364.69/291.48	WORST_CASE(Omega(n^1), ?)
364.69/291.49	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
364.69/291.49	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
364.69/291.49	
364.69/291.49	
364.69/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.69/291.49	
364.69/291.49	(0) CpxTRS
364.69/291.49	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
364.69/291.49	(2) TRS for Loop Detection
364.69/291.49	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
364.69/291.49	(4) BEST
364.69/291.49	    (5) proven lower bound
364.69/291.49	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
364.69/291.49	        (7) BOUNDS(n^1, INF)
364.69/291.49	    (8) TRS for Loop Detection
364.69/291.49	
364.69/291.49	
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(0)
364.69/291.49	Obligation:
364.69/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.69/291.49	
364.69/291.49	
364.69/291.49	The TRS R consists of the following rules:
364.69/291.49	
364.69/291.49	   not(not(x)) -> x
364.69/291.49	   not(or(x, y)) -> and(not(x), not(y))
364.69/291.49	   not(and(x, y)) -> or(not(x), not(y))
364.69/291.49	   and(x, or(y, z)) -> or(and(x, y), and(x, z))
364.69/291.49	   and(or(y, z), x) -> or(and(x, y), and(x, z))
364.69/291.49	
364.69/291.49	S is empty.
364.69/291.49	Rewrite Strategy: FULL
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
364.69/291.49	Transformed a relative TRS into a decreasing-loop problem.
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(2)
364.69/291.49	Obligation:
364.69/291.49	Analyzing the following TRS for decreasing loops:
364.69/291.49	
364.69/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.69/291.49	
364.69/291.49	
364.69/291.49	The TRS R consists of the following rules:
364.69/291.49	
364.69/291.49	   not(not(x)) -> x
364.69/291.49	   not(or(x, y)) -> and(not(x), not(y))
364.69/291.49	   not(and(x, y)) -> or(not(x), not(y))
364.69/291.49	   and(x, or(y, z)) -> or(and(x, y), and(x, z))
364.69/291.49	   and(or(y, z), x) -> or(and(x, y), and(x, z))
364.69/291.49	
364.69/291.49	S is empty.
364.69/291.49	Rewrite Strategy: FULL
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(3) DecreasingLoopProof (LOWER BOUND(ID))
364.69/291.49	The following loop(s) give(s) rise to the lower bound Omega(n^1):
364.69/291.49	
364.69/291.49	The rewrite sequence
364.69/291.49	
364.69/291.49	and(x, or(y, z)) ->^+ or(and(x, y), and(x, z))
364.69/291.49	
364.69/291.49	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
364.69/291.49	
364.69/291.49	The pumping substitution is [y / or(y, z)].
364.69/291.49	
364.69/291.49	The result substitution is [ ].
364.69/291.49	
364.69/291.49	
364.69/291.49	
364.69/291.49	
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(4)
364.69/291.49	Complex Obligation (BEST)
364.69/291.49	
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(5)
364.69/291.49	Obligation:
364.69/291.49	Proved the lower bound n^1 for the following obligation:
364.69/291.49	
364.69/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.69/291.49	
364.69/291.49	
364.69/291.49	The TRS R consists of the following rules:
364.69/291.49	
364.69/291.49	   not(not(x)) -> x
364.69/291.49	   not(or(x, y)) -> and(not(x), not(y))
364.69/291.49	   not(and(x, y)) -> or(not(x), not(y))
364.69/291.49	   and(x, or(y, z)) -> or(and(x, y), and(x, z))
364.69/291.49	   and(or(y, z), x) -> or(and(x, y), and(x, z))
364.69/291.49	
364.69/291.49	S is empty.
364.69/291.49	Rewrite Strategy: FULL
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(6) LowerBoundPropagationProof (FINISHED)
364.69/291.49	Propagated lower bound.
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(7)
364.69/291.49	BOUNDS(n^1, INF)
364.69/291.49	
364.69/291.49	----------------------------------------
364.69/291.49	
364.69/291.49	(8)
364.69/291.49	Obligation:
364.69/291.49	Analyzing the following TRS for decreasing loops:
364.69/291.49	
364.69/291.49	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
364.69/291.49	
364.69/291.49	
364.69/291.49	The TRS R consists of the following rules:
364.69/291.49	
364.69/291.49	   not(not(x)) -> x
364.69/291.49	   not(or(x, y)) -> and(not(x), not(y))
364.69/291.49	   not(and(x, y)) -> or(not(x), not(y))
364.69/291.49	   and(x, or(y, z)) -> or(and(x, y), and(x, z))
364.69/291.49	   and(or(y, z), x) -> or(and(x, y), and(x, z))
364.69/291.49	
364.69/291.49	S is empty.
364.69/291.49	Rewrite Strategy: FULL
364.69/291.54	EOF