303.35/291.50	WORST_CASE(Omega(n^1), ?)
303.35/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
303.35/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.35/291.51	
303.35/291.51	
303.35/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.35/291.51	
303.35/291.51	(0) CpxTRS
303.35/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.35/291.51	(2) TRS for Loop Detection
303.35/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
303.35/291.51	(4) BEST
303.35/291.51	    (5) proven lower bound
303.35/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.35/291.51	        (7) BOUNDS(n^1, INF)
303.35/291.51	    (8) TRS for Loop Detection
303.35/291.51	
303.35/291.51	
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(0)
303.35/291.51	Obligation:
303.35/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.35/291.51	
303.35/291.51	
303.35/291.51	The TRS R consists of the following rules:
303.35/291.51	
303.35/291.51	   cond(true, x) -> cond(odd(x), p(p(p(x))))
303.35/291.51	   odd(0) -> false
303.35/291.51	   odd(s(0)) -> true
303.35/291.51	   odd(s(s(x))) -> odd(x)
303.35/291.51	   p(0) -> 0
303.35/291.51	   p(s(x)) -> x
303.35/291.51	
303.35/291.51	S is empty.
303.35/291.51	Rewrite Strategy: FULL
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.35/291.51	Transformed a relative TRS into a decreasing-loop problem.
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(2)
303.35/291.51	Obligation:
303.35/291.51	Analyzing the following TRS for decreasing loops:
303.35/291.51	
303.35/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.35/291.51	
303.35/291.51	
303.35/291.51	The TRS R consists of the following rules:
303.35/291.51	
303.35/291.51	   cond(true, x) -> cond(odd(x), p(p(p(x))))
303.35/291.51	   odd(0) -> false
303.35/291.51	   odd(s(0)) -> true
303.35/291.51	   odd(s(s(x))) -> odd(x)
303.35/291.51	   p(0) -> 0
303.35/291.51	   p(s(x)) -> x
303.35/291.51	
303.35/291.51	S is empty.
303.35/291.51	Rewrite Strategy: FULL
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.35/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.35/291.51	
303.35/291.51	The rewrite sequence
303.35/291.51	
303.35/291.51	odd(s(s(x))) ->^+ odd(x)
303.35/291.51	
303.35/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
303.35/291.51	
303.35/291.51	The pumping substitution is [x / s(s(x))].
303.35/291.51	
303.35/291.51	The result substitution is [ ].
303.35/291.51	
303.35/291.51	
303.35/291.51	
303.35/291.51	
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(4)
303.35/291.51	Complex Obligation (BEST)
303.35/291.51	
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(5)
303.35/291.51	Obligation:
303.35/291.51	Proved the lower bound n^1 for the following obligation:
303.35/291.51	
303.35/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.35/291.51	
303.35/291.51	
303.35/291.51	The TRS R consists of the following rules:
303.35/291.51	
303.35/291.51	   cond(true, x) -> cond(odd(x), p(p(p(x))))
303.35/291.51	   odd(0) -> false
303.35/291.51	   odd(s(0)) -> true
303.35/291.51	   odd(s(s(x))) -> odd(x)
303.35/291.51	   p(0) -> 0
303.35/291.51	   p(s(x)) -> x
303.35/291.51	
303.35/291.51	S is empty.
303.35/291.51	Rewrite Strategy: FULL
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(6) LowerBoundPropagationProof (FINISHED)
303.35/291.51	Propagated lower bound.
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(7)
303.35/291.51	BOUNDS(n^1, INF)
303.35/291.51	
303.35/291.51	----------------------------------------
303.35/291.51	
303.35/291.51	(8)
303.35/291.51	Obligation:
303.35/291.51	Analyzing the following TRS for decreasing loops:
303.35/291.51	
303.35/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.35/291.51	
303.35/291.51	
303.35/291.51	The TRS R consists of the following rules:
303.35/291.51	
303.35/291.51	   cond(true, x) -> cond(odd(x), p(p(p(x))))
303.35/291.51	   odd(0) -> false
303.35/291.51	   odd(s(0)) -> true
303.35/291.51	   odd(s(s(x))) -> odd(x)
303.35/291.51	   p(0) -> 0
303.35/291.51	   p(s(x)) -> x
303.35/291.51	
303.35/291.51	S is empty.
303.35/291.51	Rewrite Strategy: FULL
303.41/291.53	EOF