947.60/291.49	WORST_CASE(Omega(n^1), ?)
947.60/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
947.60/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
947.60/291.50	
947.60/291.50	
947.60/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
947.60/291.50	
947.60/291.50	(0) CpxTRS
947.60/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
947.60/291.50	(2) TRS for Loop Detection
947.60/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
947.60/291.50	(4) BEST
947.60/291.50	    (5) proven lower bound
947.60/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
947.60/291.50	        (7) BOUNDS(n^1, INF)
947.60/291.50	    (8) TRS for Loop Detection
947.60/291.50	
947.60/291.50	
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(0)
947.60/291.50	Obligation:
947.60/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
947.60/291.50	
947.60/291.50	
947.60/291.50	The TRS R consists of the following rules:
947.60/291.50	
947.60/291.50	   plus(s(s(x)), y) -> s(plus(x, s(y)))
947.60/291.50	   plus(x, s(s(y))) -> s(plus(s(x), y))
947.60/291.50	   plus(s(0), y) -> s(y)
947.60/291.50	   plus(0, y) -> y
947.60/291.50	   ack(0, y) -> s(y)
947.60/291.50	   ack(s(x), 0) -> ack(x, s(0))
947.60/291.50	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
947.60/291.50	
947.60/291.50	S is empty.
947.60/291.50	Rewrite Strategy: INNERMOST
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
947.60/291.50	Transformed a relative TRS into a decreasing-loop problem.
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(2)
947.60/291.50	Obligation:
947.60/291.50	Analyzing the following TRS for decreasing loops:
947.60/291.50	
947.60/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
947.60/291.50	
947.60/291.50	
947.60/291.50	The TRS R consists of the following rules:
947.60/291.50	
947.60/291.50	   plus(s(s(x)), y) -> s(plus(x, s(y)))
947.60/291.50	   plus(x, s(s(y))) -> s(plus(s(x), y))
947.60/291.50	   plus(s(0), y) -> s(y)
947.60/291.50	   plus(0, y) -> y
947.60/291.50	   ack(0, y) -> s(y)
947.60/291.50	   ack(s(x), 0) -> ack(x, s(0))
947.60/291.50	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
947.60/291.50	
947.60/291.50	S is empty.
947.60/291.50	Rewrite Strategy: INNERMOST
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
947.60/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
947.60/291.50	
947.60/291.50	The rewrite sequence
947.60/291.50	
947.60/291.50	plus(x, s(s(y))) ->^+ s(plus(s(x), y))
947.60/291.50	
947.60/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
947.60/291.50	
947.60/291.50	The pumping substitution is [y / s(s(y))].
947.60/291.50	
947.60/291.50	The result substitution is [x / s(x)].
947.60/291.50	
947.60/291.50	
947.60/291.50	
947.60/291.50	
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(4)
947.60/291.50	Complex Obligation (BEST)
947.60/291.50	
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(5)
947.60/291.50	Obligation:
947.60/291.50	Proved the lower bound n^1 for the following obligation:
947.60/291.50	
947.60/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
947.60/291.50	
947.60/291.50	
947.60/291.50	The TRS R consists of the following rules:
947.60/291.50	
947.60/291.50	   plus(s(s(x)), y) -> s(plus(x, s(y)))
947.60/291.50	   plus(x, s(s(y))) -> s(plus(s(x), y))
947.60/291.50	   plus(s(0), y) -> s(y)
947.60/291.50	   plus(0, y) -> y
947.60/291.50	   ack(0, y) -> s(y)
947.60/291.50	   ack(s(x), 0) -> ack(x, s(0))
947.60/291.50	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
947.60/291.50	
947.60/291.50	S is empty.
947.60/291.50	Rewrite Strategy: INNERMOST
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(6) LowerBoundPropagationProof (FINISHED)
947.60/291.50	Propagated lower bound.
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(7)
947.60/291.50	BOUNDS(n^1, INF)
947.60/291.50	
947.60/291.50	----------------------------------------
947.60/291.50	
947.60/291.50	(8)
947.60/291.50	Obligation:
947.60/291.50	Analyzing the following TRS for decreasing loops:
947.60/291.50	
947.60/291.50	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
947.60/291.50	
947.60/291.50	
947.60/291.50	The TRS R consists of the following rules:
947.60/291.50	
947.60/291.50	   plus(s(s(x)), y) -> s(plus(x, s(y)))
947.60/291.50	   plus(x, s(s(y))) -> s(plus(s(x), y))
947.60/291.50	   plus(s(0), y) -> s(y)
947.60/291.50	   plus(0, y) -> y
947.60/291.50	   ack(0, y) -> s(y)
947.60/291.50	   ack(s(x), 0) -> ack(x, s(0))
947.60/291.50	   ack(s(x), s(y)) -> ack(x, plus(y, ack(s(x), y)))
947.60/291.50	
947.60/291.50	S is empty.
947.60/291.50	Rewrite Strategy: INNERMOST
948.00/291.60	EOF