305.67/291.50	WORST_CASE(Omega(n^1), ?)
305.67/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
305.67/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
305.67/291.51	
305.67/291.51	
305.67/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.67/291.51	
305.67/291.51	(0) CpxTRS
305.67/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
305.67/291.51	(2) TRS for Loop Detection
305.67/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
305.67/291.51	(4) BEST
305.67/291.51	    (5) proven lower bound
305.67/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
305.67/291.51	        (7) BOUNDS(n^1, INF)
305.67/291.51	    (8) TRS for Loop Detection
305.67/291.51	
305.67/291.51	
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(0)
305.67/291.51	Obligation:
305.67/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.67/291.51	
305.67/291.51	
305.67/291.51	The TRS R consists of the following rules:
305.67/291.51	
305.67/291.51	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
305.67/291.51	   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
305.67/291.51	   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
305.67/291.51	   gr(0, x) -> false
305.67/291.51	   gr(s(x), 0) -> true
305.67/291.51	   gr(s(x), s(y)) -> gr(x, y)
305.67/291.51	   p(0) -> 0
305.67/291.51	   p(s(x)) -> x
305.67/291.51	
305.67/291.51	S is empty.
305.67/291.51	Rewrite Strategy: FULL
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
305.67/291.51	Transformed a relative TRS into a decreasing-loop problem.
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(2)
305.67/291.51	Obligation:
305.67/291.51	Analyzing the following TRS for decreasing loops:
305.67/291.51	
305.67/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.67/291.51	
305.67/291.51	
305.67/291.51	The TRS R consists of the following rules:
305.67/291.51	
305.67/291.51	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
305.67/291.51	   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
305.67/291.51	   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
305.67/291.51	   gr(0, x) -> false
305.67/291.51	   gr(s(x), 0) -> true
305.67/291.51	   gr(s(x), s(y)) -> gr(x, y)
305.67/291.51	   p(0) -> 0
305.67/291.51	   p(s(x)) -> x
305.67/291.51	
305.67/291.51	S is empty.
305.67/291.51	Rewrite Strategy: FULL
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
305.67/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
305.67/291.51	
305.67/291.51	The rewrite sequence
305.67/291.51	
305.67/291.51	gr(s(x), s(y)) ->^+ gr(x, y)
305.67/291.51	
305.67/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
305.67/291.51	
305.67/291.51	The pumping substitution is [x / s(x), y / s(y)].
305.67/291.51	
305.67/291.51	The result substitution is [ ].
305.67/291.51	
305.67/291.51	
305.67/291.51	
305.67/291.51	
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(4)
305.67/291.51	Complex Obligation (BEST)
305.67/291.51	
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(5)
305.67/291.51	Obligation:
305.67/291.51	Proved the lower bound n^1 for the following obligation:
305.67/291.51	
305.67/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.67/291.51	
305.67/291.51	
305.67/291.51	The TRS R consists of the following rules:
305.67/291.51	
305.67/291.51	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
305.67/291.51	   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
305.67/291.51	   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
305.67/291.51	   gr(0, x) -> false
305.67/291.51	   gr(s(x), 0) -> true
305.67/291.51	   gr(s(x), s(y)) -> gr(x, y)
305.67/291.51	   p(0) -> 0
305.67/291.51	   p(s(x)) -> x
305.67/291.51	
305.67/291.51	S is empty.
305.67/291.51	Rewrite Strategy: FULL
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(6) LowerBoundPropagationProof (FINISHED)
305.67/291.51	Propagated lower bound.
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(7)
305.67/291.51	BOUNDS(n^1, INF)
305.67/291.51	
305.67/291.51	----------------------------------------
305.67/291.51	
305.67/291.51	(8)
305.67/291.51	Obligation:
305.67/291.51	Analyzing the following TRS for decreasing loops:
305.67/291.51	
305.67/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
305.67/291.51	
305.67/291.51	
305.67/291.51	The TRS R consists of the following rules:
305.67/291.51	
305.67/291.51	   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
305.67/291.51	   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
305.67/291.51	   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
305.67/291.51	   gr(0, x) -> false
305.67/291.51	   gr(s(x), 0) -> true
305.67/291.51	   gr(s(x), s(y)) -> gr(x, y)
305.67/291.51	   p(0) -> 0
305.67/291.51	   p(s(x)) -> x
305.67/291.51	
305.67/291.51	S is empty.
305.67/291.51	Rewrite Strategy: FULL
305.76/291.55	EOF