317.81/291.46	WORST_CASE(Omega(n^1), ?)
317.81/291.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
317.81/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
317.81/291.47	
317.81/291.47	
317.81/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.81/291.47	
317.81/291.47	(0) CpxTRS
317.81/291.47	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
317.81/291.47	(2) TRS for Loop Detection
317.81/291.47	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
317.81/291.47	(4) BEST
317.81/291.47	    (5) proven lower bound
317.81/291.47	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
317.81/291.47	        (7) BOUNDS(n^1, INF)
317.81/291.47	    (8) TRS for Loop Detection
317.81/291.47	
317.81/291.47	
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(0)
317.81/291.47	Obligation:
317.81/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.81/291.47	
317.81/291.47	
317.81/291.47	The TRS R consists of the following rules:
317.81/291.47	
317.81/291.47	   check(0) -> zero
317.81/291.47	   check(s(0)) -> odd
317.81/291.47	   check(s(s(0))) -> even
317.81/291.47	   check(s(s(s(x)))) -> check(s(x))
317.81/291.47	   half(0) -> 0
317.81/291.47	   half(s(0)) -> 0
317.81/291.47	   half(s(s(x))) -> s(half(x))
317.81/291.47	   plus(0, y) -> y
317.81/291.47	   plus(s(x), y) -> s(plus(x, y))
317.81/291.47	   times(x, y) -> timesIter(x, y, 0)
317.81/291.47	   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
317.81/291.47	   p(s(x)) -> x
317.81/291.47	   p(0) -> 0
317.81/291.47	   if(zero, x, y, z, u) -> z
317.81/291.47	   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
317.81/291.47	   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))
317.81/291.47	
317.81/291.47	S is empty.
317.81/291.47	Rewrite Strategy: FULL
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
317.81/291.47	Transformed a relative TRS into a decreasing-loop problem.
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(2)
317.81/291.47	Obligation:
317.81/291.47	Analyzing the following TRS for decreasing loops:
317.81/291.47	
317.81/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.81/291.47	
317.81/291.47	
317.81/291.47	The TRS R consists of the following rules:
317.81/291.47	
317.81/291.47	   check(0) -> zero
317.81/291.47	   check(s(0)) -> odd
317.81/291.47	   check(s(s(0))) -> even
317.81/291.47	   check(s(s(s(x)))) -> check(s(x))
317.81/291.47	   half(0) -> 0
317.81/291.47	   half(s(0)) -> 0
317.81/291.47	   half(s(s(x))) -> s(half(x))
317.81/291.47	   plus(0, y) -> y
317.81/291.47	   plus(s(x), y) -> s(plus(x, y))
317.81/291.47	   times(x, y) -> timesIter(x, y, 0)
317.81/291.47	   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
317.81/291.47	   p(s(x)) -> x
317.81/291.47	   p(0) -> 0
317.81/291.47	   if(zero, x, y, z, u) -> z
317.81/291.47	   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
317.81/291.47	   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))
317.81/291.47	
317.81/291.47	S is empty.
317.81/291.47	Rewrite Strategy: FULL
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(3) DecreasingLoopProof (LOWER BOUND(ID))
317.81/291.47	The following loop(s) give(s) rise to the lower bound Omega(n^1):
317.81/291.47	
317.81/291.47	The rewrite sequence
317.81/291.47	
317.81/291.47	check(s(s(s(x)))) ->^+ check(s(x))
317.81/291.47	
317.81/291.47	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
317.81/291.47	
317.81/291.47	The pumping substitution is [x / s(s(x))].
317.81/291.47	
317.81/291.47	The result substitution is [ ].
317.81/291.47	
317.81/291.47	
317.81/291.47	
317.81/291.47	
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(4)
317.81/291.47	Complex Obligation (BEST)
317.81/291.47	
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(5)
317.81/291.47	Obligation:
317.81/291.47	Proved the lower bound n^1 for the following obligation:
317.81/291.47	
317.81/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.81/291.47	
317.81/291.47	
317.81/291.47	The TRS R consists of the following rules:
317.81/291.47	
317.81/291.47	   check(0) -> zero
317.81/291.47	   check(s(0)) -> odd
317.81/291.47	   check(s(s(0))) -> even
317.81/291.47	   check(s(s(s(x)))) -> check(s(x))
317.81/291.47	   half(0) -> 0
317.81/291.47	   half(s(0)) -> 0
317.81/291.47	   half(s(s(x))) -> s(half(x))
317.81/291.47	   plus(0, y) -> y
317.81/291.47	   plus(s(x), y) -> s(plus(x, y))
317.81/291.47	   times(x, y) -> timesIter(x, y, 0)
317.81/291.47	   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
317.81/291.47	   p(s(x)) -> x
317.81/291.47	   p(0) -> 0
317.81/291.47	   if(zero, x, y, z, u) -> z
317.81/291.47	   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
317.81/291.47	   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))
317.81/291.47	
317.81/291.47	S is empty.
317.81/291.47	Rewrite Strategy: FULL
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(6) LowerBoundPropagationProof (FINISHED)
317.81/291.47	Propagated lower bound.
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(7)
317.81/291.47	BOUNDS(n^1, INF)
317.81/291.47	
317.81/291.47	----------------------------------------
317.81/291.47	
317.81/291.47	(8)
317.81/291.47	Obligation:
317.81/291.47	Analyzing the following TRS for decreasing loops:
317.81/291.47	
317.81/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
317.81/291.47	
317.81/291.47	
317.81/291.47	The TRS R consists of the following rules:
317.81/291.47	
317.81/291.47	   check(0) -> zero
317.81/291.47	   check(s(0)) -> odd
317.81/291.47	   check(s(s(0))) -> even
317.81/291.47	   check(s(s(s(x)))) -> check(s(x))
317.81/291.47	   half(0) -> 0
317.81/291.47	   half(s(0)) -> 0
317.81/291.47	   half(s(s(x))) -> s(half(x))
317.81/291.47	   plus(0, y) -> y
317.81/291.47	   plus(s(x), y) -> s(plus(x, y))
317.81/291.47	   times(x, y) -> timesIter(x, y, 0)
317.81/291.47	   timesIter(x, y, z) -> if(check(x), x, y, z, plus(z, y))
317.81/291.47	   p(s(x)) -> x
317.81/291.47	   p(0) -> 0
317.81/291.47	   if(zero, x, y, z, u) -> z
317.81/291.47	   if(odd, x, y, z, u) -> timesIter(p(x), y, u)
317.81/291.47	   if(even, x, y, z, u) -> plus(timesIter(half(x), y, half(z)), timesIter(half(x), y, half(s(z))))
317.81/291.47	
317.81/291.47	S is empty.
317.81/291.47	Rewrite Strategy: FULL
317.91/291.50	EOF