642.74/291.62	WORST_CASE(Omega(n^1), ?)
642.92/291.63	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
642.92/291.63	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
642.92/291.63	
642.92/291.63	
642.92/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
642.92/291.63	
642.92/291.63	(0) CpxTRS
642.92/291.63	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
642.92/291.63	(2) TRS for Loop Detection
642.92/291.63	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
642.92/291.63	(4) BEST
642.92/291.63	    (5) proven lower bound
642.92/291.63	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
642.92/291.63	        (7) BOUNDS(n^1, INF)
642.92/291.63	    (8) TRS for Loop Detection
642.92/291.63	
642.92/291.63	
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(0)
642.92/291.63	Obligation:
642.92/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
642.92/291.63	
642.92/291.63	
642.92/291.63	The TRS R consists of the following rules:
642.92/291.63	
642.92/291.63	   f(0, 1, x) -> f(g(x), g(x), x)
642.92/291.63	   f(g(x), y, z) -> g(f(x, y, z))
642.92/291.63	   f(x, g(y), z) -> g(f(x, y, z))
642.92/291.63	   f(x, y, g(z)) -> g(f(x, y, z))
642.92/291.63	
642.92/291.63	S is empty.
642.92/291.63	Rewrite Strategy: FULL
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
642.92/291.63	Transformed a relative TRS into a decreasing-loop problem.
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(2)
642.92/291.63	Obligation:
642.92/291.63	Analyzing the following TRS for decreasing loops:
642.92/291.63	
642.92/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
642.92/291.63	
642.92/291.63	
642.92/291.63	The TRS R consists of the following rules:
642.92/291.63	
642.92/291.63	   f(0, 1, x) -> f(g(x), g(x), x)
642.92/291.63	   f(g(x), y, z) -> g(f(x, y, z))
642.92/291.63	   f(x, g(y), z) -> g(f(x, y, z))
642.92/291.63	   f(x, y, g(z)) -> g(f(x, y, z))
642.92/291.63	
642.92/291.63	S is empty.
642.92/291.63	Rewrite Strategy: FULL
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(3) DecreasingLoopProof (LOWER BOUND(ID))
642.92/291.63	The following loop(s) give(s) rise to the lower bound Omega(n^1):
642.92/291.63	
642.92/291.63	The rewrite sequence
642.92/291.63	
642.92/291.63	f(g(x), y, z) ->^+ g(f(x, y, z))
642.92/291.63	
642.92/291.63	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
642.92/291.63	
642.92/291.63	The pumping substitution is [x / g(x)].
642.92/291.63	
642.92/291.63	The result substitution is [ ].
642.92/291.63	
642.92/291.63	
642.92/291.63	
642.92/291.63	
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(4)
642.92/291.63	Complex Obligation (BEST)
642.92/291.63	
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(5)
642.92/291.63	Obligation:
642.92/291.63	Proved the lower bound n^1 for the following obligation:
642.92/291.63	
642.92/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
642.92/291.63	
642.92/291.63	
642.92/291.63	The TRS R consists of the following rules:
642.92/291.63	
642.92/291.63	   f(0, 1, x) -> f(g(x), g(x), x)
642.92/291.63	   f(g(x), y, z) -> g(f(x, y, z))
642.92/291.63	   f(x, g(y), z) -> g(f(x, y, z))
642.92/291.63	   f(x, y, g(z)) -> g(f(x, y, z))
642.92/291.63	
642.92/291.63	S is empty.
642.92/291.63	Rewrite Strategy: FULL
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(6) LowerBoundPropagationProof (FINISHED)
642.92/291.63	Propagated lower bound.
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(7)
642.92/291.63	BOUNDS(n^1, INF)
642.92/291.63	
642.92/291.63	----------------------------------------
642.92/291.63	
642.92/291.63	(8)
642.92/291.63	Obligation:
642.92/291.63	Analyzing the following TRS for decreasing loops:
642.92/291.63	
642.92/291.63	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
642.92/291.63	
642.92/291.63	
642.92/291.63	The TRS R consists of the following rules:
642.92/291.63	
642.92/291.63	   f(0, 1, x) -> f(g(x), g(x), x)
642.92/291.63	   f(g(x), y, z) -> g(f(x, y, z))
642.92/291.63	   f(x, g(y), z) -> g(f(x, y, z))
642.92/291.63	   f(x, y, g(z)) -> g(f(x, y, z))
642.92/291.63	
642.92/291.63	S is empty.
642.92/291.63	Rewrite Strategy: FULL
642.92/291.69	EOF