302.26/291.50	WORST_CASE(Omega(n^1), ?)
302.26/291.50	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
302.26/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
302.26/291.50	
302.26/291.50	
302.26/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.26/291.50	
302.26/291.50	(0) CpxTRS
302.26/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
302.26/291.50	(2) TRS for Loop Detection
302.26/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
302.26/291.50	(4) BEST
302.26/291.50	    (5) proven lower bound
302.26/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
302.26/291.50	        (7) BOUNDS(n^1, INF)
302.26/291.50	    (8) TRS for Loop Detection
302.26/291.50	
302.26/291.50	
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(0)
302.26/291.50	Obligation:
302.26/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.26/291.50	
302.26/291.50	
302.26/291.50	The TRS R consists of the following rules:
302.26/291.50	
302.26/291.50	   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
302.26/291.50	   and(x, false) -> false
302.26/291.50	   and(false, x) -> false
302.26/291.50	   and(true, true) -> true
302.26/291.50	   even(0) -> true
302.26/291.50	   even(s(0)) -> false
302.26/291.50	   even(s(s(x))) -> even(x)
302.26/291.50	   gr(0, x) -> false
302.26/291.50	   gr(s(x), 0) -> true
302.26/291.50	   gr(s(x), s(y)) -> gr(x, y)
302.26/291.50	   p(0) -> 0
302.26/291.50	   p(s(x)) -> x
302.26/291.50	
302.26/291.50	S is empty.
302.26/291.50	Rewrite Strategy: FULL
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
302.26/291.50	Transformed a relative TRS into a decreasing-loop problem.
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(2)
302.26/291.50	Obligation:
302.26/291.50	Analyzing the following TRS for decreasing loops:
302.26/291.50	
302.26/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.26/291.50	
302.26/291.50	
302.26/291.50	The TRS R consists of the following rules:
302.26/291.50	
302.26/291.50	   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
302.26/291.50	   and(x, false) -> false
302.26/291.50	   and(false, x) -> false
302.26/291.50	   and(true, true) -> true
302.26/291.50	   even(0) -> true
302.26/291.50	   even(s(0)) -> false
302.26/291.50	   even(s(s(x))) -> even(x)
302.26/291.50	   gr(0, x) -> false
302.26/291.50	   gr(s(x), 0) -> true
302.26/291.50	   gr(s(x), s(y)) -> gr(x, y)
302.26/291.50	   p(0) -> 0
302.26/291.50	   p(s(x)) -> x
302.26/291.50	
302.26/291.50	S is empty.
302.26/291.50	Rewrite Strategy: FULL
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
302.26/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
302.26/291.50	
302.26/291.50	The rewrite sequence
302.26/291.50	
302.26/291.50	even(s(s(x))) ->^+ even(x)
302.26/291.50	
302.26/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
302.26/291.50	
302.26/291.50	The pumping substitution is [x / s(s(x))].
302.26/291.50	
302.26/291.50	The result substitution is [ ].
302.26/291.50	
302.26/291.50	
302.26/291.50	
302.26/291.50	
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(4)
302.26/291.50	Complex Obligation (BEST)
302.26/291.50	
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(5)
302.26/291.50	Obligation:
302.26/291.50	Proved the lower bound n^1 for the following obligation:
302.26/291.50	
302.26/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.26/291.50	
302.26/291.50	
302.26/291.50	The TRS R consists of the following rules:
302.26/291.50	
302.26/291.50	   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
302.26/291.50	   and(x, false) -> false
302.26/291.50	   and(false, x) -> false
302.26/291.50	   and(true, true) -> true
302.26/291.50	   even(0) -> true
302.26/291.50	   even(s(0)) -> false
302.26/291.50	   even(s(s(x))) -> even(x)
302.26/291.50	   gr(0, x) -> false
302.26/291.50	   gr(s(x), 0) -> true
302.26/291.50	   gr(s(x), s(y)) -> gr(x, y)
302.26/291.50	   p(0) -> 0
302.26/291.50	   p(s(x)) -> x
302.26/291.50	
302.26/291.50	S is empty.
302.26/291.50	Rewrite Strategy: FULL
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(6) LowerBoundPropagationProof (FINISHED)
302.26/291.50	Propagated lower bound.
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(7)
302.26/291.50	BOUNDS(n^1, INF)
302.26/291.50	
302.26/291.50	----------------------------------------
302.26/291.50	
302.26/291.50	(8)
302.26/291.50	Obligation:
302.26/291.50	Analyzing the following TRS for decreasing loops:
302.26/291.50	
302.26/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
302.26/291.50	
302.26/291.50	
302.26/291.50	The TRS R consists of the following rules:
302.26/291.50	
302.26/291.50	   cond(true, x) -> cond(and(even(x), gr(x, 0)), p(x))
302.26/291.50	   and(x, false) -> false
302.26/291.50	   and(false, x) -> false
302.26/291.50	   and(true, true) -> true
302.26/291.50	   even(0) -> true
302.26/291.50	   even(s(0)) -> false
302.26/291.50	   even(s(s(x))) -> even(x)
302.26/291.50	   gr(0, x) -> false
302.26/291.50	   gr(s(x), 0) -> true
302.26/291.50	   gr(s(x), s(y)) -> gr(x, y)
302.26/291.50	   p(0) -> 0
302.26/291.50	   p(s(x)) -> x
302.26/291.50	
302.26/291.50	S is empty.
302.26/291.50	Rewrite Strategy: FULL
302.26/291.54	EOF