309.55/291.46	WORST_CASE(Omega(n^1), ?)
309.73/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
309.73/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
309.73/291.51	
309.73/291.51	
309.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.73/291.51	
309.73/291.51	(0) CpxTRS
309.73/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
309.73/291.51	(2) TRS for Loop Detection
309.73/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
309.73/291.51	(4) BEST
309.73/291.51	    (5) proven lower bound
309.73/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
309.73/291.51	        (7) BOUNDS(n^1, INF)
309.73/291.51	    (8) TRS for Loop Detection
309.73/291.51	
309.73/291.51	
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(0)
309.73/291.51	Obligation:
309.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.73/291.51	
309.73/291.51	
309.73/291.51	The TRS R consists of the following rules:
309.73/291.51	
309.73/291.51	   p(0) -> s(s(0))
309.73/291.51	   p(s(x)) -> x
309.73/291.51	   p(p(s(x))) -> p(x)
309.73/291.51	   le(p(s(x)), x) -> le(x, x)
309.73/291.51	   le(0, y) -> true
309.73/291.51	   le(s(x), 0) -> false
309.73/291.51	   le(s(x), s(y)) -> le(x, y)
309.73/291.51	   minus(x, y) -> if(le(x, y), x, y)
309.73/291.51	   if(true, x, y) -> 0
309.73/291.51	   if(false, x, y) -> s(minus(p(x), y))
309.73/291.51	
309.73/291.51	S is empty.
309.73/291.51	Rewrite Strategy: FULL
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
309.73/291.51	Transformed a relative TRS into a decreasing-loop problem.
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(2)
309.73/291.51	Obligation:
309.73/291.51	Analyzing the following TRS for decreasing loops:
309.73/291.51	
309.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.73/291.51	
309.73/291.51	
309.73/291.51	The TRS R consists of the following rules:
309.73/291.51	
309.73/291.51	   p(0) -> s(s(0))
309.73/291.51	   p(s(x)) -> x
309.73/291.51	   p(p(s(x))) -> p(x)
309.73/291.51	   le(p(s(x)), x) -> le(x, x)
309.73/291.51	   le(0, y) -> true
309.73/291.51	   le(s(x), 0) -> false
309.73/291.51	   le(s(x), s(y)) -> le(x, y)
309.73/291.51	   minus(x, y) -> if(le(x, y), x, y)
309.73/291.51	   if(true, x, y) -> 0
309.73/291.51	   if(false, x, y) -> s(minus(p(x), y))
309.73/291.51	
309.73/291.51	S is empty.
309.73/291.51	Rewrite Strategy: FULL
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
309.73/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
309.73/291.51	
309.73/291.51	The rewrite sequence
309.73/291.51	
309.73/291.51	le(s(x), s(y)) ->^+ le(x, y)
309.73/291.51	
309.73/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
309.73/291.51	
309.73/291.51	The pumping substitution is [x / s(x), y / s(y)].
309.73/291.51	
309.73/291.51	The result substitution is [ ].
309.73/291.51	
309.73/291.51	
309.73/291.51	
309.73/291.51	
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(4)
309.73/291.51	Complex Obligation (BEST)
309.73/291.51	
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(5)
309.73/291.51	Obligation:
309.73/291.51	Proved the lower bound n^1 for the following obligation:
309.73/291.51	
309.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.73/291.51	
309.73/291.51	
309.73/291.51	The TRS R consists of the following rules:
309.73/291.51	
309.73/291.51	   p(0) -> s(s(0))
309.73/291.51	   p(s(x)) -> x
309.73/291.51	   p(p(s(x))) -> p(x)
309.73/291.51	   le(p(s(x)), x) -> le(x, x)
309.73/291.51	   le(0, y) -> true
309.73/291.51	   le(s(x), 0) -> false
309.73/291.51	   le(s(x), s(y)) -> le(x, y)
309.73/291.51	   minus(x, y) -> if(le(x, y), x, y)
309.73/291.51	   if(true, x, y) -> 0
309.73/291.51	   if(false, x, y) -> s(minus(p(x), y))
309.73/291.51	
309.73/291.51	S is empty.
309.73/291.51	Rewrite Strategy: FULL
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(6) LowerBoundPropagationProof (FINISHED)
309.73/291.51	Propagated lower bound.
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(7)
309.73/291.51	BOUNDS(n^1, INF)
309.73/291.51	
309.73/291.51	----------------------------------------
309.73/291.51	
309.73/291.51	(8)
309.73/291.51	Obligation:
309.73/291.51	Analyzing the following TRS for decreasing loops:
309.73/291.51	
309.73/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
309.73/291.51	
309.73/291.51	
309.73/291.51	The TRS R consists of the following rules:
309.73/291.51	
309.73/291.51	   p(0) -> s(s(0))
309.73/291.51	   p(s(x)) -> x
309.73/291.51	   p(p(s(x))) -> p(x)
309.73/291.51	   le(p(s(x)), x) -> le(x, x)
309.73/291.51	   le(0, y) -> true
309.73/291.51	   le(s(x), 0) -> false
309.73/291.51	   le(s(x), s(y)) -> le(x, y)
309.73/291.51	   minus(x, y) -> if(le(x, y), x, y)
309.73/291.51	   if(true, x, y) -> 0
309.73/291.51	   if(false, x, y) -> s(minus(p(x), y))
309.73/291.51	
309.73/291.51	S is empty.
309.73/291.51	Rewrite Strategy: FULL
309.74/291.55	EOF