308.13/291.55	WORST_CASE(Omega(n^1), ?)
308.14/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
308.14/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.14/291.56	
308.14/291.56	
308.14/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.14/291.56	
308.14/291.56	(0) CpxTRS
308.14/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
308.14/291.56	(2) TRS for Loop Detection
308.14/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
308.14/291.56	(4) BEST
308.14/291.56	    (5) proven lower bound
308.14/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
308.14/291.56	        (7) BOUNDS(n^1, INF)
308.14/291.56	    (8) TRS for Loop Detection
308.14/291.56	
308.14/291.56	
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(0)
308.14/291.56	Obligation:
308.14/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.14/291.56	
308.14/291.56	
308.14/291.56	The TRS R consists of the following rules:
308.14/291.56	
308.14/291.56	   cond1(true, x) -> cond2(even(x), x)
308.14/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
308.14/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
308.14/291.56	   neq(0, 0) -> false
308.14/291.56	   neq(0, s(x)) -> true
308.14/291.56	   neq(s(x), 0) -> true
308.14/291.56	   neq(s(x), s(y)) -> neq(x, y)
308.14/291.56	   even(0) -> true
308.14/291.56	   even(s(0)) -> false
308.14/291.56	   even(s(s(x))) -> even(x)
308.14/291.56	   div2(0) -> 0
308.14/291.56	   div2(s(0)) -> 0
308.14/291.56	   div2(s(s(x))) -> s(div2(x))
308.14/291.56	   p(0) -> 0
308.14/291.56	   p(s(x)) -> x
308.14/291.56	
308.14/291.56	S is empty.
308.14/291.56	Rewrite Strategy: FULL
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
308.14/291.56	Transformed a relative TRS into a decreasing-loop problem.
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(2)
308.14/291.56	Obligation:
308.14/291.56	Analyzing the following TRS for decreasing loops:
308.14/291.56	
308.14/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.14/291.56	
308.14/291.56	
308.14/291.56	The TRS R consists of the following rules:
308.14/291.56	
308.14/291.56	   cond1(true, x) -> cond2(even(x), x)
308.14/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
308.14/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
308.14/291.56	   neq(0, 0) -> false
308.14/291.56	   neq(0, s(x)) -> true
308.14/291.56	   neq(s(x), 0) -> true
308.14/291.56	   neq(s(x), s(y)) -> neq(x, y)
308.14/291.56	   even(0) -> true
308.14/291.56	   even(s(0)) -> false
308.14/291.56	   even(s(s(x))) -> even(x)
308.14/291.56	   div2(0) -> 0
308.14/291.56	   div2(s(0)) -> 0
308.14/291.56	   div2(s(s(x))) -> s(div2(x))
308.14/291.56	   p(0) -> 0
308.14/291.56	   p(s(x)) -> x
308.14/291.56	
308.14/291.56	S is empty.
308.14/291.56	Rewrite Strategy: FULL
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
308.14/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
308.14/291.56	
308.14/291.56	The rewrite sequence
308.14/291.56	
308.14/291.56	even(s(s(x))) ->^+ even(x)
308.14/291.56	
308.14/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
308.14/291.56	
308.14/291.56	The pumping substitution is [x / s(s(x))].
308.14/291.56	
308.14/291.56	The result substitution is [ ].
308.14/291.56	
308.14/291.56	
308.14/291.56	
308.14/291.56	
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(4)
308.14/291.56	Complex Obligation (BEST)
308.14/291.56	
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(5)
308.14/291.56	Obligation:
308.14/291.56	Proved the lower bound n^1 for the following obligation:
308.14/291.56	
308.14/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.14/291.56	
308.14/291.56	
308.14/291.56	The TRS R consists of the following rules:
308.14/291.56	
308.14/291.56	   cond1(true, x) -> cond2(even(x), x)
308.14/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
308.14/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
308.14/291.56	   neq(0, 0) -> false
308.14/291.56	   neq(0, s(x)) -> true
308.14/291.56	   neq(s(x), 0) -> true
308.14/291.56	   neq(s(x), s(y)) -> neq(x, y)
308.14/291.56	   even(0) -> true
308.14/291.56	   even(s(0)) -> false
308.14/291.56	   even(s(s(x))) -> even(x)
308.14/291.56	   div2(0) -> 0
308.14/291.56	   div2(s(0)) -> 0
308.14/291.56	   div2(s(s(x))) -> s(div2(x))
308.14/291.56	   p(0) -> 0
308.14/291.56	   p(s(x)) -> x
308.14/291.56	
308.14/291.56	S is empty.
308.14/291.56	Rewrite Strategy: FULL
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(6) LowerBoundPropagationProof (FINISHED)
308.14/291.56	Propagated lower bound.
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(7)
308.14/291.56	BOUNDS(n^1, INF)
308.14/291.56	
308.14/291.56	----------------------------------------
308.14/291.56	
308.14/291.56	(8)
308.14/291.56	Obligation:
308.14/291.56	Analyzing the following TRS for decreasing loops:
308.14/291.56	
308.14/291.56	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.14/291.56	
308.14/291.56	
308.14/291.56	The TRS R consists of the following rules:
308.14/291.56	
308.14/291.56	   cond1(true, x) -> cond2(even(x), x)
308.14/291.56	   cond2(true, x) -> cond1(neq(x, 0), div2(x))
308.14/291.56	   cond2(false, x) -> cond1(neq(x, 0), p(x))
308.14/291.56	   neq(0, 0) -> false
308.14/291.56	   neq(0, s(x)) -> true
308.14/291.56	   neq(s(x), 0) -> true
308.14/291.56	   neq(s(x), s(y)) -> neq(x, y)
308.14/291.56	   even(0) -> true
308.14/291.56	   even(s(0)) -> false
308.14/291.56	   even(s(s(x))) -> even(x)
308.14/291.56	   div2(0) -> 0
308.14/291.56	   div2(s(0)) -> 0
308.14/291.56	   div2(s(s(x))) -> s(div2(x))
308.14/291.56	   p(0) -> 0
308.14/291.56	   p(s(x)) -> x
308.14/291.56	
308.14/291.56	S is empty.
308.14/291.56	Rewrite Strategy: FULL
308.14/291.59	EOF