307.71/291.51	WORST_CASE(Omega(n^1), ?)
307.71/291.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
307.71/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
307.71/291.55	
307.71/291.55	
307.71/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.71/291.55	
307.71/291.55	(0) CpxTRS
307.71/291.55	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
307.71/291.55	(2) TRS for Loop Detection
307.71/291.55	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
307.71/291.55	(4) BEST
307.71/291.55	    (5) proven lower bound
307.71/291.55	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
307.71/291.55	        (7) BOUNDS(n^1, INF)
307.71/291.55	    (8) TRS for Loop Detection
307.71/291.55	
307.71/291.55	
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(0)
307.71/291.55	Obligation:
307.71/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.71/291.55	
307.71/291.55	
307.71/291.55	The TRS R consists of the following rules:
307.71/291.55	
307.71/291.55	   f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
307.71/291.55	   f(s(x), y, z) -> f(x, s(c(y)), c(z))
307.71/291.55	   f(c(x), x, y) -> c(y)
307.71/291.55	   g(x, y) -> x
307.71/291.55	   g(x, y) -> y
307.71/291.55	
307.71/291.55	S is empty.
307.71/291.55	Rewrite Strategy: FULL
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
307.71/291.55	Transformed a relative TRS into a decreasing-loop problem.
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(2)
307.71/291.55	Obligation:
307.71/291.55	Analyzing the following TRS for decreasing loops:
307.71/291.55	
307.71/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.71/291.55	
307.71/291.55	
307.71/291.55	The TRS R consists of the following rules:
307.71/291.55	
307.71/291.55	   f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
307.71/291.55	   f(s(x), y, z) -> f(x, s(c(y)), c(z))
307.71/291.55	   f(c(x), x, y) -> c(y)
307.71/291.55	   g(x, y) -> x
307.71/291.55	   g(x, y) -> y
307.71/291.55	
307.71/291.55	S is empty.
307.71/291.55	Rewrite Strategy: FULL
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(3) DecreasingLoopProof (LOWER BOUND(ID))
307.71/291.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
307.71/291.55	
307.71/291.55	The rewrite sequence
307.71/291.55	
307.71/291.55	f(s(x), y, z) ->^+ f(x, s(c(y)), c(z))
307.71/291.55	
307.71/291.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
307.71/291.55	
307.71/291.55	The pumping substitution is [x / s(x)].
307.71/291.55	
307.71/291.55	The result substitution is [y / s(c(y)), z / c(z)].
307.71/291.55	
307.71/291.55	
307.71/291.55	
307.71/291.55	
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(4)
307.71/291.55	Complex Obligation (BEST)
307.71/291.55	
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(5)
307.71/291.55	Obligation:
307.71/291.55	Proved the lower bound n^1 for the following obligation:
307.71/291.55	
307.71/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.71/291.55	
307.71/291.55	
307.71/291.55	The TRS R consists of the following rules:
307.71/291.55	
307.71/291.55	   f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
307.71/291.55	   f(s(x), y, z) -> f(x, s(c(y)), c(z))
307.71/291.55	   f(c(x), x, y) -> c(y)
307.71/291.55	   g(x, y) -> x
307.71/291.55	   g(x, y) -> y
307.71/291.55	
307.71/291.55	S is empty.
307.71/291.55	Rewrite Strategy: FULL
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(6) LowerBoundPropagationProof (FINISHED)
307.71/291.55	Propagated lower bound.
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(7)
307.71/291.55	BOUNDS(n^1, INF)
307.71/291.55	
307.71/291.55	----------------------------------------
307.71/291.55	
307.71/291.55	(8)
307.71/291.55	Obligation:
307.71/291.55	Analyzing the following TRS for decreasing loops:
307.71/291.55	
307.71/291.55	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.71/291.55	
307.71/291.55	
307.71/291.55	The TRS R consists of the following rules:
307.71/291.55	
307.71/291.55	   f(x, c(x), c(y)) -> f(y, y, f(y, x, y))
307.71/291.55	   f(s(x), y, z) -> f(x, s(c(y)), c(z))
307.71/291.55	   f(c(x), x, y) -> c(y)
307.71/291.55	   g(x, y) -> x
307.71/291.55	   g(x, y) -> y
307.71/291.55	
307.71/291.55	S is empty.
307.71/291.55	Rewrite Strategy: FULL
307.71/291.58	EOF