307.05/291.55	WORST_CASE(Omega(n^1), ?)
307.05/291.56	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
307.05/291.56	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
307.05/291.56	
307.05/291.56	
307.05/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.05/291.56	
307.05/291.56	(0) CpxTRS
307.05/291.56	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
307.05/291.56	(2) TRS for Loop Detection
307.05/291.56	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
307.05/291.56	(4) BEST
307.05/291.56	    (5) proven lower bound
307.05/291.56	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
307.05/291.56	        (7) BOUNDS(n^1, INF)
307.05/291.56	    (8) TRS for Loop Detection
307.05/291.56	
307.05/291.56	
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(0)
307.05/291.56	Obligation:
307.05/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.05/291.56	
307.05/291.56	
307.05/291.56	The TRS R consists of the following rules:
307.05/291.56	
307.05/291.56	   f(h(x), y) -> h(f(y, f(x, h(a))))
307.05/291.56	
307.05/291.56	S is empty.
307.05/291.56	Rewrite Strategy: INNERMOST
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
307.05/291.56	Transformed a relative TRS into a decreasing-loop problem.
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(2)
307.05/291.56	Obligation:
307.05/291.56	Analyzing the following TRS for decreasing loops:
307.05/291.56	
307.05/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.05/291.56	
307.05/291.56	
307.05/291.56	The TRS R consists of the following rules:
307.05/291.56	
307.05/291.56	   f(h(x), y) -> h(f(y, f(x, h(a))))
307.05/291.56	
307.05/291.56	S is empty.
307.05/291.56	Rewrite Strategy: INNERMOST
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(3) DecreasingLoopProof (LOWER BOUND(ID))
307.05/291.56	The following loop(s) give(s) rise to the lower bound Omega(n^1):
307.05/291.56	
307.05/291.56	The rewrite sequence
307.05/291.56	
307.05/291.56	f(h(x), y) ->^+ h(f(y, f(x, h(a))))
307.05/291.56	
307.05/291.56	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
307.05/291.56	
307.05/291.56	The pumping substitution is [x / h(x)].
307.05/291.56	
307.05/291.56	The result substitution is [y / h(a)].
307.05/291.56	
307.05/291.56	
307.05/291.56	
307.05/291.56	
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(4)
307.05/291.56	Complex Obligation (BEST)
307.05/291.56	
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(5)
307.05/291.56	Obligation:
307.05/291.56	Proved the lower bound n^1 for the following obligation:
307.05/291.56	
307.05/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.05/291.56	
307.05/291.56	
307.05/291.56	The TRS R consists of the following rules:
307.05/291.56	
307.05/291.56	   f(h(x), y) -> h(f(y, f(x, h(a))))
307.05/291.56	
307.05/291.56	S is empty.
307.05/291.56	Rewrite Strategy: INNERMOST
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(6) LowerBoundPropagationProof (FINISHED)
307.05/291.56	Propagated lower bound.
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(7)
307.05/291.56	BOUNDS(n^1, INF)
307.05/291.56	
307.05/291.56	----------------------------------------
307.05/291.56	
307.05/291.56	(8)
307.05/291.56	Obligation:
307.05/291.56	Analyzing the following TRS for decreasing loops:
307.05/291.56	
307.05/291.56	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
307.05/291.56	
307.05/291.56	
307.05/291.56	The TRS R consists of the following rules:
307.05/291.56	
307.05/291.56	   f(h(x), y) -> h(f(y, f(x, h(a))))
307.05/291.56	
307.05/291.56	S is empty.
307.05/291.56	Rewrite Strategy: INNERMOST
307.05/291.59	EOF