301.15/291.52	WORST_CASE(Omega(n^1), ?)
301.15/291.53	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
301.15/291.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
301.15/291.53	
301.15/291.53	
301.15/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.15/291.53	
301.15/291.53	(0) CpxTRS
301.15/291.53	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
301.15/291.53	(2) TRS for Loop Detection
301.15/291.53	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
301.15/291.53	(4) BEST
301.15/291.53	    (5) proven lower bound
301.15/291.53	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
301.15/291.53	        (7) BOUNDS(n^1, INF)
301.15/291.53	    (8) TRS for Loop Detection
301.15/291.53	
301.15/291.53	
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(0)
301.15/291.53	Obligation:
301.15/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.15/291.53	
301.15/291.53	
301.15/291.53	The TRS R consists of the following rules:
301.15/291.53	
301.15/291.53	   minus(x, 0) -> x
301.15/291.53	   minus(s(x), s(y)) -> minus(x, y)
301.15/291.53	   quot(0, s(y)) -> 0
301.15/291.53	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
301.15/291.53	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.15/291.53	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.15/291.53	   plus(zero, y) -> y
301.15/291.53	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.15/291.53	   id(x) -> x
301.15/291.53	   if(true, x, y) -> x
301.15/291.53	   if(false, x, y) -> y
301.15/291.53	   not(x) -> if(x, false, true)
301.15/291.53	   gt(s(x), zero) -> true
301.15/291.53	   gt(zero, y) -> false
301.15/291.53	   gt(s(x), s(y)) -> gt(x, y)
301.15/291.53	
301.15/291.53	S is empty.
301.15/291.53	Rewrite Strategy: FULL
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
301.15/291.53	Transformed a relative TRS into a decreasing-loop problem.
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(2)
301.15/291.53	Obligation:
301.15/291.53	Analyzing the following TRS for decreasing loops:
301.15/291.53	
301.15/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.15/291.53	
301.15/291.53	
301.15/291.53	The TRS R consists of the following rules:
301.15/291.53	
301.15/291.53	   minus(x, 0) -> x
301.15/291.53	   minus(s(x), s(y)) -> minus(x, y)
301.15/291.53	   quot(0, s(y)) -> 0
301.15/291.53	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
301.15/291.53	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.15/291.53	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.15/291.53	   plus(zero, y) -> y
301.15/291.53	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.15/291.53	   id(x) -> x
301.15/291.53	   if(true, x, y) -> x
301.15/291.53	   if(false, x, y) -> y
301.15/291.53	   not(x) -> if(x, false, true)
301.15/291.53	   gt(s(x), zero) -> true
301.15/291.53	   gt(zero, y) -> false
301.15/291.53	   gt(s(x), s(y)) -> gt(x, y)
301.15/291.53	
301.15/291.53	S is empty.
301.15/291.53	Rewrite Strategy: FULL
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(3) DecreasingLoopProof (LOWER BOUND(ID))
301.15/291.53	The following loop(s) give(s) rise to the lower bound Omega(n^1):
301.15/291.53	
301.15/291.53	The rewrite sequence
301.15/291.53	
301.15/291.53	minus(s(x), s(y)) ->^+ minus(x, y)
301.15/291.53	
301.15/291.53	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
301.15/291.53	
301.15/291.53	The pumping substitution is [x / s(x), y / s(y)].
301.15/291.53	
301.15/291.53	The result substitution is [ ].
301.15/291.53	
301.15/291.53	
301.15/291.53	
301.15/291.53	
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(4)
301.15/291.53	Complex Obligation (BEST)
301.15/291.53	
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(5)
301.15/291.53	Obligation:
301.15/291.53	Proved the lower bound n^1 for the following obligation:
301.15/291.53	
301.15/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.15/291.53	
301.15/291.53	
301.15/291.53	The TRS R consists of the following rules:
301.15/291.53	
301.15/291.53	   minus(x, 0) -> x
301.15/291.53	   minus(s(x), s(y)) -> minus(x, y)
301.15/291.53	   quot(0, s(y)) -> 0
301.15/291.53	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
301.15/291.53	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.15/291.53	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.15/291.53	   plus(zero, y) -> y
301.15/291.53	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.15/291.53	   id(x) -> x
301.15/291.53	   if(true, x, y) -> x
301.15/291.53	   if(false, x, y) -> y
301.15/291.53	   not(x) -> if(x, false, true)
301.15/291.53	   gt(s(x), zero) -> true
301.15/291.53	   gt(zero, y) -> false
301.15/291.53	   gt(s(x), s(y)) -> gt(x, y)
301.15/291.53	
301.15/291.53	S is empty.
301.15/291.53	Rewrite Strategy: FULL
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(6) LowerBoundPropagationProof (FINISHED)
301.15/291.53	Propagated lower bound.
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(7)
301.15/291.53	BOUNDS(n^1, INF)
301.15/291.53	
301.15/291.53	----------------------------------------
301.15/291.53	
301.15/291.53	(8)
301.15/291.53	Obligation:
301.15/291.53	Analyzing the following TRS for decreasing loops:
301.15/291.53	
301.15/291.53	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
301.15/291.53	
301.15/291.53	
301.15/291.53	The TRS R consists of the following rules:
301.15/291.53	
301.15/291.53	   minus(x, 0) -> x
301.15/291.53	   minus(s(x), s(y)) -> minus(x, y)
301.15/291.53	   quot(0, s(y)) -> 0
301.15/291.53	   quot(s(x), s(y)) -> s(quot(minus(x, y), s(y)))
301.15/291.53	   plus(s(x), s(y)) -> s(s(plus(if(gt(x, y), x, y), if(not(gt(x, y)), id(x), id(y)))))
301.15/291.53	   plus(s(x), x) -> plus(if(gt(x, x), id(x), id(x)), s(x))
301.15/291.53	   plus(zero, y) -> y
301.15/291.53	   plus(id(x), s(y)) -> s(plus(x, if(gt(s(y), y), y, s(y))))
301.15/291.53	   id(x) -> x
301.15/291.53	   if(true, x, y) -> x
301.15/291.53	   if(false, x, y) -> y
301.15/291.53	   not(x) -> if(x, false, true)
301.15/291.53	   gt(s(x), zero) -> true
301.15/291.53	   gt(zero, y) -> false
301.15/291.53	   gt(s(x), s(y)) -> gt(x, y)
301.15/291.53	
301.15/291.53	S is empty.
301.15/291.53	Rewrite Strategy: FULL
301.15/291.55	EOF