912.09/291.48	WORST_CASE(Omega(n^1), ?)
912.09/291.48	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
912.09/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
912.09/291.48	
912.09/291.48	
912.09/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
912.09/291.48	
912.09/291.48	(0) CpxTRS
912.09/291.48	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
912.09/291.48	(2) TRS for Loop Detection
912.09/291.48	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
912.09/291.48	(4) BEST
912.09/291.48	    (5) proven lower bound
912.09/291.48	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
912.09/291.48	        (7) BOUNDS(n^1, INF)
912.09/291.48	    (8) TRS for Loop Detection
912.09/291.48	
912.09/291.48	
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(0)
912.09/291.48	Obligation:
912.09/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
912.09/291.48	
912.09/291.48	
912.09/291.48	The TRS R consists of the following rules:
912.09/291.48	
912.09/291.48	   f(c(s(x), y)) -> f(c(x, s(y)))
912.09/291.48	   f(c(s(x), s(y))) -> g(c(x, y))
912.09/291.48	   g(c(x, s(y))) -> g(c(s(x), y))
912.09/291.48	   g(c(s(x), s(y))) -> f(c(x, y))
912.09/291.48	
912.09/291.48	S is empty.
912.09/291.48	Rewrite Strategy: FULL
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
912.09/291.48	Transformed a relative TRS into a decreasing-loop problem.
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(2)
912.09/291.48	Obligation:
912.09/291.48	Analyzing the following TRS for decreasing loops:
912.09/291.48	
912.09/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
912.09/291.48	
912.09/291.48	
912.09/291.48	The TRS R consists of the following rules:
912.09/291.48	
912.09/291.48	   f(c(s(x), y)) -> f(c(x, s(y)))
912.09/291.48	   f(c(s(x), s(y))) -> g(c(x, y))
912.09/291.48	   g(c(x, s(y))) -> g(c(s(x), y))
912.09/291.48	   g(c(s(x), s(y))) -> f(c(x, y))
912.09/291.48	
912.09/291.48	S is empty.
912.09/291.48	Rewrite Strategy: FULL
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(3) DecreasingLoopProof (LOWER BOUND(ID))
912.09/291.48	The following loop(s) give(s) rise to the lower bound Omega(n^1):
912.09/291.48	
912.09/291.48	The rewrite sequence
912.09/291.48	
912.09/291.48	g(c(x, s(y))) ->^+ g(c(s(x), y))
912.09/291.48	
912.09/291.48	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
912.09/291.48	
912.09/291.48	The pumping substitution is [y / s(y)].
912.09/291.48	
912.09/291.48	The result substitution is [x / s(x)].
912.09/291.48	
912.09/291.48	
912.09/291.48	
912.09/291.48	
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(4)
912.09/291.48	Complex Obligation (BEST)
912.09/291.48	
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(5)
912.09/291.48	Obligation:
912.09/291.48	Proved the lower bound n^1 for the following obligation:
912.09/291.48	
912.09/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
912.09/291.48	
912.09/291.48	
912.09/291.48	The TRS R consists of the following rules:
912.09/291.48	
912.09/291.48	   f(c(s(x), y)) -> f(c(x, s(y)))
912.09/291.48	   f(c(s(x), s(y))) -> g(c(x, y))
912.09/291.48	   g(c(x, s(y))) -> g(c(s(x), y))
912.09/291.48	   g(c(s(x), s(y))) -> f(c(x, y))
912.09/291.48	
912.09/291.48	S is empty.
912.09/291.48	Rewrite Strategy: FULL
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(6) LowerBoundPropagationProof (FINISHED)
912.09/291.48	Propagated lower bound.
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(7)
912.09/291.48	BOUNDS(n^1, INF)
912.09/291.48	
912.09/291.48	----------------------------------------
912.09/291.48	
912.09/291.48	(8)
912.09/291.48	Obligation:
912.09/291.48	Analyzing the following TRS for decreasing loops:
912.09/291.48	
912.09/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
912.09/291.48	
912.09/291.48	
912.09/291.48	The TRS R consists of the following rules:
912.09/291.48	
912.09/291.48	   f(c(s(x), y)) -> f(c(x, s(y)))
912.09/291.48	   f(c(s(x), s(y))) -> g(c(x, y))
912.09/291.48	   g(c(x, s(y))) -> g(c(s(x), y))
912.09/291.48	   g(c(s(x), s(y))) -> f(c(x, y))
912.09/291.48	
912.09/291.48	S is empty.
912.09/291.48	Rewrite Strategy: FULL
912.20/291.53	EOF