1130.41/291.58	WORST_CASE(Omega(n^1), ?)
1130.41/291.59	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1130.41/291.59	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1130.41/291.59	
1130.41/291.59	(0) CpxTRS
1130.41/291.59	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1130.41/291.59	(2) TRS for Loop Detection
1130.41/291.59	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1130.41/291.59	(4) BEST
1130.41/291.59	    (5) proven lower bound
1130.41/291.59	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1130.41/291.59	        (7) BOUNDS(n^1, INF)
1130.41/291.59	    (8) TRS for Loop Detection
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(0)
1130.41/291.59	Obligation:
1130.41/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	The TRS R consists of the following rules:
1130.41/291.59	
1130.41/291.59	   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
1130.41/291.59	   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
1130.41/291.59	   if(true, x, y, z, u) -> z
1130.41/291.59	   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
1130.41/291.59	   if2(true, x, y, z, u) -> z
1130.41/291.59	   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
1130.41/291.59	   plus(0, y) -> y
1130.41/291.59	   plus(s(x), y) -> s(plus(x, y))
1130.41/291.59	   times(x, y) -> ifTimes(ge(0, x), x, y)
1130.41/291.59	   ifTimes(true, x, y) -> 0
1130.41/291.59	   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
1130.41/291.59	   p(s(x)) -> x
1130.41/291.59	   p(0) -> s(s(0))
1130.41/291.59	   ge(x, 0) -> true
1130.41/291.59	   ge(0, s(y)) -> false
1130.41/291.59	   ge(s(x), s(y)) -> ge(x, y)
1130.41/291.59	   or(true, y) -> true
1130.41/291.59	   or(false, y) -> y
1130.41/291.59	   divisible(0, s(y)) -> true
1130.41/291.59	   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
1130.41/291.59	   div(x, y, 0) -> divisible(x, y)
1130.41/291.59	   div(0, y, s(z)) -> false
1130.41/291.59	   div(s(x), y, s(z)) -> div(x, y, z)
1130.41/291.59	   a -> b
1130.41/291.59	   a -> c
1130.41/291.59	
1130.41/291.59	S is empty.
1130.41/291.59	Rewrite Strategy: INNERMOST
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1130.41/291.59	Transformed a relative TRS into a decreasing-loop problem.
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(2)
1130.41/291.59	Obligation:
1130.41/291.59	Analyzing the following TRS for decreasing loops:
1130.41/291.59	
1130.41/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	The TRS R consists of the following rules:
1130.41/291.59	
1130.41/291.59	   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
1130.41/291.59	   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
1130.41/291.59	   if(true, x, y, z, u) -> z
1130.41/291.59	   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
1130.41/291.59	   if2(true, x, y, z, u) -> z
1130.41/291.59	   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
1130.41/291.59	   plus(0, y) -> y
1130.41/291.59	   plus(s(x), y) -> s(plus(x, y))
1130.41/291.59	   times(x, y) -> ifTimes(ge(0, x), x, y)
1130.41/291.59	   ifTimes(true, x, y) -> 0
1130.41/291.59	   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
1130.41/291.59	   p(s(x)) -> x
1130.41/291.59	   p(0) -> s(s(0))
1130.41/291.59	   ge(x, 0) -> true
1130.41/291.59	   ge(0, s(y)) -> false
1130.41/291.59	   ge(s(x), s(y)) -> ge(x, y)
1130.41/291.59	   or(true, y) -> true
1130.41/291.59	   or(false, y) -> y
1130.41/291.59	   divisible(0, s(y)) -> true
1130.41/291.59	   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
1130.41/291.59	   div(x, y, 0) -> divisible(x, y)
1130.41/291.59	   div(0, y, s(z)) -> false
1130.41/291.59	   div(s(x), y, s(z)) -> div(x, y, z)
1130.41/291.59	   a -> b
1130.41/291.59	   a -> c
1130.41/291.59	
1130.41/291.59	S is empty.
1130.41/291.59	Rewrite Strategy: INNERMOST
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(3) DecreasingLoopProof (LOWER BOUND(ID))
1130.41/291.59	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1130.41/291.59	
1130.41/291.59	The rewrite sequence
1130.41/291.59	
1130.41/291.59	div(s(x), y, s(z)) ->^+ div(x, y, z)
1130.41/291.59	
1130.41/291.59	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1130.41/291.59	
1130.41/291.59	The pumping substitution is [x / s(x), z / s(z)].
1130.41/291.59	
1130.41/291.59	The result substitution is [ ].
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(4)
1130.41/291.59	Complex Obligation (BEST)
1130.41/291.59	
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(5)
1130.41/291.59	Obligation:
1130.41/291.59	Proved the lower bound n^1 for the following obligation:
1130.41/291.59	
1130.41/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	The TRS R consists of the following rules:
1130.41/291.59	
1130.41/291.59	   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
1130.41/291.59	   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
1130.41/291.59	   if(true, x, y, z, u) -> z
1130.41/291.59	   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
1130.41/291.59	   if2(true, x, y, z, u) -> z
1130.41/291.59	   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
1130.41/291.59	   plus(0, y) -> y
1130.41/291.59	   plus(s(x), y) -> s(plus(x, y))
1130.41/291.59	   times(x, y) -> ifTimes(ge(0, x), x, y)
1130.41/291.59	   ifTimes(true, x, y) -> 0
1130.41/291.59	   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
1130.41/291.59	   p(s(x)) -> x
1130.41/291.59	   p(0) -> s(s(0))
1130.41/291.59	   ge(x, 0) -> true
1130.41/291.59	   ge(0, s(y)) -> false
1130.41/291.59	   ge(s(x), s(y)) -> ge(x, y)
1130.41/291.59	   or(true, y) -> true
1130.41/291.59	   or(false, y) -> y
1130.41/291.59	   divisible(0, s(y)) -> true
1130.41/291.59	   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
1130.41/291.59	   div(x, y, 0) -> divisible(x, y)
1130.41/291.59	   div(0, y, s(z)) -> false
1130.41/291.59	   div(s(x), y, s(z)) -> div(x, y, z)
1130.41/291.59	   a -> b
1130.41/291.59	   a -> c
1130.41/291.59	
1130.41/291.59	S is empty.
1130.41/291.59	Rewrite Strategy: INNERMOST
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(6) LowerBoundPropagationProof (FINISHED)
1130.41/291.59	Propagated lower bound.
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(7)
1130.41/291.59	BOUNDS(n^1, INF)
1130.41/291.59	
1130.41/291.59	----------------------------------------
1130.41/291.59	
1130.41/291.59	(8)
1130.41/291.59	Obligation:
1130.41/291.59	Analyzing the following TRS for decreasing loops:
1130.41/291.59	
1130.41/291.59	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1130.41/291.59	
1130.41/291.59	
1130.41/291.59	The TRS R consists of the following rules:
1130.41/291.59	
1130.41/291.59	   lcm(x, y) -> lcmIter(x, y, 0, times(x, y))
1130.41/291.59	   lcmIter(x, y, z, u) -> if(or(ge(0, x), ge(z, u)), x, y, z, u)
1130.41/291.59	   if(true, x, y, z, u) -> z
1130.41/291.59	   if(false, x, y, z, u) -> if2(divisible(z, y), x, y, z, u)
1130.41/291.59	   if2(true, x, y, z, u) -> z
1130.41/291.59	   if2(false, x, y, z, u) -> lcmIter(x, y, plus(x, z), u)
1130.41/291.59	   plus(0, y) -> y
1130.41/291.59	   plus(s(x), y) -> s(plus(x, y))
1130.41/291.59	   times(x, y) -> ifTimes(ge(0, x), x, y)
1130.41/291.59	   ifTimes(true, x, y) -> 0
1130.41/291.59	   ifTimes(false, x, y) -> plus(y, times(y, p(x)))
1130.41/291.59	   p(s(x)) -> x
1130.41/291.59	   p(0) -> s(s(0))
1130.41/291.59	   ge(x, 0) -> true
1130.41/291.59	   ge(0, s(y)) -> false
1130.41/291.59	   ge(s(x), s(y)) -> ge(x, y)
1130.41/291.59	   or(true, y) -> true
1130.41/291.59	   or(false, y) -> y
1130.41/291.59	   divisible(0, s(y)) -> true
1130.41/291.59	   divisible(s(x), s(y)) -> div(s(x), s(y), s(y))
1130.41/291.59	   div(x, y, 0) -> divisible(x, y)
1130.41/291.59	   div(0, y, s(z)) -> false
1130.41/291.59	   div(s(x), y, s(z)) -> div(x, y, z)
1130.41/291.59	   a -> b
1130.41/291.59	   a -> c
1130.41/291.59	
1130.41/291.59	S is empty.
1130.41/291.59	Rewrite Strategy: INNERMOST
1130.60/291.65	EOF