304.92/291.51	WORST_CASE(Omega(n^1), ?)
304.92/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
304.92/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.92/291.52	
304.92/291.52	
304.92/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.52	
304.92/291.52	(0) CpxTRS
304.92/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
304.92/291.52	(2) TRS for Loop Detection
304.92/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
304.92/291.52	(4) BEST
304.92/291.52	    (5) proven lower bound
304.92/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
304.92/291.52	        (7) BOUNDS(n^1, INF)
304.92/291.52	    (8) TRS for Loop Detection
304.92/291.52	
304.92/291.52	
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(0)
304.92/291.52	Obligation:
304.92/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.52	
304.92/291.52	
304.92/291.52	The TRS R consists of the following rules:
304.92/291.52	
304.92/291.52	   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
304.92/291.52	   gr(0, x) -> false
304.92/291.52	   gr(s(x), 0) -> true
304.92/291.52	   gr(s(x), s(y)) -> gr(x, y)
304.92/291.52	   add(0, x) -> x
304.92/291.52	   add(s(x), y) -> s(add(x, y))
304.92/291.52	
304.92/291.52	S is empty.
304.92/291.52	Rewrite Strategy: FULL
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
304.92/291.52	Transformed a relative TRS into a decreasing-loop problem.
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(2)
304.92/291.52	Obligation:
304.92/291.52	Analyzing the following TRS for decreasing loops:
304.92/291.52	
304.92/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.52	
304.92/291.52	
304.92/291.52	The TRS R consists of the following rules:
304.92/291.52	
304.92/291.52	   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
304.92/291.52	   gr(0, x) -> false
304.92/291.52	   gr(s(x), 0) -> true
304.92/291.52	   gr(s(x), s(y)) -> gr(x, y)
304.92/291.52	   add(0, x) -> x
304.92/291.52	   add(s(x), y) -> s(add(x, y))
304.92/291.52	
304.92/291.52	S is empty.
304.92/291.52	Rewrite Strategy: FULL
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
304.92/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
304.92/291.52	
304.92/291.52	The rewrite sequence
304.92/291.52	
304.92/291.52	add(s(x), y) ->^+ s(add(x, y))
304.92/291.52	
304.92/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
304.92/291.52	
304.92/291.52	The pumping substitution is [x / s(x)].
304.92/291.52	
304.92/291.52	The result substitution is [ ].
304.92/291.52	
304.92/291.52	
304.92/291.52	
304.92/291.52	
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(4)
304.92/291.52	Complex Obligation (BEST)
304.92/291.52	
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(5)
304.92/291.52	Obligation:
304.92/291.52	Proved the lower bound n^1 for the following obligation:
304.92/291.52	
304.92/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.52	
304.92/291.52	
304.92/291.52	The TRS R consists of the following rules:
304.92/291.52	
304.92/291.52	   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
304.92/291.52	   gr(0, x) -> false
304.92/291.52	   gr(s(x), 0) -> true
304.92/291.52	   gr(s(x), s(y)) -> gr(x, y)
304.92/291.52	   add(0, x) -> x
304.92/291.52	   add(s(x), y) -> s(add(x, y))
304.92/291.52	
304.92/291.52	S is empty.
304.92/291.52	Rewrite Strategy: FULL
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(6) LowerBoundPropagationProof (FINISHED)
304.92/291.52	Propagated lower bound.
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(7)
304.92/291.52	BOUNDS(n^1, INF)
304.92/291.52	
304.92/291.52	----------------------------------------
304.92/291.52	
304.92/291.52	(8)
304.92/291.52	Obligation:
304.92/291.52	Analyzing the following TRS for decreasing loops:
304.92/291.52	
304.92/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.52	
304.92/291.52	
304.92/291.52	The TRS R consists of the following rules:
304.92/291.52	
304.92/291.52	   cond(true, x, y) -> cond(gr(x, y), x, add(x, y))
304.92/291.52	   gr(0, x) -> false
304.92/291.52	   gr(s(x), 0) -> true
304.92/291.52	   gr(s(x), s(y)) -> gr(x, y)
304.92/291.52	   add(0, x) -> x
304.92/291.52	   add(s(x), y) -> s(add(x, y))
304.92/291.52	
304.92/291.52	S is empty.
304.92/291.52	Rewrite Strategy: FULL
304.98/291.54	EOF