313.12/291.51	WORST_CASE(Omega(n^1), ?)
313.12/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
313.12/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
313.12/291.52	
313.12/291.52	
313.12/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.12/291.52	
313.12/291.52	(0) CpxTRS
313.12/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
313.12/291.52	(2) TRS for Loop Detection
313.12/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
313.12/291.52	(4) BEST
313.12/291.52	    (5) proven lower bound
313.12/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
313.12/291.52	        (7) BOUNDS(n^1, INF)
313.12/291.52	    (8) TRS for Loop Detection
313.12/291.52	
313.12/291.52	
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(0)
313.12/291.52	Obligation:
313.12/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.12/291.52	
313.12/291.52	
313.12/291.52	The TRS R consists of the following rules:
313.12/291.52	
313.12/291.52	   double(x) -> permute(x, x, a)
313.12/291.52	   permute(x, y, a) -> permute(isZero(x), x, b)
313.12/291.52	   permute(false, x, b) -> permute(ack(x, x), p(x), c)
313.12/291.52	   permute(true, x, b) -> 0
313.12/291.52	   permute(y, x, c) -> s(s(permute(x, y, a)))
313.12/291.52	   p(0) -> 0
313.12/291.52	   p(s(x)) -> x
313.12/291.52	   ack(0, x) -> plus(x, s(0))
313.12/291.52	   ack(s(x), 0) -> ack(x, s(0))
313.12/291.52	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
313.12/291.52	   plus(0, y) -> y
313.12/291.52	   plus(s(x), y) -> plus(x, s(y))
313.12/291.52	   plus(x, s(s(y))) -> s(plus(s(x), y))
313.12/291.52	   plus(x, s(0)) -> s(x)
313.12/291.52	   plus(x, 0) -> x
313.12/291.52	   isZero(0) -> true
313.12/291.52	   isZero(s(x)) -> false
313.12/291.52	
313.12/291.52	S is empty.
313.12/291.52	Rewrite Strategy: FULL
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
313.12/291.52	Transformed a relative TRS into a decreasing-loop problem.
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(2)
313.12/291.52	Obligation:
313.12/291.52	Analyzing the following TRS for decreasing loops:
313.12/291.52	
313.12/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.12/291.52	
313.12/291.52	
313.12/291.52	The TRS R consists of the following rules:
313.12/291.52	
313.12/291.52	   double(x) -> permute(x, x, a)
313.12/291.52	   permute(x, y, a) -> permute(isZero(x), x, b)
313.12/291.52	   permute(false, x, b) -> permute(ack(x, x), p(x), c)
313.12/291.52	   permute(true, x, b) -> 0
313.12/291.52	   permute(y, x, c) -> s(s(permute(x, y, a)))
313.12/291.52	   p(0) -> 0
313.12/291.52	   p(s(x)) -> x
313.12/291.52	   ack(0, x) -> plus(x, s(0))
313.12/291.52	   ack(s(x), 0) -> ack(x, s(0))
313.12/291.52	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
313.12/291.52	   plus(0, y) -> y
313.12/291.52	   plus(s(x), y) -> plus(x, s(y))
313.12/291.52	   plus(x, s(s(y))) -> s(plus(s(x), y))
313.12/291.52	   plus(x, s(0)) -> s(x)
313.12/291.52	   plus(x, 0) -> x
313.12/291.52	   isZero(0) -> true
313.12/291.52	   isZero(s(x)) -> false
313.12/291.52	
313.12/291.52	S is empty.
313.12/291.52	Rewrite Strategy: FULL
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
313.12/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
313.12/291.52	
313.12/291.52	The rewrite sequence
313.12/291.52	
313.12/291.52	plus(x, s(s(y))) ->^+ s(plus(s(x), y))
313.12/291.52	
313.12/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
313.12/291.52	
313.12/291.52	The pumping substitution is [y / s(s(y))].
313.12/291.52	
313.12/291.52	The result substitution is [x / s(x)].
313.12/291.52	
313.12/291.52	
313.12/291.52	
313.12/291.52	
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(4)
313.12/291.52	Complex Obligation (BEST)
313.12/291.52	
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(5)
313.12/291.52	Obligation:
313.12/291.52	Proved the lower bound n^1 for the following obligation:
313.12/291.52	
313.12/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.12/291.52	
313.12/291.52	
313.12/291.52	The TRS R consists of the following rules:
313.12/291.52	
313.12/291.52	   double(x) -> permute(x, x, a)
313.12/291.52	   permute(x, y, a) -> permute(isZero(x), x, b)
313.12/291.52	   permute(false, x, b) -> permute(ack(x, x), p(x), c)
313.12/291.52	   permute(true, x, b) -> 0
313.12/291.52	   permute(y, x, c) -> s(s(permute(x, y, a)))
313.12/291.52	   p(0) -> 0
313.12/291.52	   p(s(x)) -> x
313.12/291.52	   ack(0, x) -> plus(x, s(0))
313.12/291.52	   ack(s(x), 0) -> ack(x, s(0))
313.12/291.52	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
313.12/291.52	   plus(0, y) -> y
313.12/291.52	   plus(s(x), y) -> plus(x, s(y))
313.12/291.52	   plus(x, s(s(y))) -> s(plus(s(x), y))
313.12/291.52	   plus(x, s(0)) -> s(x)
313.12/291.52	   plus(x, 0) -> x
313.12/291.52	   isZero(0) -> true
313.12/291.52	   isZero(s(x)) -> false
313.12/291.52	
313.12/291.52	S is empty.
313.12/291.52	Rewrite Strategy: FULL
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(6) LowerBoundPropagationProof (FINISHED)
313.12/291.52	Propagated lower bound.
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(7)
313.12/291.52	BOUNDS(n^1, INF)
313.12/291.52	
313.12/291.52	----------------------------------------
313.12/291.52	
313.12/291.52	(8)
313.12/291.52	Obligation:
313.12/291.52	Analyzing the following TRS for decreasing loops:
313.12/291.52	
313.12/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
313.12/291.52	
313.12/291.52	
313.12/291.52	The TRS R consists of the following rules:
313.12/291.52	
313.12/291.52	   double(x) -> permute(x, x, a)
313.12/291.52	   permute(x, y, a) -> permute(isZero(x), x, b)
313.12/291.52	   permute(false, x, b) -> permute(ack(x, x), p(x), c)
313.12/291.52	   permute(true, x, b) -> 0
313.12/291.52	   permute(y, x, c) -> s(s(permute(x, y, a)))
313.12/291.52	   p(0) -> 0
313.12/291.52	   p(s(x)) -> x
313.12/291.52	   ack(0, x) -> plus(x, s(0))
313.12/291.52	   ack(s(x), 0) -> ack(x, s(0))
313.12/291.52	   ack(s(x), s(y)) -> ack(x, ack(s(x), y))
313.12/291.52	   plus(0, y) -> y
313.12/291.52	   plus(s(x), y) -> plus(x, s(y))
313.12/291.52	   plus(x, s(s(y))) -> s(plus(s(x), y))
313.12/291.52	   plus(x, s(0)) -> s(x)
313.12/291.52	   plus(x, 0) -> x
313.12/291.52	   isZero(0) -> true
313.12/291.52	   isZero(s(x)) -> false
313.12/291.52	
313.12/291.52	S is empty.
313.12/291.52	Rewrite Strategy: FULL
313.16/291.56	EOF