302.99/291.52	WORST_CASE(Omega(n^1), ?)
302.99/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
302.99/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
302.99/291.53	
302.99/291.53	
302.99/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.99/291.53	
302.99/291.53	(0) CpxTRS
302.99/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
302.99/291.53	(2) TRS for Loop Detection
302.99/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
302.99/291.53	(4) BEST
302.99/291.53	    (5) proven lower bound
302.99/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
302.99/291.53	        (7) BOUNDS(n^1, INF)
302.99/291.53	    (8) TRS for Loop Detection
302.99/291.53	
302.99/291.53	
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(0)
302.99/291.53	Obligation:
302.99/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.99/291.53	
302.99/291.53	
302.99/291.53	The TRS R consists of the following rules:
302.99/291.53	
302.99/291.53	   -(x, 0) -> x
302.99/291.53	   -(s(x), s(y)) -> -(x, y)
302.99/291.53	   p(s(x)) -> x
302.99/291.53	   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
302.99/291.53	   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))
302.99/291.53	
302.99/291.53	S is empty.
302.99/291.53	Rewrite Strategy: FULL
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
302.99/291.53	Transformed a relative TRS into a decreasing-loop problem.
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(2)
302.99/291.53	Obligation:
302.99/291.53	Analyzing the following TRS for decreasing loops:
302.99/291.53	
302.99/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.99/291.53	
302.99/291.53	
302.99/291.53	The TRS R consists of the following rules:
302.99/291.53	
302.99/291.53	   -(x, 0) -> x
302.99/291.53	   -(s(x), s(y)) -> -(x, y)
302.99/291.53	   p(s(x)) -> x
302.99/291.53	   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
302.99/291.53	   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))
302.99/291.53	
302.99/291.53	S is empty.
302.99/291.53	Rewrite Strategy: FULL
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
302.99/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
302.99/291.53	
302.99/291.53	The rewrite sequence
302.99/291.53	
302.99/291.53	-(s(x), s(y)) ->^+ -(x, y)
302.99/291.53	
302.99/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
302.99/291.53	
302.99/291.53	The pumping substitution is [x / s(x), y / s(y)].
302.99/291.53	
302.99/291.53	The result substitution is [ ].
302.99/291.53	
302.99/291.53	
302.99/291.53	
302.99/291.53	
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(4)
302.99/291.53	Complex Obligation (BEST)
302.99/291.53	
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(5)
302.99/291.53	Obligation:
302.99/291.53	Proved the lower bound n^1 for the following obligation:
302.99/291.53	
302.99/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.99/291.53	
302.99/291.53	
302.99/291.53	The TRS R consists of the following rules:
302.99/291.53	
302.99/291.53	   -(x, 0) -> x
302.99/291.53	   -(s(x), s(y)) -> -(x, y)
302.99/291.53	   p(s(x)) -> x
302.99/291.53	   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
302.99/291.53	   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))
302.99/291.53	
302.99/291.53	S is empty.
302.99/291.53	Rewrite Strategy: FULL
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(6) LowerBoundPropagationProof (FINISHED)
302.99/291.53	Propagated lower bound.
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(7)
302.99/291.53	BOUNDS(n^1, INF)
302.99/291.53	
302.99/291.53	----------------------------------------
302.99/291.53	
302.99/291.53	(8)
302.99/291.53	Obligation:
302.99/291.53	Analyzing the following TRS for decreasing loops:
302.99/291.53	
302.99/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.99/291.53	
302.99/291.53	
302.99/291.53	The TRS R consists of the following rules:
302.99/291.53	
302.99/291.53	   -(x, 0) -> x
302.99/291.53	   -(s(x), s(y)) -> -(x, y)
302.99/291.53	   p(s(x)) -> x
302.99/291.53	   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
302.99/291.53	   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))
302.99/291.53	
302.99/291.53	S is empty.
302.99/291.53	Rewrite Strategy: FULL
303.08/291.56	EOF