303.84/291.49	WORST_CASE(Omega(n^1), ?)
303.84/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
303.84/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.84/291.49	
303.84/291.49	
303.84/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.84/291.49	
303.84/291.49	(0) CpxTRS
303.84/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.84/291.49	(2) TRS for Loop Detection
303.84/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
303.84/291.49	(4) BEST
303.84/291.49	    (5) proven lower bound
303.84/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.84/291.49	        (7) BOUNDS(n^1, INF)
303.84/291.49	    (8) TRS for Loop Detection
303.84/291.49	
303.84/291.49	
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(0)
303.84/291.49	Obligation:
303.84/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.84/291.49	
303.84/291.49	
303.84/291.49	The TRS R consists of the following rules:
303.84/291.49	
303.84/291.49	   p(0) -> 0
303.84/291.49	   p(s(x)) -> x
303.84/291.49	   le(0, y) -> true
303.84/291.49	   le(s(x), 0) -> false
303.84/291.49	   le(s(x), s(y)) -> le(x, y)
303.84/291.49	   minus(x, y) -> if(le(x, y), x, y)
303.84/291.49	   if(true, x, y) -> 0
303.84/291.49	   if(false, x, y) -> s(minus(p(x), y))
303.84/291.49	
303.84/291.49	S is empty.
303.84/291.49	Rewrite Strategy: FULL
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.84/291.49	Transformed a relative TRS into a decreasing-loop problem.
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(2)
303.84/291.49	Obligation:
303.84/291.49	Analyzing the following TRS for decreasing loops:
303.84/291.49	
303.84/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.84/291.49	
303.84/291.49	
303.84/291.49	The TRS R consists of the following rules:
303.84/291.49	
303.84/291.49	   p(0) -> 0
303.84/291.49	   p(s(x)) -> x
303.84/291.49	   le(0, y) -> true
303.84/291.49	   le(s(x), 0) -> false
303.84/291.49	   le(s(x), s(y)) -> le(x, y)
303.84/291.49	   minus(x, y) -> if(le(x, y), x, y)
303.84/291.49	   if(true, x, y) -> 0
303.84/291.49	   if(false, x, y) -> s(minus(p(x), y))
303.84/291.49	
303.84/291.49	S is empty.
303.84/291.49	Rewrite Strategy: FULL
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.84/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.84/291.49	
303.84/291.49	The rewrite sequence
303.84/291.49	
303.84/291.49	le(s(x), s(y)) ->^+ le(x, y)
303.84/291.49	
303.84/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
303.84/291.49	
303.84/291.49	The pumping substitution is [x / s(x), y / s(y)].
303.84/291.49	
303.84/291.49	The result substitution is [ ].
303.84/291.49	
303.84/291.49	
303.84/291.49	
303.84/291.49	
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(4)
303.84/291.49	Complex Obligation (BEST)
303.84/291.49	
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(5)
303.84/291.49	Obligation:
303.84/291.49	Proved the lower bound n^1 for the following obligation:
303.84/291.49	
303.84/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.84/291.49	
303.84/291.49	
303.84/291.49	The TRS R consists of the following rules:
303.84/291.49	
303.84/291.49	   p(0) -> 0
303.84/291.49	   p(s(x)) -> x
303.84/291.49	   le(0, y) -> true
303.84/291.49	   le(s(x), 0) -> false
303.84/291.49	   le(s(x), s(y)) -> le(x, y)
303.84/291.49	   minus(x, y) -> if(le(x, y), x, y)
303.84/291.49	   if(true, x, y) -> 0
303.84/291.49	   if(false, x, y) -> s(minus(p(x), y))
303.84/291.49	
303.84/291.49	S is empty.
303.84/291.49	Rewrite Strategy: FULL
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(6) LowerBoundPropagationProof (FINISHED)
303.84/291.49	Propagated lower bound.
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(7)
303.84/291.49	BOUNDS(n^1, INF)
303.84/291.49	
303.84/291.49	----------------------------------------
303.84/291.49	
303.84/291.49	(8)
303.84/291.49	Obligation:
303.84/291.49	Analyzing the following TRS for decreasing loops:
303.84/291.49	
303.84/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.84/291.49	
303.84/291.49	
303.84/291.49	The TRS R consists of the following rules:
303.84/291.49	
303.84/291.49	   p(0) -> 0
303.84/291.49	   p(s(x)) -> x
303.84/291.49	   le(0, y) -> true
303.84/291.49	   le(s(x), 0) -> false
303.84/291.49	   le(s(x), s(y)) -> le(x, y)
303.84/291.49	   minus(x, y) -> if(le(x, y), x, y)
303.84/291.49	   if(true, x, y) -> 0
303.84/291.49	   if(false, x, y) -> s(minus(p(x), y))
303.84/291.49	
303.84/291.49	S is empty.
303.84/291.49	Rewrite Strategy: FULL
303.93/291.52	EOF