910.87/291.46	WORST_CASE(Omega(n^1), ?)
910.96/291.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
910.96/291.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
910.96/291.46	
910.96/291.46	
910.96/291.46	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
910.96/291.46	
910.96/291.46	(0) CpxTRS
910.96/291.46	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
910.96/291.46	(2) TRS for Loop Detection
910.96/291.46	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
910.96/291.46	(4) BEST
910.96/291.46	    (5) proven lower bound
910.96/291.46	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
910.96/291.46	        (7) BOUNDS(n^1, INF)
910.96/291.46	    (8) TRS for Loop Detection
910.96/291.46	
910.96/291.46	
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(0)
910.96/291.46	Obligation:
910.96/291.46	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
910.96/291.46	
910.96/291.46	
910.96/291.46	The TRS R consists of the following rules:
910.96/291.46	
910.96/291.46	   f(c(s(x), y)) -> f(c(x, s(y)))
910.96/291.46	   f(c(s(x), s(y))) -> g(c(x, y))
910.96/291.46	   g(c(x, s(y))) -> g(c(s(x), y))
910.96/291.46	   g(c(s(x), s(y))) -> f(c(x, y))
910.96/291.46	
910.96/291.46	S is empty.
910.96/291.46	Rewrite Strategy: INNERMOST
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
910.96/291.46	Transformed a relative TRS into a decreasing-loop problem.
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(2)
910.96/291.46	Obligation:
910.96/291.46	Analyzing the following TRS for decreasing loops:
910.96/291.46	
910.96/291.46	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
910.96/291.46	
910.96/291.46	
910.96/291.46	The TRS R consists of the following rules:
910.96/291.46	
910.96/291.46	   f(c(s(x), y)) -> f(c(x, s(y)))
910.96/291.46	   f(c(s(x), s(y))) -> g(c(x, y))
910.96/291.46	   g(c(x, s(y))) -> g(c(s(x), y))
910.96/291.46	   g(c(s(x), s(y))) -> f(c(x, y))
910.96/291.46	
910.96/291.46	S is empty.
910.96/291.46	Rewrite Strategy: INNERMOST
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(3) DecreasingLoopProof (LOWER BOUND(ID))
910.96/291.46	The following loop(s) give(s) rise to the lower bound Omega(n^1):
910.96/291.46	
910.96/291.46	The rewrite sequence
910.96/291.46	
910.96/291.46	g(c(x, s(y))) ->^+ g(c(s(x), y))
910.96/291.46	
910.96/291.46	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
910.96/291.46	
910.96/291.46	The pumping substitution is [y / s(y)].
910.96/291.46	
910.96/291.46	The result substitution is [x / s(x)].
910.96/291.46	
910.96/291.46	
910.96/291.46	
910.96/291.46	
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(4)
910.96/291.46	Complex Obligation (BEST)
910.96/291.46	
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(5)
910.96/291.46	Obligation:
910.96/291.46	Proved the lower bound n^1 for the following obligation:
910.96/291.46	
910.96/291.46	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
910.96/291.46	
910.96/291.46	
910.96/291.46	The TRS R consists of the following rules:
910.96/291.46	
910.96/291.46	   f(c(s(x), y)) -> f(c(x, s(y)))
910.96/291.46	   f(c(s(x), s(y))) -> g(c(x, y))
910.96/291.46	   g(c(x, s(y))) -> g(c(s(x), y))
910.96/291.46	   g(c(s(x), s(y))) -> f(c(x, y))
910.96/291.46	
910.96/291.46	S is empty.
910.96/291.46	Rewrite Strategy: INNERMOST
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(6) LowerBoundPropagationProof (FINISHED)
910.96/291.46	Propagated lower bound.
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(7)
910.96/291.46	BOUNDS(n^1, INF)
910.96/291.46	
910.96/291.46	----------------------------------------
910.96/291.46	
910.96/291.46	(8)
910.96/291.46	Obligation:
910.96/291.46	Analyzing the following TRS for decreasing loops:
910.96/291.46	
910.96/291.46	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
910.96/291.46	
910.96/291.46	
910.96/291.46	The TRS R consists of the following rules:
910.96/291.46	
910.96/291.46	   f(c(s(x), y)) -> f(c(x, s(y)))
910.96/291.46	   f(c(s(x), s(y))) -> g(c(x, y))
910.96/291.46	   g(c(x, s(y))) -> g(c(s(x), y))
910.96/291.46	   g(c(s(x), s(y))) -> f(c(x, y))
910.96/291.46	
910.96/291.46	S is empty.
910.96/291.46	Rewrite Strategy: INNERMOST
910.96/291.52	EOF