318.14/291.51	WORST_CASE(Omega(n^1), ?)
318.14/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
318.14/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
318.14/291.52	
318.14/291.52	
318.14/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.52	
318.14/291.52	(0) CpxTRS
318.14/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
318.14/291.52	(2) TRS for Loop Detection
318.14/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
318.14/291.52	(4) BEST
318.14/291.52	    (5) proven lower bound
318.14/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
318.14/291.52	        (7) BOUNDS(n^1, INF)
318.14/291.52	    (8) TRS for Loop Detection
318.14/291.52	
318.14/291.52	
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(0)
318.14/291.52	Obligation:
318.14/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.52	
318.14/291.52	
318.14/291.52	The TRS R consists of the following rules:
318.14/291.52	
318.14/291.52	   ge(0, 0) -> true
318.14/291.52	   ge(s(x), 0) -> ge(x, 0)
318.14/291.52	   ge(0, s(0)) -> false
318.14/291.52	   ge(0, s(s(x))) -> ge(0, s(x))
318.14/291.52	   ge(s(x), s(y)) -> ge(x, y)
318.14/291.52	   minus(0, 0) -> 0
318.14/291.52	   minus(0, s(x)) -> minus(0, x)
318.14/291.52	   minus(s(x), 0) -> s(minus(x, 0))
318.14/291.52	   minus(s(x), s(y)) -> minus(x, y)
318.14/291.52	   plus(0, 0) -> 0
318.14/291.52	   plus(0, s(x)) -> s(plus(0, x))
318.14/291.52	   plus(s(x), y) -> s(plus(x, y))
318.14/291.52	   div(x, y) -> ify(ge(y, s(0)), x, y)
318.14/291.52	   ify(false, x, y) -> divByZeroError
318.14/291.52	   ify(true, x, y) -> if(ge(x, y), x, y)
318.14/291.52	   if(false, x, y) -> 0
318.14/291.52	   if(true, x, y) -> s(div(minus(x, y), y))
318.14/291.52	
318.14/291.52	S is empty.
318.14/291.52	Rewrite Strategy: FULL
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
318.14/291.52	Transformed a relative TRS into a decreasing-loop problem.
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(2)
318.14/291.52	Obligation:
318.14/291.52	Analyzing the following TRS for decreasing loops:
318.14/291.52	
318.14/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.52	
318.14/291.52	
318.14/291.52	The TRS R consists of the following rules:
318.14/291.52	
318.14/291.52	   ge(0, 0) -> true
318.14/291.52	   ge(s(x), 0) -> ge(x, 0)
318.14/291.52	   ge(0, s(0)) -> false
318.14/291.52	   ge(0, s(s(x))) -> ge(0, s(x))
318.14/291.52	   ge(s(x), s(y)) -> ge(x, y)
318.14/291.52	   minus(0, 0) -> 0
318.14/291.52	   minus(0, s(x)) -> minus(0, x)
318.14/291.52	   minus(s(x), 0) -> s(minus(x, 0))
318.14/291.52	   minus(s(x), s(y)) -> minus(x, y)
318.14/291.52	   plus(0, 0) -> 0
318.14/291.52	   plus(0, s(x)) -> s(plus(0, x))
318.14/291.52	   plus(s(x), y) -> s(plus(x, y))
318.14/291.52	   div(x, y) -> ify(ge(y, s(0)), x, y)
318.14/291.52	   ify(false, x, y) -> divByZeroError
318.14/291.52	   ify(true, x, y) -> if(ge(x, y), x, y)
318.14/291.52	   if(false, x, y) -> 0
318.14/291.52	   if(true, x, y) -> s(div(minus(x, y), y))
318.14/291.52	
318.14/291.52	S is empty.
318.14/291.52	Rewrite Strategy: FULL
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
318.14/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
318.14/291.52	
318.14/291.52	The rewrite sequence
318.14/291.52	
318.14/291.52	minus(s(x), 0) ->^+ s(minus(x, 0))
318.14/291.52	
318.14/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
318.14/291.52	
318.14/291.52	The pumping substitution is [x / s(x)].
318.14/291.52	
318.14/291.52	The result substitution is [ ].
318.14/291.52	
318.14/291.52	
318.14/291.52	
318.14/291.52	
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(4)
318.14/291.52	Complex Obligation (BEST)
318.14/291.52	
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(5)
318.14/291.52	Obligation:
318.14/291.52	Proved the lower bound n^1 for the following obligation:
318.14/291.52	
318.14/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.52	
318.14/291.52	
318.14/291.52	The TRS R consists of the following rules:
318.14/291.52	
318.14/291.52	   ge(0, 0) -> true
318.14/291.52	   ge(s(x), 0) -> ge(x, 0)
318.14/291.52	   ge(0, s(0)) -> false
318.14/291.52	   ge(0, s(s(x))) -> ge(0, s(x))
318.14/291.52	   ge(s(x), s(y)) -> ge(x, y)
318.14/291.52	   minus(0, 0) -> 0
318.14/291.52	   minus(0, s(x)) -> minus(0, x)
318.14/291.52	   minus(s(x), 0) -> s(minus(x, 0))
318.14/291.52	   minus(s(x), s(y)) -> minus(x, y)
318.14/291.52	   plus(0, 0) -> 0
318.14/291.52	   plus(0, s(x)) -> s(plus(0, x))
318.14/291.52	   plus(s(x), y) -> s(plus(x, y))
318.14/291.52	   div(x, y) -> ify(ge(y, s(0)), x, y)
318.14/291.52	   ify(false, x, y) -> divByZeroError
318.14/291.52	   ify(true, x, y) -> if(ge(x, y), x, y)
318.14/291.52	   if(false, x, y) -> 0
318.14/291.52	   if(true, x, y) -> s(div(minus(x, y), y))
318.14/291.52	
318.14/291.52	S is empty.
318.14/291.52	Rewrite Strategy: FULL
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(6) LowerBoundPropagationProof (FINISHED)
318.14/291.52	Propagated lower bound.
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(7)
318.14/291.52	BOUNDS(n^1, INF)
318.14/291.52	
318.14/291.52	----------------------------------------
318.14/291.52	
318.14/291.52	(8)
318.14/291.52	Obligation:
318.14/291.52	Analyzing the following TRS for decreasing loops:
318.14/291.52	
318.14/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
318.14/291.52	
318.14/291.52	
318.14/291.52	The TRS R consists of the following rules:
318.14/291.52	
318.14/291.52	   ge(0, 0) -> true
318.14/291.52	   ge(s(x), 0) -> ge(x, 0)
318.14/291.52	   ge(0, s(0)) -> false
318.14/291.52	   ge(0, s(s(x))) -> ge(0, s(x))
318.14/291.52	   ge(s(x), s(y)) -> ge(x, y)
318.14/291.52	   minus(0, 0) -> 0
318.14/291.52	   minus(0, s(x)) -> minus(0, x)
318.14/291.52	   minus(s(x), 0) -> s(minus(x, 0))
318.14/291.52	   minus(s(x), s(y)) -> minus(x, y)
318.14/291.52	   plus(0, 0) -> 0
318.14/291.52	   plus(0, s(x)) -> s(plus(0, x))
318.14/291.52	   plus(s(x), y) -> s(plus(x, y))
318.14/291.52	   div(x, y) -> ify(ge(y, s(0)), x, y)
318.14/291.52	   ify(false, x, y) -> divByZeroError
318.14/291.52	   ify(true, x, y) -> if(ge(x, y), x, y)
318.14/291.52	   if(false, x, y) -> 0
318.14/291.52	   if(true, x, y) -> s(div(minus(x, y), y))
318.14/291.52	
318.14/291.52	S is empty.
318.14/291.52	Rewrite Strategy: FULL
318.14/291.55	EOF