303.59/291.52	WORST_CASE(Omega(n^1), ?)
303.59/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
303.59/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.59/291.52	
303.59/291.52	
303.59/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.59/291.52	
303.59/291.52	(0) CpxTRS
303.59/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.59/291.52	(2) TRS for Loop Detection
303.59/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
303.59/291.52	(4) BEST
303.59/291.52	    (5) proven lower bound
303.59/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.59/291.52	        (7) BOUNDS(n^1, INF)
303.59/291.52	    (8) TRS for Loop Detection
303.59/291.52	
303.59/291.52	
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(0)
303.59/291.52	Obligation:
303.59/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.59/291.52	
303.59/291.52	
303.59/291.52	The TRS R consists of the following rules:
303.59/291.52	
303.59/291.52	   minus(x, y) -> if(gt(x, y), x, y)
303.59/291.52	   if(true, x, y) -> s(minus(p(x), y))
303.59/291.52	   if(false, x, y) -> 0
303.59/291.52	   p(0) -> 0
303.59/291.52	   p(s(x)) -> x
303.59/291.52	   ge(x, 0) -> true
303.59/291.52	   ge(0, s(x)) -> false
303.59/291.52	   ge(s(x), s(y)) -> ge(x, y)
303.59/291.52	   gt(0, y) -> false
303.59/291.52	   gt(s(x), 0) -> true
303.59/291.52	   gt(s(x), s(y)) -> gt(x, y)
303.59/291.52	   div(x, y) -> if1(ge(x, y), x, y)
303.59/291.52	   if1(true, x, y) -> if2(gt(y, 0), x, y)
303.59/291.52	   if1(false, x, y) -> 0
303.59/291.52	   if2(true, x, y) -> s(div(minus(x, y), y))
303.59/291.52	   if2(false, x, y) -> 0
303.59/291.52	
303.59/291.52	S is empty.
303.59/291.52	Rewrite Strategy: FULL
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.59/291.52	Transformed a relative TRS into a decreasing-loop problem.
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(2)
303.59/291.52	Obligation:
303.59/291.52	Analyzing the following TRS for decreasing loops:
303.59/291.52	
303.59/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.59/291.52	
303.59/291.52	
303.59/291.52	The TRS R consists of the following rules:
303.59/291.52	
303.59/291.52	   minus(x, y) -> if(gt(x, y), x, y)
303.59/291.52	   if(true, x, y) -> s(minus(p(x), y))
303.59/291.52	   if(false, x, y) -> 0
303.59/291.52	   p(0) -> 0
303.59/291.52	   p(s(x)) -> x
303.59/291.52	   ge(x, 0) -> true
303.59/291.52	   ge(0, s(x)) -> false
303.59/291.52	   ge(s(x), s(y)) -> ge(x, y)
303.59/291.52	   gt(0, y) -> false
303.59/291.52	   gt(s(x), 0) -> true
303.59/291.52	   gt(s(x), s(y)) -> gt(x, y)
303.59/291.52	   div(x, y) -> if1(ge(x, y), x, y)
303.59/291.52	   if1(true, x, y) -> if2(gt(y, 0), x, y)
303.59/291.52	   if1(false, x, y) -> 0
303.59/291.52	   if2(true, x, y) -> s(div(minus(x, y), y))
303.59/291.52	   if2(false, x, y) -> 0
303.59/291.52	
303.59/291.52	S is empty.
303.59/291.52	Rewrite Strategy: FULL
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.59/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.59/291.52	
303.59/291.52	The rewrite sequence
303.59/291.52	
303.59/291.52	gt(s(x), s(y)) ->^+ gt(x, y)
303.59/291.52	
303.59/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
303.59/291.52	
303.59/291.52	The pumping substitution is [x / s(x), y / s(y)].
303.59/291.52	
303.59/291.52	The result substitution is [ ].
303.59/291.52	
303.59/291.52	
303.59/291.52	
303.59/291.52	
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(4)
303.59/291.52	Complex Obligation (BEST)
303.59/291.52	
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(5)
303.59/291.52	Obligation:
303.59/291.52	Proved the lower bound n^1 for the following obligation:
303.59/291.52	
303.59/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.59/291.52	
303.59/291.52	
303.59/291.52	The TRS R consists of the following rules:
303.59/291.52	
303.59/291.52	   minus(x, y) -> if(gt(x, y), x, y)
303.59/291.52	   if(true, x, y) -> s(minus(p(x), y))
303.59/291.52	   if(false, x, y) -> 0
303.59/291.52	   p(0) -> 0
303.59/291.52	   p(s(x)) -> x
303.59/291.52	   ge(x, 0) -> true
303.59/291.52	   ge(0, s(x)) -> false
303.59/291.52	   ge(s(x), s(y)) -> ge(x, y)
303.59/291.52	   gt(0, y) -> false
303.59/291.52	   gt(s(x), 0) -> true
303.59/291.52	   gt(s(x), s(y)) -> gt(x, y)
303.59/291.52	   div(x, y) -> if1(ge(x, y), x, y)
303.59/291.52	   if1(true, x, y) -> if2(gt(y, 0), x, y)
303.59/291.52	   if1(false, x, y) -> 0
303.59/291.52	   if2(true, x, y) -> s(div(minus(x, y), y))
303.59/291.52	   if2(false, x, y) -> 0
303.59/291.52	
303.59/291.52	S is empty.
303.59/291.52	Rewrite Strategy: FULL
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(6) LowerBoundPropagationProof (FINISHED)
303.59/291.52	Propagated lower bound.
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(7)
303.59/291.52	BOUNDS(n^1, INF)
303.59/291.52	
303.59/291.52	----------------------------------------
303.59/291.52	
303.59/291.52	(8)
303.59/291.52	Obligation:
303.59/291.52	Analyzing the following TRS for decreasing loops:
303.59/291.52	
303.59/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.59/291.52	
303.59/291.52	
303.59/291.52	The TRS R consists of the following rules:
303.59/291.52	
303.59/291.52	   minus(x, y) -> if(gt(x, y), x, y)
303.59/291.52	   if(true, x, y) -> s(minus(p(x), y))
303.59/291.52	   if(false, x, y) -> 0
303.59/291.52	   p(0) -> 0
303.59/291.52	   p(s(x)) -> x
303.59/291.52	   ge(x, 0) -> true
303.59/291.52	   ge(0, s(x)) -> false
303.59/291.52	   ge(s(x), s(y)) -> ge(x, y)
303.59/291.52	   gt(0, y) -> false
303.59/291.52	   gt(s(x), 0) -> true
303.59/291.52	   gt(s(x), s(y)) -> gt(x, y)
303.59/291.52	   div(x, y) -> if1(ge(x, y), x, y)
303.59/291.52	   if1(true, x, y) -> if2(gt(y, 0), x, y)
303.59/291.52	   if1(false, x, y) -> 0
303.59/291.52	   if2(true, x, y) -> s(div(minus(x, y), y))
303.59/291.52	   if2(false, x, y) -> 0
303.59/291.52	
303.59/291.52	S is empty.
303.59/291.52	Rewrite Strategy: FULL
303.66/291.56	EOF