301.90/291.49	WORST_CASE(Omega(n^1), ?)
301.90/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
301.90/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.90/291.50	
301.90/291.50	
301.90/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.90/291.50	
301.90/291.50	(0) CpxTRS
301.90/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.90/291.50	(2) TRS for Loop Detection
301.90/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
301.90/291.50	(4) BEST
301.90/291.50	    (5) proven lower bound
301.90/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.90/291.50	        (7) BOUNDS(n^1, INF)
301.90/291.50	    (8) TRS for Loop Detection
301.90/291.50	
301.90/291.50	
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(0)
301.90/291.50	Obligation:
301.90/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.90/291.50	
301.90/291.50	
301.90/291.50	The TRS R consists of the following rules:
301.90/291.50	
301.90/291.50	   minus(x, 0) -> x
301.90/291.50	   minus(s(x), s(y)) -> minus(x, y)
301.90/291.50	   double(0) -> 0
301.90/291.50	   double(s(x)) -> s(s(double(x)))
301.90/291.50	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.90/291.50	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.90/291.50	   plus(zero, y) -> y
301.90/291.50	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.90/291.50	   id(x) -> x
301.90/291.50	   if(true, x, y) -> x
301.90/291.50	   if(false, x, y) -> y
301.90/291.50	   not(x) -> if(x, false, true)
301.90/291.50	   gt(s(x), zero) -> true
301.90/291.50	   gt(zero, y) -> false
301.90/291.50	   gt(s(x), s(y)) -> gt(x, y)
301.90/291.50	
301.90/291.50	S is empty.
301.90/291.50	Rewrite Strategy: FULL
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.90/291.50	Transformed a relative TRS into a decreasing-loop problem.
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(2)
301.90/291.50	Obligation:
301.90/291.50	Analyzing the following TRS for decreasing loops:
301.90/291.50	
301.90/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.90/291.50	
301.90/291.50	
301.90/291.50	The TRS R consists of the following rules:
301.90/291.50	
301.90/291.50	   minus(x, 0) -> x
301.90/291.50	   minus(s(x), s(y)) -> minus(x, y)
301.90/291.50	   double(0) -> 0
301.90/291.50	   double(s(x)) -> s(s(double(x)))
301.90/291.50	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.90/291.50	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.90/291.50	   plus(zero, y) -> y
301.90/291.50	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.90/291.50	   id(x) -> x
301.90/291.50	   if(true, x, y) -> x
301.90/291.50	   if(false, x, y) -> y
301.90/291.50	   not(x) -> if(x, false, true)
301.90/291.50	   gt(s(x), zero) -> true
301.90/291.50	   gt(zero, y) -> false
301.90/291.50	   gt(s(x), s(y)) -> gt(x, y)
301.90/291.50	
301.90/291.50	S is empty.
301.90/291.50	Rewrite Strategy: FULL
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.90/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.90/291.50	
301.90/291.50	The rewrite sequence
301.90/291.50	
301.90/291.50	double(s(x)) ->^+ s(s(double(x)))
301.90/291.50	
301.90/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
301.90/291.50	
301.90/291.50	The pumping substitution is [x / s(x)].
301.90/291.50	
301.90/291.50	The result substitution is [ ].
301.90/291.50	
301.90/291.50	
301.90/291.50	
301.90/291.50	
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(4)
301.90/291.50	Complex Obligation (BEST)
301.90/291.50	
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(5)
301.90/291.50	Obligation:
301.90/291.50	Proved the lower bound n^1 for the following obligation:
301.90/291.50	
301.90/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.90/291.50	
301.90/291.50	
301.90/291.50	The TRS R consists of the following rules:
301.90/291.50	
301.90/291.50	   minus(x, 0) -> x
301.90/291.50	   minus(s(x), s(y)) -> minus(x, y)
301.90/291.50	   double(0) -> 0
301.90/291.50	   double(s(x)) -> s(s(double(x)))
301.90/291.50	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.90/291.50	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.90/291.50	   plus(zero, y) -> y
301.90/291.50	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.90/291.50	   id(x) -> x
301.90/291.50	   if(true, x, y) -> x
301.90/291.50	   if(false, x, y) -> y
301.90/291.50	   not(x) -> if(x, false, true)
301.90/291.50	   gt(s(x), zero) -> true
301.90/291.50	   gt(zero, y) -> false
301.90/291.50	   gt(s(x), s(y)) -> gt(x, y)
301.90/291.50	
301.90/291.50	S is empty.
301.90/291.50	Rewrite Strategy: FULL
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(6) LowerBoundPropagationProof (FINISHED)
301.90/291.50	Propagated lower bound.
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(7)
301.90/291.50	BOUNDS(n^1, INF)
301.90/291.50	
301.90/291.50	----------------------------------------
301.90/291.50	
301.90/291.50	(8)
301.90/291.50	Obligation:
301.90/291.50	Analyzing the following TRS for decreasing loops:
301.90/291.50	
301.90/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.90/291.50	
301.90/291.50	
301.90/291.50	The TRS R consists of the following rules:
301.90/291.50	
301.90/291.50	   minus(x, 0) -> x
301.90/291.50	   minus(s(x), s(y)) -> minus(x, y)
301.90/291.50	   double(0) -> 0
301.90/291.50	   double(s(x)) -> s(s(double(x)))
301.90/291.50	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.90/291.50	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.90/291.50	   plus(zero, y) -> y
301.90/291.50	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.90/291.50	   id(x) -> x
301.90/291.50	   if(true, x, y) -> x
301.90/291.50	   if(false, x, y) -> y
301.90/291.50	   not(x) -> if(x, false, true)
301.90/291.50	   gt(s(x), zero) -> true
301.90/291.50	   gt(zero, y) -> false
301.90/291.50	   gt(s(x), s(y)) -> gt(x, y)
301.90/291.50	
301.90/291.50	S is empty.
301.90/291.50	Rewrite Strategy: FULL
301.90/291.53	EOF