704.74/291.52	WORST_CASE(Omega(n^1), ?)
704.82/291.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
704.82/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
704.82/291.53	
704.82/291.53	
704.82/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
704.82/291.53	
704.82/291.53	(0) CpxTRS
704.82/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
704.82/291.53	(2) TRS for Loop Detection
704.82/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
704.82/291.53	(4) BEST
704.82/291.53	    (5) proven lower bound
704.82/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
704.82/291.53	        (7) BOUNDS(n^1, INF)
704.82/291.53	    (8) TRS for Loop Detection
704.82/291.53	
704.82/291.53	
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(0)
704.82/291.53	Obligation:
704.82/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
704.82/291.53	
704.82/291.53	
704.82/291.53	The TRS R consists of the following rules:
704.82/291.53	
704.82/291.53	   f(x) -> s(x)
704.82/291.53	   f(s(s(x))) -> s(f(f(x)))
704.82/291.53	
704.82/291.53	S is empty.
704.82/291.53	Rewrite Strategy: FULL
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
704.82/291.53	Transformed a relative TRS into a decreasing-loop problem.
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(2)
704.82/291.53	Obligation:
704.82/291.53	Analyzing the following TRS for decreasing loops:
704.82/291.53	
704.82/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
704.82/291.53	
704.82/291.53	
704.82/291.53	The TRS R consists of the following rules:
704.82/291.53	
704.82/291.53	   f(x) -> s(x)
704.82/291.53	   f(s(s(x))) -> s(f(f(x)))
704.82/291.53	
704.82/291.53	S is empty.
704.82/291.53	Rewrite Strategy: FULL
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
704.82/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
704.82/291.53	
704.82/291.53	The rewrite sequence
704.82/291.53	
704.82/291.53	f(s(s(x))) ->^+ s(f(f(x)))
704.82/291.53	
704.82/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
704.82/291.53	
704.82/291.53	The pumping substitution is [x / s(s(x))].
704.82/291.53	
704.82/291.53	The result substitution is [ ].
704.82/291.53	
704.82/291.53	
704.82/291.53	
704.82/291.53	
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(4)
704.82/291.53	Complex Obligation (BEST)
704.82/291.53	
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(5)
704.82/291.53	Obligation:
704.82/291.53	Proved the lower bound n^1 for the following obligation:
704.82/291.53	
704.82/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
704.82/291.53	
704.82/291.53	
704.82/291.53	The TRS R consists of the following rules:
704.82/291.53	
704.82/291.53	   f(x) -> s(x)
704.82/291.53	   f(s(s(x))) -> s(f(f(x)))
704.82/291.53	
704.82/291.53	S is empty.
704.82/291.53	Rewrite Strategy: FULL
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(6) LowerBoundPropagationProof (FINISHED)
704.82/291.53	Propagated lower bound.
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(7)
704.82/291.53	BOUNDS(n^1, INF)
704.82/291.53	
704.82/291.53	----------------------------------------
704.82/291.53	
704.82/291.53	(8)
704.82/291.53	Obligation:
704.82/291.53	Analyzing the following TRS for decreasing loops:
704.82/291.53	
704.82/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
704.82/291.53	
704.82/291.53	
704.82/291.53	The TRS R consists of the following rules:
704.82/291.53	
704.82/291.53	   f(x) -> s(x)
704.82/291.53	   f(s(s(x))) -> s(f(f(x)))
704.82/291.53	
704.82/291.53	S is empty.
704.82/291.53	Rewrite Strategy: FULL
704.84/291.57	EOF