318.05/291.48	WORST_CASE(Omega(n^1), ?)
318.05/291.49	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
318.05/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
318.05/291.49	
318.05/291.49	
318.05/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.05/291.49	
318.05/291.49	(0) CpxTRS
318.05/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
318.05/291.49	(2) TRS for Loop Detection
318.05/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
318.05/291.49	(4) BEST
318.05/291.49	    (5) proven lower bound
318.05/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
318.05/291.49	        (7) BOUNDS(n^1, INF)
318.05/291.49	    (8) TRS for Loop Detection
318.05/291.49	
318.05/291.49	
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(0)
318.05/291.49	Obligation:
318.05/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.05/291.49	
318.05/291.49	
318.05/291.49	The TRS R consists of the following rules:
318.05/291.49	
318.05/291.49	   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
318.05/291.49	   ifPlus(true, x, y) -> p(y)
318.05/291.49	   ifPlus(false, x, y) -> plus(p(x), y)
318.05/291.49	   times(x, y) -> timesIter(0, x, y, 0)
318.05/291.49	   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
318.05/291.49	   ifTimes(true, i, x, y, z) -> z
318.05/291.49	   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
318.05/291.49	   isZero(0) -> true
318.05/291.49	   isZero(s(0)) -> false
318.05/291.49	   isZero(s(s(x))) -> isZero(s(x))
318.05/291.49	   inc(0) -> s(0)
318.05/291.49	   inc(s(x)) -> s(inc(x))
318.05/291.49	   inc(x) -> s(x)
318.05/291.49	   p(0) -> 0
318.05/291.49	   p(s(x)) -> x
318.05/291.49	   p(s(s(x))) -> s(p(s(x)))
318.05/291.49	   ge(x, 0) -> true
318.05/291.49	   ge(0, s(y)) -> false
318.05/291.49	   ge(s(x), s(y)) -> ge(x, y)
318.05/291.49	   f0(0, y, x) -> f1(x, y, x)
318.05/291.49	   f1(x, y, z) -> f2(x, y, z)
318.05/291.49	   f2(x, 1, z) -> f0(x, z, z)
318.05/291.49	   f0(x, y, z) -> d
318.05/291.49	   f1(x, y, z) -> c
318.05/291.49	
318.05/291.49	S is empty.
318.05/291.49	Rewrite Strategy: FULL
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
318.05/291.49	Transformed a relative TRS into a decreasing-loop problem.
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(2)
318.05/291.49	Obligation:
318.05/291.49	Analyzing the following TRS for decreasing loops:
318.05/291.49	
318.05/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.05/291.49	
318.05/291.49	
318.05/291.49	The TRS R consists of the following rules:
318.05/291.49	
318.05/291.49	   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
318.05/291.49	   ifPlus(true, x, y) -> p(y)
318.05/291.49	   ifPlus(false, x, y) -> plus(p(x), y)
318.05/291.49	   times(x, y) -> timesIter(0, x, y, 0)
318.05/291.49	   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
318.05/291.49	   ifTimes(true, i, x, y, z) -> z
318.05/291.49	   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
318.05/291.49	   isZero(0) -> true
318.05/291.49	   isZero(s(0)) -> false
318.05/291.49	   isZero(s(s(x))) -> isZero(s(x))
318.05/291.49	   inc(0) -> s(0)
318.05/291.49	   inc(s(x)) -> s(inc(x))
318.05/291.49	   inc(x) -> s(x)
318.05/291.49	   p(0) -> 0
318.05/291.49	   p(s(x)) -> x
318.05/291.49	   p(s(s(x))) -> s(p(s(x)))
318.05/291.49	   ge(x, 0) -> true
318.05/291.49	   ge(0, s(y)) -> false
318.05/291.49	   ge(s(x), s(y)) -> ge(x, y)
318.05/291.49	   f0(0, y, x) -> f1(x, y, x)
318.05/291.49	   f1(x, y, z) -> f2(x, y, z)
318.05/291.49	   f2(x, 1, z) -> f0(x, z, z)
318.05/291.49	   f0(x, y, z) -> d
318.05/291.49	   f1(x, y, z) -> c
318.05/291.49	
318.05/291.49	S is empty.
318.05/291.49	Rewrite Strategy: FULL
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
318.05/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
318.05/291.49	
318.05/291.49	The rewrite sequence
318.05/291.49	
318.05/291.49	isZero(s(s(x))) ->^+ isZero(s(x))
318.05/291.49	
318.05/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
318.05/291.49	
318.05/291.49	The pumping substitution is [x / s(x)].
318.05/291.49	
318.05/291.49	The result substitution is [ ].
318.05/291.49	
318.05/291.49	
318.05/291.49	
318.05/291.49	
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(4)
318.05/291.49	Complex Obligation (BEST)
318.05/291.49	
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(5)
318.05/291.49	Obligation:
318.05/291.49	Proved the lower bound n^1 for the following obligation:
318.05/291.49	
318.05/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.05/291.49	
318.05/291.49	
318.05/291.49	The TRS R consists of the following rules:
318.05/291.49	
318.05/291.49	   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
318.05/291.49	   ifPlus(true, x, y) -> p(y)
318.05/291.49	   ifPlus(false, x, y) -> plus(p(x), y)
318.05/291.49	   times(x, y) -> timesIter(0, x, y, 0)
318.05/291.49	   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
318.05/291.49	   ifTimes(true, i, x, y, z) -> z
318.05/291.49	   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
318.05/291.49	   isZero(0) -> true
318.05/291.49	   isZero(s(0)) -> false
318.05/291.49	   isZero(s(s(x))) -> isZero(s(x))
318.05/291.49	   inc(0) -> s(0)
318.05/291.49	   inc(s(x)) -> s(inc(x))
318.05/291.49	   inc(x) -> s(x)
318.05/291.49	   p(0) -> 0
318.05/291.49	   p(s(x)) -> x
318.05/291.49	   p(s(s(x))) -> s(p(s(x)))
318.05/291.49	   ge(x, 0) -> true
318.05/291.49	   ge(0, s(y)) -> false
318.05/291.49	   ge(s(x), s(y)) -> ge(x, y)
318.05/291.49	   f0(0, y, x) -> f1(x, y, x)
318.05/291.49	   f1(x, y, z) -> f2(x, y, z)
318.05/291.49	   f2(x, 1, z) -> f0(x, z, z)
318.05/291.49	   f0(x, y, z) -> d
318.05/291.49	   f1(x, y, z) -> c
318.05/291.49	
318.05/291.49	S is empty.
318.05/291.49	Rewrite Strategy: FULL
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(6) LowerBoundPropagationProof (FINISHED)
318.05/291.49	Propagated lower bound.
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(7)
318.05/291.49	BOUNDS(n^1, INF)
318.05/291.49	
318.05/291.49	----------------------------------------
318.05/291.49	
318.05/291.49	(8)
318.05/291.49	Obligation:
318.05/291.49	Analyzing the following TRS for decreasing loops:
318.05/291.49	
318.05/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.05/291.49	
318.05/291.49	
318.05/291.49	The TRS R consists of the following rules:
318.05/291.49	
318.05/291.49	   plus(x, y) -> ifPlus(isZero(x), x, inc(y))
318.05/291.49	   ifPlus(true, x, y) -> p(y)
318.05/291.49	   ifPlus(false, x, y) -> plus(p(x), y)
318.05/291.49	   times(x, y) -> timesIter(0, x, y, 0)
318.05/291.49	   timesIter(i, x, y, z) -> ifTimes(ge(i, x), i, x, y, z)
318.05/291.49	   ifTimes(true, i, x, y, z) -> z
318.05/291.49	   ifTimes(false, i, x, y, z) -> timesIter(inc(i), x, y, plus(z, y))
318.05/291.49	   isZero(0) -> true
318.05/291.49	   isZero(s(0)) -> false
318.05/291.49	   isZero(s(s(x))) -> isZero(s(x))
318.05/291.49	   inc(0) -> s(0)
318.05/291.49	   inc(s(x)) -> s(inc(x))
318.05/291.49	   inc(x) -> s(x)
318.05/291.49	   p(0) -> 0
318.05/291.49	   p(s(x)) -> x
318.05/291.49	   p(s(s(x))) -> s(p(s(x)))
318.05/291.49	   ge(x, 0) -> true
318.05/291.49	   ge(0, s(y)) -> false
318.05/291.49	   ge(s(x), s(y)) -> ge(x, y)
318.05/291.49	   f0(0, y, x) -> f1(x, y, x)
318.05/291.49	   f1(x, y, z) -> f2(x, y, z)
318.05/291.49	   f2(x, 1, z) -> f0(x, z, z)
318.05/291.49	   f0(x, y, z) -> d
318.05/291.49	   f1(x, y, z) -> c
318.05/291.49	
318.05/291.49	S is empty.
318.05/291.49	Rewrite Strategy: FULL
318.13/291.51	EOF