306.07/291.47	WORST_CASE(Omega(n^1), ?)
306.07/291.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
306.07/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
306.07/291.47	
306.07/291.47	
306.07/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.07/291.47	
306.07/291.47	(0) CpxTRS
306.07/291.47	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
306.07/291.47	(2) TRS for Loop Detection
306.07/291.47	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
306.07/291.47	(4) BEST
306.07/291.47	    (5) proven lower bound
306.07/291.47	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
306.07/291.47	        (7) BOUNDS(n^1, INF)
306.07/291.47	    (8) TRS for Loop Detection
306.07/291.47	
306.07/291.47	
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(0)
306.07/291.47	Obligation:
306.07/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.07/291.47	
306.07/291.47	
306.07/291.47	The TRS R consists of the following rules:
306.07/291.47	
306.07/291.47	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
306.07/291.47	   cond2(true, x, y) -> cond1(gr0(x), y, y)
306.07/291.47	   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
306.07/291.47	   gr(0, x) -> false
306.07/291.47	   gr(s(x), 0) -> true
306.07/291.47	   gr(s(x), s(y)) -> gr(x, y)
306.07/291.47	   gr0(0) -> false
306.07/291.47	   gr0(s(x)) -> true
306.07/291.47	   p(0) -> 0
306.07/291.47	   p(s(x)) -> x
306.07/291.47	
306.07/291.47	S is empty.
306.07/291.47	Rewrite Strategy: FULL
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
306.07/291.47	Transformed a relative TRS into a decreasing-loop problem.
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(2)
306.07/291.47	Obligation:
306.07/291.47	Analyzing the following TRS for decreasing loops:
306.07/291.47	
306.07/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.07/291.47	
306.07/291.47	
306.07/291.47	The TRS R consists of the following rules:
306.07/291.47	
306.07/291.47	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
306.07/291.47	   cond2(true, x, y) -> cond1(gr0(x), y, y)
306.07/291.47	   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
306.07/291.47	   gr(0, x) -> false
306.07/291.47	   gr(s(x), 0) -> true
306.07/291.47	   gr(s(x), s(y)) -> gr(x, y)
306.07/291.47	   gr0(0) -> false
306.07/291.47	   gr0(s(x)) -> true
306.07/291.47	   p(0) -> 0
306.07/291.47	   p(s(x)) -> x
306.07/291.47	
306.07/291.47	S is empty.
306.07/291.47	Rewrite Strategy: FULL
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(3) DecreasingLoopProof (LOWER BOUND(ID))
306.07/291.47	The following loop(s) give(s) rise to the lower bound Omega(n^1):
306.07/291.47	
306.07/291.47	The rewrite sequence
306.07/291.47	
306.07/291.47	gr(s(x), s(y)) ->^+ gr(x, y)
306.07/291.47	
306.07/291.47	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
306.07/291.47	
306.07/291.47	The pumping substitution is [x / s(x), y / s(y)].
306.07/291.47	
306.07/291.47	The result substitution is [ ].
306.07/291.47	
306.07/291.47	
306.07/291.47	
306.07/291.47	
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(4)
306.07/291.47	Complex Obligation (BEST)
306.07/291.47	
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(5)
306.07/291.47	Obligation:
306.07/291.47	Proved the lower bound n^1 for the following obligation:
306.07/291.47	
306.07/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.07/291.47	
306.07/291.47	
306.07/291.47	The TRS R consists of the following rules:
306.07/291.47	
306.07/291.47	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
306.07/291.47	   cond2(true, x, y) -> cond1(gr0(x), y, y)
306.07/291.47	   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
306.07/291.47	   gr(0, x) -> false
306.07/291.47	   gr(s(x), 0) -> true
306.07/291.47	   gr(s(x), s(y)) -> gr(x, y)
306.07/291.47	   gr0(0) -> false
306.07/291.47	   gr0(s(x)) -> true
306.07/291.47	   p(0) -> 0
306.07/291.47	   p(s(x)) -> x
306.07/291.47	
306.07/291.47	S is empty.
306.07/291.47	Rewrite Strategy: FULL
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(6) LowerBoundPropagationProof (FINISHED)
306.07/291.47	Propagated lower bound.
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(7)
306.07/291.47	BOUNDS(n^1, INF)
306.07/291.47	
306.07/291.47	----------------------------------------
306.07/291.47	
306.07/291.47	(8)
306.07/291.47	Obligation:
306.07/291.47	Analyzing the following TRS for decreasing loops:
306.07/291.47	
306.07/291.47	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.07/291.47	
306.07/291.47	
306.07/291.47	The TRS R consists of the following rules:
306.07/291.47	
306.07/291.47	   cond1(true, x, y) -> cond2(gr(x, y), x, y)
306.07/291.47	   cond2(true, x, y) -> cond1(gr0(x), y, y)
306.07/291.47	   cond2(false, x, y) -> cond1(gr0(x), p(x), y)
306.07/291.47	   gr(0, x) -> false
306.07/291.47	   gr(s(x), 0) -> true
306.07/291.47	   gr(s(x), s(y)) -> gr(x, y)
306.07/291.47	   gr0(0) -> false
306.07/291.47	   gr0(s(x)) -> true
306.07/291.47	   p(0) -> 0
306.07/291.47	   p(s(x)) -> x
306.07/291.47	
306.07/291.47	S is empty.
306.07/291.47	Rewrite Strategy: FULL
306.16/291.50	EOF