301.97/291.48	WORST_CASE(Omega(n^1), ?)
301.97/291.48	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
301.97/291.48	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.97/291.48	
301.97/291.48	
301.97/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.97/291.48	
301.97/291.48	(0) CpxTRS
301.97/291.48	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.97/291.48	(2) TRS for Loop Detection
301.97/291.48	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
301.97/291.48	(4) BEST
301.97/291.48	    (5) proven lower bound
301.97/291.48	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.97/291.48	        (7) BOUNDS(n^1, INF)
301.97/291.48	    (8) TRS for Loop Detection
301.97/291.48	
301.97/291.48	
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(0)
301.97/291.48	Obligation:
301.97/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.97/291.48	
301.97/291.48	
301.97/291.48	The TRS R consists of the following rules:
301.97/291.48	
301.97/291.48	   f(a, g(y)) -> g(g(y))
301.97/291.48	   f(g(x), a) -> f(x, g(a))
301.97/291.48	   f(g(x), g(y)) -> h(g(y), x, g(y))
301.97/291.48	   h(g(x), y, z) -> f(y, h(x, y, z))
301.97/291.48	   h(a, y, z) -> z
301.97/291.48	
301.97/291.48	S is empty.
301.97/291.48	Rewrite Strategy: FULL
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.97/291.48	Transformed a relative TRS into a decreasing-loop problem.
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(2)
301.97/291.48	Obligation:
301.97/291.48	Analyzing the following TRS for decreasing loops:
301.97/291.48	
301.97/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.97/291.48	
301.97/291.48	
301.97/291.48	The TRS R consists of the following rules:
301.97/291.48	
301.97/291.48	   f(a, g(y)) -> g(g(y))
301.97/291.48	   f(g(x), a) -> f(x, g(a))
301.97/291.48	   f(g(x), g(y)) -> h(g(y), x, g(y))
301.97/291.48	   h(g(x), y, z) -> f(y, h(x, y, z))
301.97/291.48	   h(a, y, z) -> z
301.97/291.48	
301.97/291.48	S is empty.
301.97/291.48	Rewrite Strategy: FULL
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.97/291.48	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.97/291.48	
301.97/291.48	The rewrite sequence
301.97/291.48	
301.97/291.48	h(g(x), y, z) ->^+ f(y, h(x, y, z))
301.97/291.48	
301.97/291.48	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
301.97/291.48	
301.97/291.48	The pumping substitution is [x / g(x)].
301.97/291.48	
301.97/291.48	The result substitution is [ ].
301.97/291.48	
301.97/291.48	
301.97/291.48	
301.97/291.48	
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(4)
301.97/291.48	Complex Obligation (BEST)
301.97/291.48	
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(5)
301.97/291.48	Obligation:
301.97/291.48	Proved the lower bound n^1 for the following obligation:
301.97/291.48	
301.97/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.97/291.48	
301.97/291.48	
301.97/291.48	The TRS R consists of the following rules:
301.97/291.48	
301.97/291.48	   f(a, g(y)) -> g(g(y))
301.97/291.48	   f(g(x), a) -> f(x, g(a))
301.97/291.48	   f(g(x), g(y)) -> h(g(y), x, g(y))
301.97/291.48	   h(g(x), y, z) -> f(y, h(x, y, z))
301.97/291.48	   h(a, y, z) -> z
301.97/291.48	
301.97/291.48	S is empty.
301.97/291.48	Rewrite Strategy: FULL
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(6) LowerBoundPropagationProof (FINISHED)
301.97/291.48	Propagated lower bound.
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(7)
301.97/291.48	BOUNDS(n^1, INF)
301.97/291.48	
301.97/291.48	----------------------------------------
301.97/291.48	
301.97/291.48	(8)
301.97/291.48	Obligation:
301.97/291.48	Analyzing the following TRS for decreasing loops:
301.97/291.48	
301.97/291.48	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.97/291.48	
301.97/291.48	
301.97/291.48	The TRS R consists of the following rules:
301.97/291.48	
301.97/291.48	   f(a, g(y)) -> g(g(y))
301.97/291.48	   f(g(x), a) -> f(x, g(a))
301.97/291.48	   f(g(x), g(y)) -> h(g(y), x, g(y))
301.97/291.48	   h(g(x), y, z) -> f(y, h(x, y, z))
301.97/291.48	   h(a, y, z) -> z
301.97/291.48	
301.97/291.48	S is empty.
301.97/291.48	Rewrite Strategy: FULL
301.97/291.51	EOF