606.82/291.47	WORST_CASE(Omega(n^1), ?)
606.94/291.47	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
606.94/291.47	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
606.94/291.47	
606.94/291.47	
606.94/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
606.94/291.47	
606.94/291.47	(0) CpxTRS
606.94/291.47	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
606.94/291.47	(2) TRS for Loop Detection
606.94/291.47	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
606.94/291.47	(4) BEST
606.94/291.47	    (5) proven lower bound
606.94/291.47	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
606.94/291.47	        (7) BOUNDS(n^1, INF)
606.94/291.47	    (8) TRS for Loop Detection
606.94/291.47	
606.94/291.47	
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(0)
606.94/291.47	Obligation:
606.94/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
606.94/291.47	
606.94/291.47	
606.94/291.47	The TRS R consists of the following rules:
606.94/291.47	
606.94/291.47	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
606.94/291.47	
606.94/291.47	S is empty.
606.94/291.47	Rewrite Strategy: INNERMOST
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
606.94/291.47	Transformed a relative TRS into a decreasing-loop problem.
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(2)
606.94/291.47	Obligation:
606.94/291.47	Analyzing the following TRS for decreasing loops:
606.94/291.47	
606.94/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
606.94/291.47	
606.94/291.47	
606.94/291.47	The TRS R consists of the following rules:
606.94/291.47	
606.94/291.47	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
606.94/291.47	
606.94/291.47	S is empty.
606.94/291.47	Rewrite Strategy: INNERMOST
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(3) DecreasingLoopProof (LOWER BOUND(ID))
606.94/291.47	The following loop(s) give(s) rise to the lower bound Omega(n^1):
606.94/291.47	
606.94/291.47	The rewrite sequence
606.94/291.47	
606.94/291.47	f(h(x), y) ->^+ h(f(y, f(x, h(f(a, a)))))
606.94/291.47	
606.94/291.47	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1].
606.94/291.47	
606.94/291.47	The pumping substitution is [x / h(x)].
606.94/291.47	
606.94/291.47	The result substitution is [y / h(f(a, a))].
606.94/291.47	
606.94/291.47	
606.94/291.47	
606.94/291.47	
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(4)
606.94/291.47	Complex Obligation (BEST)
606.94/291.47	
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(5)
606.94/291.47	Obligation:
606.94/291.47	Proved the lower bound n^1 for the following obligation:
606.94/291.47	
606.94/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
606.94/291.47	
606.94/291.47	
606.94/291.47	The TRS R consists of the following rules:
606.94/291.47	
606.94/291.47	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
606.94/291.47	
606.94/291.47	S is empty.
606.94/291.47	Rewrite Strategy: INNERMOST
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(6) LowerBoundPropagationProof (FINISHED)
606.94/291.47	Propagated lower bound.
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(7)
606.94/291.47	BOUNDS(n^1, INF)
606.94/291.47	
606.94/291.47	----------------------------------------
606.94/291.47	
606.94/291.47	(8)
606.94/291.47	Obligation:
606.94/291.47	Analyzing the following TRS for decreasing loops:
606.94/291.47	
606.94/291.47	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
606.94/291.47	
606.94/291.47	
606.94/291.47	The TRS R consists of the following rules:
606.94/291.47	
606.94/291.47	   f(h(x), y) -> h(f(y, f(x, h(f(a, a)))))
606.94/291.47	
606.94/291.47	S is empty.
606.94/291.47	Rewrite Strategy: INNERMOST
606.94/291.50	EOF