0, s(y)) -> 0
1115.00/291.57	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1115.00/291.57	   log(s(0), s(s(y))) -> 0
1115.00/291.57	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1115.00/291.57	
1115.00/291.57	S is empty.
1115.00/291.57	Rewrite Strategy: FULL
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(6) LowerBoundPropagationProof (FINISHED)
1115.00/291.57	Propagated lower bound.
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(7)
1115.00/291.57	BOUNDS(n^1, INF)
1115.00/291.57	
1115.00/291.57	----------------------------------------
1115.00/291.57	
1115.00/291.57	(8)
1115.00/291.57	Obligation:
1115.00/291.57	Analyzing the following TRS for decreasing loops:
1115.00/291.57	
1115.00/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1115.00/291.57	
1115.00/291.57	
1115.00/291.57	The TRS R consists of the following rules:
1115.00/291.57	
1115.00/291.57	   p(0) -> 0
1115.00/291.57	   p(s(x)) -> x
1115.00/291.57	   minus(x, 0) -> x
1115.00/291.57	   minus(s(x), s(y)) -> minus(x, y)
1115.00/291.57	   minus(x, s(y)) -> p(minus(x, y))
1115.00/291.57	   div(0, s(y)) -> 0
1115.00/291.57	   div(s(x), s(y)) -> s(div(minus(s(x), s(y)), s(y)))
1115.00/291.57	   log(s(0), s(s(y))) -> 0
1115.00/291.57	   log(s(s(x)), s(s(y))) -> s(log(div(minus(x, y), s(s(y))), s(s(y))))
1115.00/291.57	
1115.00/291.57	S is empty.
1115.00/291.57	Rewrite Strategy: FULL
1115.09/291.66	EOF