303.78/291.47	WORST_CASE(Omega(n^1), ?)
303.78/291.48	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
303.78/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.78/291.48	
303.78/291.48	
303.78/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.78/291.48	
303.78/291.48	(0) CpxTRS
303.78/291.48	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.78/291.48	(2) TRS for Loop Detection
303.78/291.48	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
303.78/291.48	(4) BEST
303.78/291.48	    (5) proven lower bound
303.78/291.48	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.78/291.48	        (7) BOUNDS(n^1, INF)
303.78/291.48	    (8) TRS for Loop Detection
303.78/291.48	
303.78/291.48	
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(0)
303.78/291.48	Obligation:
303.78/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.78/291.48	
303.78/291.48	
303.78/291.48	The TRS R consists of the following rules:
303.78/291.48	
303.78/291.48	   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
303.78/291.48	   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
303.78/291.48	   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
303.78/291.48	   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
303.78/291.48	   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
303.78/291.48	   gr(0, x) -> false
303.78/291.48	   gr(s(x), 0) -> true
303.78/291.48	   gr(s(x), s(y)) -> gr(x, y)
303.78/291.48	   or(false, false) -> false
303.78/291.48	   or(true, x) -> true
303.78/291.48	   or(x, true) -> true
303.78/291.48	   p(0) -> 0
303.78/291.48	   p(s(x)) -> x
303.78/291.48	
303.78/291.48	S is empty.
303.78/291.48	Rewrite Strategy: FULL
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.78/291.48	Transformed a relative TRS into a decreasing-loop problem.
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(2)
303.78/291.48	Obligation:
303.78/291.48	Analyzing the following TRS for decreasing loops:
303.78/291.48	
303.78/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.78/291.48	
303.78/291.48	
303.78/291.48	The TRS R consists of the following rules:
303.78/291.48	
303.78/291.48	   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
303.78/291.48	   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
303.78/291.48	   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
303.78/291.48	   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
303.78/291.48	   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
303.78/291.48	   gr(0, x) -> false
303.78/291.48	   gr(s(x), 0) -> true
303.78/291.48	   gr(s(x), s(y)) -> gr(x, y)
303.78/291.48	   or(false, false) -> false
303.78/291.48	   or(true, x) -> true
303.78/291.48	   or(x, true) -> true
303.78/291.48	   p(0) -> 0
303.78/291.48	   p(s(x)) -> x
303.78/291.48	
303.78/291.48	S is empty.
303.78/291.48	Rewrite Strategy: FULL
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.78/291.48	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.78/291.48	
303.78/291.48	The rewrite sequence
303.78/291.48	
303.78/291.48	gr(s(x), s(y)) ->^+ gr(x, y)
303.78/291.48	
303.78/291.48	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
303.78/291.48	
303.78/291.48	The pumping substitution is [x / s(x), y / s(y)].
303.78/291.48	
303.78/291.48	The result substitution is [ ].
303.78/291.48	
303.78/291.48	
303.78/291.48	
303.78/291.48	
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(4)
303.78/291.48	Complex Obligation (BEST)
303.78/291.48	
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(5)
303.78/291.48	Obligation:
303.78/291.48	Proved the lower bound n^1 for the following obligation:
303.78/291.48	
303.78/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.78/291.48	
303.78/291.48	
303.78/291.48	The TRS R consists of the following rules:
303.78/291.48	
303.78/291.48	   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
303.78/291.48	   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
303.78/291.48	   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
303.78/291.48	   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
303.78/291.48	   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
303.78/291.48	   gr(0, x) -> false
303.78/291.48	   gr(s(x), 0) -> true
303.78/291.48	   gr(s(x), s(y)) -> gr(x, y)
303.78/291.48	   or(false, false) -> false
303.78/291.48	   or(true, x) -> true
303.78/291.48	   or(x, true) -> true
303.78/291.48	   p(0) -> 0
303.78/291.48	   p(s(x)) -> x
303.78/291.48	
303.78/291.48	S is empty.
303.78/291.48	Rewrite Strategy: FULL
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(6) LowerBoundPropagationProof (FINISHED)
303.78/291.48	Propagated lower bound.
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(7)
303.78/291.48	BOUNDS(n^1, INF)
303.78/291.48	
303.78/291.48	----------------------------------------
303.78/291.48	
303.78/291.48	(8)
303.78/291.48	Obligation:
303.78/291.48	Analyzing the following TRS for decreasing loops:
303.78/291.48	
303.78/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.78/291.48	
303.78/291.48	
303.78/291.48	The TRS R consists of the following rules:
303.78/291.48	
303.78/291.48	   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
303.78/291.48	   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
303.78/291.48	   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
303.78/291.48	   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
303.78/291.48	   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
303.78/291.48	   gr(0, x) -> false
303.78/291.48	   gr(s(x), 0) -> true
303.78/291.48	   gr(s(x), s(y)) -> gr(x, y)
303.78/291.48	   or(false, false) -> false
303.78/291.48	   or(true, x) -> true
303.78/291.48	   or(x, true) -> true
303.78/291.48	   p(0) -> 0
303.78/291.48	   p(s(x)) -> x
303.78/291.48	
303.78/291.48	S is empty.
303.78/291.48	Rewrite Strategy: FULL
303.89/291.51	EOF