316.39/291.52	WORST_CASE(Omega(n^1), ?)
316.39/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
316.39/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
316.39/291.53	
316.39/291.53	
316.39/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.39/291.53	
316.39/291.53	(0) CpxTRS
316.39/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
316.39/291.53	(2) TRS for Loop Detection
316.39/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
316.39/291.53	(4) BEST
316.39/291.53	    (5) proven lower bound
316.39/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
316.39/291.53	        (7) BOUNDS(n^1, INF)
316.39/291.53	    (8) TRS for Loop Detection
316.39/291.53	
316.39/291.53	
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(0)
316.39/291.53	Obligation:
316.39/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.39/291.53	
316.39/291.53	
316.39/291.53	The TRS R consists of the following rules:
316.39/291.53	
316.39/291.53	   ge(0, 0) -> true
316.39/291.53	   ge(s(x), 0) -> ge(x, 0)
316.39/291.53	   ge(0, s(0)) -> false
316.39/291.53	   ge(0, s(s(x))) -> ge(0, s(x))
316.39/291.53	   ge(s(x), s(y)) -> ge(x, y)
316.39/291.53	   minus(0, 0) -> 0
316.39/291.53	   minus(0, s(x)) -> minus(0, x)
316.39/291.53	   minus(s(x), 0) -> s(minus(x, 0))
316.39/291.53	   minus(s(x), s(y)) -> minus(x, y)
316.39/291.53	   plus(0, 0) -> 0
316.39/291.53	   plus(0, s(x)) -> s(plus(0, x))
316.39/291.53	   plus(s(x), y) -> s(plus(x, y))
316.39/291.53	   div(x, y) -> ify(ge(y, s(0)), x, y)
316.39/291.53	   ify(false, x, y) -> divByZeroError
316.39/291.53	   ify(true, x, y) -> if(ge(x, y), x, y)
316.39/291.53	   if(false, x, y) -> 0
316.39/291.53	   if(true, x, y) -> s(div(minus(x, y), y))
316.39/291.53	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
316.39/291.53	
316.39/291.53	S is empty.
316.39/291.53	Rewrite Strategy: FULL
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
316.39/291.53	Transformed a relative TRS into a decreasing-loop problem.
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(2)
316.39/291.53	Obligation:
316.39/291.53	Analyzing the following TRS for decreasing loops:
316.39/291.53	
316.39/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.39/291.53	
316.39/291.53	
316.39/291.53	The TRS R consists of the following rules:
316.39/291.53	
316.39/291.53	   ge(0, 0) -> true
316.39/291.53	   ge(s(x), 0) -> ge(x, 0)
316.39/291.53	   ge(0, s(0)) -> false
316.39/291.53	   ge(0, s(s(x))) -> ge(0, s(x))
316.39/291.53	   ge(s(x), s(y)) -> ge(x, y)
316.39/291.53	   minus(0, 0) -> 0
316.39/291.53	   minus(0, s(x)) -> minus(0, x)
316.39/291.53	   minus(s(x), 0) -> s(minus(x, 0))
316.39/291.53	   minus(s(x), s(y)) -> minus(x, y)
316.39/291.53	   plus(0, 0) -> 0
316.39/291.53	   plus(0, s(x)) -> s(plus(0, x))
316.39/291.53	   plus(s(x), y) -> s(plus(x, y))
316.39/291.53	   div(x, y) -> ify(ge(y, s(0)), x, y)
316.39/291.53	   ify(false, x, y) -> divByZeroError
316.39/291.53	   ify(true, x, y) -> if(ge(x, y), x, y)
316.39/291.53	   if(false, x, y) -> 0
316.39/291.53	   if(true, x, y) -> s(div(minus(x, y), y))
316.39/291.53	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
316.39/291.53	
316.39/291.53	S is empty.
316.39/291.53	Rewrite Strategy: FULL
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
316.39/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
316.39/291.53	
316.39/291.53	The rewrite sequence
316.39/291.53	
316.39/291.53	minus(s(x), 0) ->^+ s(minus(x, 0))
316.39/291.53	
316.39/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
316.39/291.53	
316.39/291.53	The pumping substitution is [x / s(x)].
316.39/291.53	
316.39/291.53	The result substitution is [ ].
316.39/291.53	
316.39/291.53	
316.39/291.53	
316.39/291.53	
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(4)
316.39/291.53	Complex Obligation (BEST)
316.39/291.53	
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(5)
316.39/291.53	Obligation:
316.39/291.53	Proved the lower bound n^1 for the following obligation:
316.39/291.53	
316.39/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.39/291.53	
316.39/291.53	
316.39/291.53	The TRS R consists of the following rules:
316.39/291.53	
316.39/291.53	   ge(0, 0) -> true
316.39/291.53	   ge(s(x), 0) -> ge(x, 0)
316.39/291.53	   ge(0, s(0)) -> false
316.39/291.53	   ge(0, s(s(x))) -> ge(0, s(x))
316.39/291.53	   ge(s(x), s(y)) -> ge(x, y)
316.39/291.53	   minus(0, 0) -> 0
316.39/291.53	   minus(0, s(x)) -> minus(0, x)
316.39/291.53	   minus(s(x), 0) -> s(minus(x, 0))
316.39/291.53	   minus(s(x), s(y)) -> minus(x, y)
316.39/291.53	   plus(0, 0) -> 0
316.39/291.53	   plus(0, s(x)) -> s(plus(0, x))
316.39/291.53	   plus(s(x), y) -> s(plus(x, y))
316.39/291.53	   div(x, y) -> ify(ge(y, s(0)), x, y)
316.39/291.53	   ify(false, x, y) -> divByZeroError
316.39/291.53	   ify(true, x, y) -> if(ge(x, y), x, y)
316.39/291.53	   if(false, x, y) -> 0
316.39/291.53	   if(true, x, y) -> s(div(minus(x, y), y))
316.39/291.53	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
316.39/291.53	
316.39/291.53	S is empty.
316.39/291.53	Rewrite Strategy: FULL
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(6) LowerBoundPropagationProof (FINISHED)
316.39/291.53	Propagated lower bound.
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(7)
316.39/291.53	BOUNDS(n^1, INF)
316.39/291.53	
316.39/291.53	----------------------------------------
316.39/291.53	
316.39/291.53	(8)
316.39/291.53	Obligation:
316.39/291.53	Analyzing the following TRS for decreasing loops:
316.39/291.53	
316.39/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
316.39/291.53	
316.39/291.53	
316.39/291.53	The TRS R consists of the following rules:
316.39/291.53	
316.39/291.53	   ge(0, 0) -> true
316.39/291.53	   ge(s(x), 0) -> ge(x, 0)
316.39/291.53	   ge(0, s(0)) -> false
316.39/291.53	   ge(0, s(s(x))) -> ge(0, s(x))
316.39/291.53	   ge(s(x), s(y)) -> ge(x, y)
316.39/291.53	   minus(0, 0) -> 0
316.39/291.53	   minus(0, s(x)) -> minus(0, x)
316.39/291.53	   minus(s(x), 0) -> s(minus(x, 0))
316.39/291.53	   minus(s(x), s(y)) -> minus(x, y)
316.39/291.53	   plus(0, 0) -> 0
316.39/291.53	   plus(0, s(x)) -> s(plus(0, x))
316.39/291.53	   plus(s(x), y) -> s(plus(x, y))
316.39/291.53	   div(x, y) -> ify(ge(y, s(0)), x, y)
316.39/291.53	   ify(false, x, y) -> divByZeroError
316.39/291.53	   ify(true, x, y) -> if(ge(x, y), x, y)
316.39/291.53	   if(false, x, y) -> 0
316.39/291.53	   if(true, x, y) -> s(div(minus(x, y), y))
316.39/291.53	   div(plus(x, y), z) -> plus(div(x, z), div(y, z))
316.39/291.53	
316.39/291.53	S is empty.
316.39/291.53	Rewrite Strategy: FULL
316.43/291.55	EOF