1147.54/291.50	WORST_CASE(Omega(n^1), ?)
1147.73/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
1147.73/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1147.73/291.52	
1147.73/291.52	(0) CpxTRS
1147.73/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1147.73/291.52	(2) TRS for Loop Detection
1147.73/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1147.73/291.52	(4) BEST
1147.73/291.52	    (5) proven lower bound
1147.73/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1147.73/291.52	        (7) BOUNDS(n^1, INF)
1147.73/291.52	    (8) TRS for Loop Detection
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(0)
1147.73/291.52	Obligation:
1147.73/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	The TRS R consists of the following rules:
1147.73/291.52	
1147.73/291.52	   min(x, 0) -> 0
1147.73/291.52	   min(0, y) -> 0
1147.73/291.52	   min(s(x), s(y)) -> s(min(x, y))
1147.73/291.52	   max(x, 0) -> x
1147.73/291.52	   max(0, y) -> y
1147.73/291.52	   max(s(x), s(y)) -> s(max(x, y))
1147.73/291.52	   -(x, 0) -> x
1147.73/291.52	   -(s(x), s(y)) -> -(x, y)
1147.73/291.52	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
1147.73/291.52	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
1147.73/291.52	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
1147.73/291.52	   gcd(x, 0, 0) -> x
1147.73/291.52	   gcd(0, y, 0) -> y
1147.73/291.52	   gcd(0, 0, z) -> z
1147.73/291.52	
1147.73/291.52	S is empty.
1147.73/291.52	Rewrite Strategy: INNERMOST
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1147.73/291.52	Transformed a relative TRS into a decreasing-loop problem.
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(2)
1147.73/291.52	Obligation:
1147.73/291.52	Analyzing the following TRS for decreasing loops:
1147.73/291.52	
1147.73/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	The TRS R consists of the following rules:
1147.73/291.52	
1147.73/291.52	   min(x, 0) -> 0
1147.73/291.52	   min(0, y) -> 0
1147.73/291.52	   min(s(x), s(y)) -> s(min(x, y))
1147.73/291.52	   max(x, 0) -> x
1147.73/291.52	   max(0, y) -> y
1147.73/291.52	   max(s(x), s(y)) -> s(max(x, y))
1147.73/291.52	   -(x, 0) -> x
1147.73/291.52	   -(s(x), s(y)) -> -(x, y)
1147.73/291.52	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
1147.73/291.52	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
1147.73/291.52	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
1147.73/291.52	   gcd(x, 0, 0) -> x
1147.73/291.52	   gcd(0, y, 0) -> y
1147.73/291.52	   gcd(0, 0, z) -> z
1147.73/291.52	
1147.73/291.52	S is empty.
1147.73/291.52	Rewrite Strategy: INNERMOST
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
1147.73/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1147.73/291.52	
1147.73/291.52	The rewrite sequence
1147.73/291.52	
1147.73/291.52	-(s(x), s(y)) ->^+ -(x, y)
1147.73/291.52	
1147.73/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1147.73/291.52	
1147.73/291.52	The pumping substitution is [x / s(x), y / s(y)].
1147.73/291.52	
1147.73/291.52	The result substitution is [ ].
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(4)
1147.73/291.52	Complex Obligation (BEST)
1147.73/291.52	
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(5)
1147.73/291.52	Obligation:
1147.73/291.52	Proved the lower bound n^1 for the following obligation:
1147.73/291.52	
1147.73/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	The TRS R consists of the following rules:
1147.73/291.52	
1147.73/291.52	   min(x, 0) -> 0
1147.73/291.52	   min(0, y) -> 0
1147.73/291.52	   min(s(x), s(y)) -> s(min(x, y))
1147.73/291.52	   max(x, 0) -> x
1147.73/291.52	   max(0, y) -> y
1147.73/291.52	   max(s(x), s(y)) -> s(max(x, y))
1147.73/291.52	   -(x, 0) -> x
1147.73/291.52	   -(s(x), s(y)) -> -(x, y)
1147.73/291.52	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
1147.73/291.52	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
1147.73/291.52	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
1147.73/291.52	   gcd(x, 0, 0) -> x
1147.73/291.52	   gcd(0, y, 0) -> y
1147.73/291.52	   gcd(0, 0, z) -> z
1147.73/291.52	
1147.73/291.52	S is empty.
1147.73/291.52	Rewrite Strategy: INNERMOST
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(6) LowerBoundPropagationProof (FINISHED)
1147.73/291.52	Propagated lower bound.
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(7)
1147.73/291.52	BOUNDS(n^1, INF)
1147.73/291.52	
1147.73/291.52	----------------------------------------
1147.73/291.52	
1147.73/291.52	(8)
1147.73/291.52	Obligation:
1147.73/291.52	Analyzing the following TRS for decreasing loops:
1147.73/291.52	
1147.73/291.52	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1147.73/291.52	
1147.73/291.52	
1147.73/291.52	The TRS R consists of the following rules:
1147.73/291.52	
1147.73/291.52	   min(x, 0) -> 0
1147.73/291.52	   min(0, y) -> 0
1147.73/291.52	   min(s(x), s(y)) -> s(min(x, y))
1147.73/291.52	   max(x, 0) -> x
1147.73/291.52	   max(0, y) -> y
1147.73/291.52	   max(s(x), s(y)) -> s(max(x, y))
1147.73/291.52	   -(x, 0) -> x
1147.73/291.52	   -(s(x), s(y)) -> -(x, y)
1147.73/291.52	   gcd(s(x), s(y), z) -> gcd(-(max(x, y), min(x, y)), s(min(x, y)), z)
1147.73/291.52	   gcd(x, s(y), s(z)) -> gcd(x, -(max(y, z), min(y, z)), s(min(y, z)))
1147.73/291.52	   gcd(s(x), y, s(z)) -> gcd(-(max(x, z), min(x, z)), y, s(min(x, z)))
1147.73/291.52	   gcd(x, 0, 0) -> x
1147.73/291.52	   gcd(0, y, 0) -> y
1147.73/291.52	   gcd(0, 0, z) -> z
1147.73/291.52	
1147.73/291.52	S is empty.
1147.73/291.52	Rewrite Strategy: INNERMOST
1147.92/291.64	EOF