301.79/291.50	WORST_CASE(Omega(n^1), ?)
301.79/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
301.79/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.79/291.51	
301.79/291.51	
301.79/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.79/291.51	
301.79/291.51	(0) CpxTRS
301.79/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.79/291.51	(2) TRS for Loop Detection
301.79/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
301.79/291.51	(4) BEST
301.79/291.51	    (5) proven lower bound
301.79/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.79/291.51	        (7) BOUNDS(n^1, INF)
301.79/291.51	    (8) TRS for Loop Detection
301.79/291.51	
301.79/291.51	
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(0)
301.79/291.51	Obligation:
301.79/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.79/291.51	
301.79/291.51	
301.79/291.51	The TRS R consists of the following rules:
301.79/291.51	
301.79/291.51	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.79/291.51	   if(true, x, y) -> s(minus(x, y))
301.79/291.51	   if(false, x, y) -> 0
301.79/291.51	   gcd(x, y) -> if1(ge(x, y), x, y)
301.79/291.51	   if1(true, x, y) -> if2(gt(y, 0), x, y)
301.79/291.51	   if1(false, x, y) -> gcd(y, x)
301.79/291.51	   if2(true, x, y) -> gcd(minus(x, y), y)
301.79/291.51	   if2(false, x, y) -> x
301.79/291.51	   gt(0, y) -> false
301.79/291.51	   gt(s(x), 0) -> true
301.79/291.51	   gt(s(x), s(y)) -> gt(x, y)
301.79/291.51	   ge(x, 0) -> true
301.79/291.51	   ge(0, s(x)) -> false
301.79/291.51	   ge(s(x), s(y)) -> ge(x, y)
301.79/291.51	
301.79/291.51	S is empty.
301.79/291.51	Rewrite Strategy: FULL
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.79/291.51	Transformed a relative TRS into a decreasing-loop problem.
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(2)
301.79/291.51	Obligation:
301.79/291.51	Analyzing the following TRS for decreasing loops:
301.79/291.51	
301.79/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.79/291.51	
301.79/291.51	
301.79/291.51	The TRS R consists of the following rules:
301.79/291.51	
301.79/291.51	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.79/291.51	   if(true, x, y) -> s(minus(x, y))
301.79/291.51	   if(false, x, y) -> 0
301.79/291.51	   gcd(x, y) -> if1(ge(x, y), x, y)
301.79/291.51	   if1(true, x, y) -> if2(gt(y, 0), x, y)
301.79/291.51	   if1(false, x, y) -> gcd(y, x)
301.79/291.51	   if2(true, x, y) -> gcd(minus(x, y), y)
301.79/291.51	   if2(false, x, y) -> x
301.79/291.51	   gt(0, y) -> false
301.79/291.51	   gt(s(x), 0) -> true
301.79/291.51	   gt(s(x), s(y)) -> gt(x, y)
301.79/291.51	   ge(x, 0) -> true
301.79/291.51	   ge(0, s(x)) -> false
301.79/291.51	   ge(s(x), s(y)) -> ge(x, y)
301.79/291.51	
301.79/291.51	S is empty.
301.79/291.51	Rewrite Strategy: FULL
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.79/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.79/291.51	
301.79/291.51	The rewrite sequence
301.79/291.51	
301.79/291.51	gt(s(x), s(y)) ->^+ gt(x, y)
301.79/291.51	
301.79/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
301.79/291.51	
301.79/291.51	The pumping substitution is [x / s(x), y / s(y)].
301.79/291.51	
301.79/291.51	The result substitution is [ ].
301.79/291.51	
301.79/291.51	
301.79/291.51	
301.79/291.51	
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(4)
301.79/291.51	Complex Obligation (BEST)
301.79/291.51	
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(5)
301.79/291.51	Obligation:
301.79/291.51	Proved the lower bound n^1 for the following obligation:
301.79/291.51	
301.79/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.79/291.51	
301.79/291.51	
301.79/291.51	The TRS R consists of the following rules:
301.79/291.51	
301.79/291.51	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.79/291.51	   if(true, x, y) -> s(minus(x, y))
301.79/291.51	   if(false, x, y) -> 0
301.79/291.51	   gcd(x, y) -> if1(ge(x, y), x, y)
301.79/291.51	   if1(true, x, y) -> if2(gt(y, 0), x, y)
301.79/291.51	   if1(false, x, y) -> gcd(y, x)
301.79/291.51	   if2(true, x, y) -> gcd(minus(x, y), y)
301.79/291.51	   if2(false, x, y) -> x
301.79/291.51	   gt(0, y) -> false
301.79/291.51	   gt(s(x), 0) -> true
301.79/291.51	   gt(s(x), s(y)) -> gt(x, y)
301.79/291.51	   ge(x, 0) -> true
301.79/291.51	   ge(0, s(x)) -> false
301.79/291.51	   ge(s(x), s(y)) -> ge(x, y)
301.79/291.51	
301.79/291.51	S is empty.
301.79/291.51	Rewrite Strategy: FULL
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(6) LowerBoundPropagationProof (FINISHED)
301.79/291.51	Propagated lower bound.
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(7)
301.79/291.51	BOUNDS(n^1, INF)
301.79/291.51	
301.79/291.51	----------------------------------------
301.79/291.51	
301.79/291.51	(8)
301.79/291.51	Obligation:
301.79/291.51	Analyzing the following TRS for decreasing loops:
301.79/291.51	
301.79/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.79/291.51	
301.79/291.51	
301.79/291.51	The TRS R consists of the following rules:
301.79/291.51	
301.79/291.51	   minus(s(x), y) -> if(gt(s(x), y), x, y)
301.79/291.51	   if(true, x, y) -> s(minus(x, y))
301.79/291.51	   if(false, x, y) -> 0
301.79/291.51	   gcd(x, y) -> if1(ge(x, y), x, y)
301.79/291.51	   if1(true, x, y) -> if2(gt(y, 0), x, y)
301.79/291.51	   if1(false, x, y) -> gcd(y, x)
301.79/291.51	   if2(true, x, y) -> gcd(minus(x, y), y)
301.79/291.51	   if2(false, x, y) -> x
301.79/291.51	   gt(0, y) -> false
301.79/291.51	   gt(s(x), 0) -> true
301.79/291.51	   gt(s(x), s(y)) -> gt(x, y)
301.79/291.51	   ge(x, 0) -> true
301.79/291.51	   ge(0, s(x)) -> false
301.79/291.51	   ge(s(x), s(y)) -> ge(x, y)
301.79/291.51	
301.79/291.51	S is empty.
301.79/291.51	Rewrite Strategy: FULL
301.88/291.55	EOF