310.93/291.52	WORST_CASE(Omega(n^1), ?)
310.93/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
310.93/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.93/291.53	
310.93/291.53	
310.93/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.93/291.53	
310.93/291.53	(0) CpxTRS
310.93/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
310.93/291.53	(2) TRS for Loop Detection
310.93/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
310.93/291.53	(4) BEST
310.93/291.53	    (5) proven lower bound
310.93/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
310.93/291.53	        (7) BOUNDS(n^1, INF)
310.93/291.53	    (8) TRS for Loop Detection
310.93/291.53	
310.93/291.53	
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(0)
310.93/291.53	Obligation:
310.93/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.93/291.53	
310.93/291.53	
310.93/291.53	The TRS R consists of the following rules:
310.93/291.53	
310.93/291.53	   le(0, y) -> true
310.93/291.53	   le(s(x), 0) -> false
310.93/291.53	   le(s(x), s(y)) -> le(x, y)
310.93/291.53	   pred(s(x)) -> x
310.93/291.53	   minus(x, 0) -> x
310.93/291.53	   minus(x, s(y)) -> pred(minus(x, y))
310.93/291.53	   gcd(0, y) -> y
310.93/291.53	   gcd(s(x), 0) -> s(x)
310.93/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.93/291.53	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
310.93/291.53	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
310.93/291.53	
310.93/291.53	S is empty.
310.93/291.53	Rewrite Strategy: FULL
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
310.93/291.53	Transformed a relative TRS into a decreasing-loop problem.
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(2)
310.93/291.53	Obligation:
310.93/291.53	Analyzing the following TRS for decreasing loops:
310.93/291.53	
310.93/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.93/291.53	
310.93/291.53	
310.93/291.53	The TRS R consists of the following rules:
310.93/291.53	
310.93/291.53	   le(0, y) -> true
310.93/291.53	   le(s(x), 0) -> false
310.93/291.53	   le(s(x), s(y)) -> le(x, y)
310.93/291.53	   pred(s(x)) -> x
310.93/291.53	   minus(x, 0) -> x
310.93/291.53	   minus(x, s(y)) -> pred(minus(x, y))
310.93/291.53	   gcd(0, y) -> y
310.93/291.53	   gcd(s(x), 0) -> s(x)
310.93/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.93/291.53	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
310.93/291.53	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
310.93/291.53	
310.93/291.53	S is empty.
310.93/291.53	Rewrite Strategy: FULL
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
310.93/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
310.93/291.53	
310.93/291.53	The rewrite sequence
310.93/291.53	
310.93/291.53	minus(x, s(y)) ->^+ pred(minus(x, y))
310.93/291.53	
310.93/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
310.93/291.53	
310.93/291.53	The pumping substitution is [y / s(y)].
310.93/291.53	
310.93/291.53	The result substitution is [ ].
310.93/291.53	
310.93/291.53	
310.93/291.53	
310.93/291.53	
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(4)
310.93/291.53	Complex Obligation (BEST)
310.93/291.53	
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(5)
310.93/291.53	Obligation:
310.93/291.53	Proved the lower bound n^1 for the following obligation:
310.93/291.53	
310.93/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.93/291.53	
310.93/291.53	
310.93/291.53	The TRS R consists of the following rules:
310.93/291.53	
310.93/291.53	   le(0, y) -> true
310.93/291.53	   le(s(x), 0) -> false
310.93/291.53	   le(s(x), s(y)) -> le(x, y)
310.93/291.53	   pred(s(x)) -> x
310.93/291.53	   minus(x, 0) -> x
310.93/291.53	   minus(x, s(y)) -> pred(minus(x, y))
310.93/291.53	   gcd(0, y) -> y
310.93/291.53	   gcd(s(x), 0) -> s(x)
310.93/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.93/291.53	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
310.93/291.53	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
310.93/291.53	
310.93/291.53	S is empty.
310.93/291.53	Rewrite Strategy: FULL
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(6) LowerBoundPropagationProof (FINISHED)
310.93/291.53	Propagated lower bound.
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(7)
310.93/291.53	BOUNDS(n^1, INF)
310.93/291.53	
310.93/291.53	----------------------------------------
310.93/291.53	
310.93/291.53	(8)
310.93/291.53	Obligation:
310.93/291.53	Analyzing the following TRS for decreasing loops:
310.93/291.53	
310.93/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.93/291.53	
310.93/291.53	
310.93/291.53	The TRS R consists of the following rules:
310.93/291.53	
310.93/291.53	   le(0, y) -> true
310.93/291.53	   le(s(x), 0) -> false
310.93/291.53	   le(s(x), s(y)) -> le(x, y)
310.93/291.53	   pred(s(x)) -> x
310.93/291.53	   minus(x, 0) -> x
310.93/291.53	   minus(x, s(y)) -> pred(minus(x, y))
310.93/291.53	   gcd(0, y) -> y
310.93/291.53	   gcd(s(x), 0) -> s(x)
310.93/291.53	   gcd(s(x), s(y)) -> if_gcd(le(y, x), s(x), s(y))
310.93/291.53	   if_gcd(true, x, y) -> gcd(minus(x, y), y)
310.93/291.53	   if_gcd(false, x, y) -> gcd(minus(y, x), x)
310.93/291.53	
310.93/291.53	S is empty.
310.93/291.53	Rewrite Strategy: FULL
310.93/291.55	EOF