304.51/291.53	WORST_CASE(Omega(n^1), ?)
304.51/291.54	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
304.51/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.51/291.54	
304.51/291.54	
304.51/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.51/291.54	
304.51/291.54	(0) CpxTRS
304.51/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
304.51/291.54	(2) TRS for Loop Detection
304.51/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
304.51/291.54	(4) BEST
304.51/291.54	    (5) proven lower bound
304.51/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
304.51/291.54	        (7) BOUNDS(n^1, INF)
304.51/291.54	    (8) TRS for Loop Detection
304.51/291.54	
304.51/291.54	
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(0)
304.51/291.54	Obligation:
304.51/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.51/291.54	
304.51/291.54	
304.51/291.54	The TRS R consists of the following rules:
304.51/291.54	
304.51/291.54	   le(0, y) -> true
304.51/291.54	   le(s(x), 0) -> false
304.51/291.54	   le(s(x), s(y)) -> le(x, y)
304.51/291.54	   minus(x, 0) -> x
304.51/291.54	   minus(s(x), s(y)) -> minus(x, y)
304.51/291.54	   gcd(0, y) -> y
304.51/291.54	   gcd(s(x), 0) -> s(x)
304.51/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.51/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.51/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.51/291.54	
304.51/291.54	S is empty.
304.51/291.54	Rewrite Strategy: FULL
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
304.51/291.54	Transformed a relative TRS into a decreasing-loop problem.
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(2)
304.51/291.54	Obligation:
304.51/291.54	Analyzing the following TRS for decreasing loops:
304.51/291.54	
304.51/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.51/291.54	
304.51/291.54	
304.51/291.54	The TRS R consists of the following rules:
304.51/291.54	
304.51/291.54	   le(0, y) -> true
304.51/291.54	   le(s(x), 0) -> false
304.51/291.54	   le(s(x), s(y)) -> le(x, y)
304.51/291.54	   minus(x, 0) -> x
304.51/291.54	   minus(s(x), s(y)) -> minus(x, y)
304.51/291.54	   gcd(0, y) -> y
304.51/291.54	   gcd(s(x), 0) -> s(x)
304.51/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.51/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.51/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.51/291.54	
304.51/291.54	S is empty.
304.51/291.54	Rewrite Strategy: FULL
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
304.51/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
304.51/291.54	
304.51/291.54	The rewrite sequence
304.51/291.54	
304.51/291.54	minus(s(x), s(y)) ->^+ minus(x, y)
304.51/291.54	
304.51/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
304.51/291.54	
304.51/291.54	The pumping substitution is [x / s(x), y / s(y)].
304.51/291.54	
304.51/291.54	The result substitution is [ ].
304.51/291.54	
304.51/291.54	
304.51/291.54	
304.51/291.54	
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(4)
304.51/291.54	Complex Obligation (BEST)
304.51/291.54	
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(5)
304.51/291.54	Obligation:
304.51/291.54	Proved the lower bound n^1 for the following obligation:
304.51/291.54	
304.51/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.51/291.54	
304.51/291.54	
304.51/291.54	The TRS R consists of the following rules:
304.51/291.54	
304.51/291.54	   le(0, y) -> true
304.51/291.54	   le(s(x), 0) -> false
304.51/291.54	   le(s(x), s(y)) -> le(x, y)
304.51/291.54	   minus(x, 0) -> x
304.51/291.54	   minus(s(x), s(y)) -> minus(x, y)
304.51/291.54	   gcd(0, y) -> y
304.51/291.54	   gcd(s(x), 0) -> s(x)
304.51/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.51/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.51/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.51/291.54	
304.51/291.54	S is empty.
304.51/291.54	Rewrite Strategy: FULL
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(6) LowerBoundPropagationProof (FINISHED)
304.51/291.54	Propagated lower bound.
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(7)
304.51/291.54	BOUNDS(n^1, INF)
304.51/291.54	
304.51/291.54	----------------------------------------
304.51/291.54	
304.51/291.54	(8)
304.51/291.54	Obligation:
304.51/291.54	Analyzing the following TRS for decreasing loops:
304.51/291.54	
304.51/291.54	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.51/291.54	
304.51/291.54	
304.51/291.54	The TRS R consists of the following rules:
304.51/291.54	
304.51/291.54	   le(0, y) -> true
304.51/291.54	   le(s(x), 0) -> false
304.51/291.54	   le(s(x), s(y)) -> le(x, y)
304.51/291.54	   minus(x, 0) -> x
304.51/291.54	   minus(s(x), s(y)) -> minus(x, y)
304.51/291.54	   gcd(0, y) -> y
304.51/291.54	   gcd(s(x), 0) -> s(x)
304.51/291.54	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
304.51/291.54	   if_gcd(true, s(x), s(y)) -> gcd(minus(x, y), s(y))
304.51/291.54	   if_gcd(false, s(x), s(y)) -> gcd(minus(y, x), s(x))
304.51/291.54	
304.51/291.54	S is empty.
304.51/291.54	Rewrite Strategy: FULL
304.59/291.57	EOF