979.20/291.47	WORST_CASE(Omega(n^1), ?)
979.20/291.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
979.20/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
979.20/291.47	
979.20/291.47	
979.20/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
979.20/291.47	
979.20/291.47	(0) CpxTRS
979.20/291.47	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
979.20/291.47	(2) TRS for Loop Detection
979.20/291.47	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
979.20/291.47	(4) BEST
979.20/291.47	    (5) proven lower bound
979.20/291.47	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
979.20/291.47	        (7) BOUNDS(n^1, INF)
979.20/291.47	    (8) TRS for Loop Detection
979.20/291.47	
979.20/291.47	
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(0)
979.20/291.47	Obligation:
979.20/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
979.20/291.47	
979.20/291.47	
979.20/291.47	The TRS R consists of the following rules:
979.20/291.47	
979.20/291.47	   ack(0, y) -> s(y)
979.20/291.47	   ack(s(x), 0) -> ack(x, s(0))
979.20/291.47	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
979.20/291.47	   f(s(x), y) -> f(x, s(x))
979.20/291.47	   f(x, s(y)) -> f(y, x)
979.20/291.47	   f(x, y) -> ack(x, y)
979.20/291.47	   ack(s(x), y) -> f(x, x)
979.20/291.47	
979.20/291.47	S is empty.
979.20/291.47	Rewrite Strategy: FULL
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
979.20/291.47	Transformed a relative TRS into a decreasing-loop problem.
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(2)
979.20/291.47	Obligation:
979.20/291.47	Analyzing the following TRS for decreasing loops:
979.20/291.47	
979.20/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
979.20/291.47	
979.20/291.47	
979.20/291.47	The TRS R consists of the following rules:
979.20/291.47	
979.20/291.47	   ack(0, y) -> s(y)
979.20/291.47	   ack(s(x), 0) -> ack(x, s(0))
979.20/291.47	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
979.20/291.47	   f(s(x), y) -> f(x, s(x))
979.20/291.47	   f(x, s(y)) -> f(y, x)
979.20/291.47	   f(x, y) -> ack(x, y)
979.20/291.47	   ack(s(x), y) -> f(x, x)
979.20/291.47	
979.20/291.47	S is empty.
979.20/291.47	Rewrite Strategy: FULL
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(3) DecreasingLoopProof (LOWER BOUND(ID))
979.20/291.47	The following loop(s) give(s) rise to the lower bound Omega(n^1):
979.20/291.47	
979.20/291.47	The rewrite sequence
979.20/291.47	
979.20/291.47	ack(s(x), s(y)) ->^+ ack(x, ack(s(x), y))
979.20/291.47	
979.20/291.47	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
979.20/291.47	
979.20/291.47	The pumping substitution is [y / s(y)].
979.20/291.47	
979.20/291.47	The result substitution is [ ].
979.20/291.47	
979.20/291.47	
979.20/291.47	
979.20/291.47	
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(4)
979.20/291.47	Complex Obligation (BEST)
979.20/291.47	
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(5)
979.20/291.47	Obligation:
979.20/291.47	Proved the lower bound n^1 for the following obligation:
979.20/291.47	
979.20/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
979.20/291.47	
979.20/291.47	
979.20/291.47	The TRS R consists of the following rules:
979.20/291.47	
979.20/291.47	   ack(0, y) -> s(y)
979.20/291.47	   ack(s(x), 0) -> ack(x, s(0))
979.20/291.47	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
979.20/291.47	   f(s(x), y) -> f(x, s(x))
979.20/291.47	   f(x, s(y)) -> f(y, x)
979.20/291.47	   f(x, y) -> ack(x, y)
979.20/291.47	   ack(s(x), y) -> f(x, x)
979.20/291.47	
979.20/291.47	S is empty.
979.20/291.47	Rewrite Strategy: FULL
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(6) LowerBoundPropagationProof (FINISHED)
979.20/291.47	Propagated lower bound.
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(7)
979.20/291.47	BOUNDS(n^1, INF)
979.20/291.47	
979.20/291.47	----------------------------------------
979.20/291.47	
979.20/291.47	(8)
979.20/291.47	Obligation:
979.20/291.47	Analyzing the following TRS for decreasing loops:
979.20/291.47	
979.20/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
979.20/291.47	
979.20/291.47	
979.20/291.47	The TRS R consists of the following rules:
979.20/291.47	
979.20/291.47	   ack(0, y) -> s(y)
979.20/291.47	   ack(s(x), 0) -> ack(x, s(0))
979.20/291.47	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
979.20/291.47	   f(s(x), y) -> f(x, s(x))
979.20/291.47	   f(x, s(y)) -> f(y, x)
979.20/291.47	   f(x, y) -> ack(x, y)
979.20/291.47	   ack(s(x), y) -> f(x, x)
979.20/291.47	
979.20/291.47	S is empty.
979.20/291.47	Rewrite Strategy: FULL
979.47/291.51	EOF