306.18/291.52	WORST_CASE(Omega(n^1), ?)
306.18/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
306.18/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
306.18/291.53	
306.18/291.53	
306.18/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.18/291.53	
306.18/291.53	(0) CpxTRS
306.18/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
306.18/291.53	(2) TRS for Loop Detection
306.18/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
306.18/291.53	(4) BEST
306.18/291.53	    (5) proven lower bound
306.18/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
306.18/291.53	        (7) BOUNDS(n^1, INF)
306.18/291.53	    (8) TRS for Loop Detection
306.18/291.53	
306.18/291.53	
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(0)
306.18/291.53	Obligation:
306.18/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.18/291.53	
306.18/291.53	
306.18/291.53	The TRS R consists of the following rules:
306.18/291.53	
306.18/291.53	   cond(true, x) -> cond(odd(x), p(x))
306.18/291.53	   odd(0) -> false
306.18/291.53	   odd(s(0)) -> true
306.18/291.53	   odd(s(s(x))) -> odd(x)
306.18/291.53	   p(0) -> 0
306.18/291.53	   p(s(x)) -> x
306.18/291.53	
306.18/291.53	S is empty.
306.18/291.53	Rewrite Strategy: FULL
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
306.18/291.53	Transformed a relative TRS into a decreasing-loop problem.
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(2)
306.18/291.53	Obligation:
306.18/291.53	Analyzing the following TRS for decreasing loops:
306.18/291.53	
306.18/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.18/291.53	
306.18/291.53	
306.18/291.53	The TRS R consists of the following rules:
306.18/291.53	
306.18/291.53	   cond(true, x) -> cond(odd(x), p(x))
306.18/291.53	   odd(0) -> false
306.18/291.53	   odd(s(0)) -> true
306.18/291.53	   odd(s(s(x))) -> odd(x)
306.18/291.53	   p(0) -> 0
306.18/291.53	   p(s(x)) -> x
306.18/291.53	
306.18/291.53	S is empty.
306.18/291.53	Rewrite Strategy: FULL
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
306.18/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
306.18/291.53	
306.18/291.53	The rewrite sequence
306.18/291.53	
306.18/291.53	odd(s(s(x))) ->^+ odd(x)
306.18/291.53	
306.18/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
306.18/291.53	
306.18/291.53	The pumping substitution is [x / s(s(x))].
306.18/291.53	
306.18/291.53	The result substitution is [ ].
306.18/291.53	
306.18/291.53	
306.18/291.53	
306.18/291.53	
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(4)
306.18/291.53	Complex Obligation (BEST)
306.18/291.53	
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(5)
306.18/291.53	Obligation:
306.18/291.53	Proved the lower bound n^1 for the following obligation:
306.18/291.53	
306.18/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.18/291.53	
306.18/291.53	
306.18/291.53	The TRS R consists of the following rules:
306.18/291.53	
306.18/291.53	   cond(true, x) -> cond(odd(x), p(x))
306.18/291.53	   odd(0) -> false
306.18/291.53	   odd(s(0)) -> true
306.18/291.53	   odd(s(s(x))) -> odd(x)
306.18/291.53	   p(0) -> 0
306.18/291.53	   p(s(x)) -> x
306.18/291.53	
306.18/291.53	S is empty.
306.18/291.53	Rewrite Strategy: FULL
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(6) LowerBoundPropagationProof (FINISHED)
306.18/291.53	Propagated lower bound.
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(7)
306.18/291.53	BOUNDS(n^1, INF)
306.18/291.53	
306.18/291.53	----------------------------------------
306.18/291.53	
306.18/291.53	(8)
306.18/291.53	Obligation:
306.18/291.53	Analyzing the following TRS for decreasing loops:
306.18/291.53	
306.18/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
306.18/291.53	
306.18/291.53	
306.18/291.53	The TRS R consists of the following rules:
306.18/291.53	
306.18/291.53	   cond(true, x) -> cond(odd(x), p(x))
306.18/291.53	   odd(0) -> false
306.18/291.53	   odd(s(0)) -> true
306.18/291.53	   odd(s(s(x))) -> odd(x)
306.18/291.53	   p(0) -> 0
306.18/291.53	   p(s(x)) -> x
306.18/291.53	
306.18/291.53	S is empty.
306.18/291.53	Rewrite Strategy: FULL
306.18/291.55	EOF