1132.50/291.53	WORST_CASE(Omega(n^1), ?)
1132.82/291.55	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1132.82/291.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.55	
1132.82/291.55	(0) CpxTRS
1132.82/291.55	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1132.82/291.55	(2) TRS for Loop Detection
1132.82/291.55	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1132.82/291.55	(4) BEST
1132.82/291.55	    (5) proven lower bound
1132.82/291.55	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1132.82/291.55	        (7) BOUNDS(n^1, INF)
1132.82/291.55	    (8) TRS for Loop Detection
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(0)
1132.82/291.55	Obligation:
1132.82/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	The TRS R consists of the following rules:
1132.82/291.55	
1132.82/291.55	   f(a, g(y)) -> g(g(y))
1132.82/291.55	   f(g(x), a) -> f(x, g(a))
1132.82/291.55	   f(g(x), g(y)) -> h(g(y), x, g(y))
1132.82/291.55	   h(g(x), y, z) -> f(y, h(x, y, z))
1132.82/291.55	   h(a, y, z) -> z
1132.82/291.55	
1132.82/291.55	S is empty.
1132.82/291.55	Rewrite Strategy: INNERMOST
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1132.82/291.55	Transformed a relative TRS into a decreasing-loop problem.
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(2)
1132.82/291.55	Obligation:
1132.82/291.55	Analyzing the following TRS for decreasing loops:
1132.82/291.55	
1132.82/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	The TRS R consists of the following rules:
1132.82/291.55	
1132.82/291.55	   f(a, g(y)) -> g(g(y))
1132.82/291.55	   f(g(x), a) -> f(x, g(a))
1132.82/291.55	   f(g(x), g(y)) -> h(g(y), x, g(y))
1132.82/291.55	   h(g(x), y, z) -> f(y, h(x, y, z))
1132.82/291.55	   h(a, y, z) -> z
1132.82/291.55	
1132.82/291.55	S is empty.
1132.82/291.55	Rewrite Strategy: INNERMOST
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(3) DecreasingLoopProof (LOWER BOUND(ID))
1132.82/291.55	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1132.82/291.55	
1132.82/291.55	The rewrite sequence
1132.82/291.55	
1132.82/291.55	h(g(x), y, z) ->^+ f(y, h(x, y, z))
1132.82/291.55	
1132.82/291.55	gives rise to a decreasing loop by considering the right hand sides subterm at position [1].
1132.82/291.55	
1132.82/291.55	The pumping substitution is [x / g(x)].
1132.82/291.55	
1132.82/291.55	The result substitution is [ ].
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(4)
1132.82/291.55	Complex Obligation (BEST)
1132.82/291.55	
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(5)
1132.82/291.55	Obligation:
1132.82/291.55	Proved the lower bound n^1 for the following obligation:
1132.82/291.55	
1132.82/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	The TRS R consists of the following rules:
1132.82/291.55	
1132.82/291.55	   f(a, g(y)) -> g(g(y))
1132.82/291.55	   f(g(x), a) -> f(x, g(a))
1132.82/291.55	   f(g(x), g(y)) -> h(g(y), x, g(y))
1132.82/291.55	   h(g(x), y, z) -> f(y, h(x, y, z))
1132.82/291.55	   h(a, y, z) -> z
1132.82/291.55	
1132.82/291.55	S is empty.
1132.82/291.55	Rewrite Strategy: INNERMOST
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(6) LowerBoundPropagationProof (FINISHED)
1132.82/291.55	Propagated lower bound.
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(7)
1132.82/291.55	BOUNDS(n^1, INF)
1132.82/291.55	
1132.82/291.55	----------------------------------------
1132.82/291.55	
1132.82/291.55	(8)
1132.82/291.55	Obligation:
1132.82/291.55	Analyzing the following TRS for decreasing loops:
1132.82/291.55	
1132.82/291.55	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1132.82/291.55	
1132.82/291.55	
1132.82/291.55	The TRS R consists of the following rules:
1132.82/291.55	
1132.82/291.55	   f(a, g(y)) -> g(g(y))
1132.82/291.55	   f(g(x), a) -> f(x, g(a))
1132.82/291.55	   f(g(x), g(y)) -> h(g(y), x, g(y))
1132.82/291.55	   h(g(x), y, z) -> f(y, h(x, y, z))
1132.82/291.55	   h(a, y, z) -> z
1132.82/291.55	
1132.82/291.55	S is empty.
1132.82/291.55	Rewrite Strategy: INNERMOST
1132.86/291.58	EOF