304.15/291.51	WORST_CASE(Omega(n^1), ?)
304.15/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
304.15/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.15/291.51	
304.15/291.51	
304.15/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.15/291.51	
304.15/291.51	(0) CpxTRS
304.15/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
304.15/291.51	(2) TRS for Loop Detection
304.15/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
304.15/291.51	(4) BEST
304.15/291.51	    (5) proven lower bound
304.15/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
304.15/291.51	        (7) BOUNDS(n^1, INF)
304.15/291.51	    (8) TRS for Loop Detection
304.15/291.51	
304.15/291.51	
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(0)
304.15/291.51	Obligation:
304.15/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.15/291.51	
304.15/291.51	
304.15/291.51	The TRS R consists of the following rules:
304.15/291.51	
304.15/291.51	   le(0, y) -> true
304.15/291.51	   le(s(x), 0) -> false
304.15/291.51	   le(s(x), s(y)) -> le(x, y)
304.15/291.51	   minus(x, x) -> 0
304.15/291.51	   minus(x, 0) -> x
304.15/291.51	   minus(0, x) -> 0
304.15/291.51	   minus(s(x), s(y)) -> minus(x, y)
304.15/291.51	   isZero(0) -> true
304.15/291.51	   isZero(s(x)) -> false
304.15/291.51	   mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y))
304.15/291.51	   if_mod(true, b, x, y, z) -> divByZeroError
304.15/291.51	   if_mod(false, false, x, y, z) -> x
304.15/291.51	   if_mod(false, true, x, y, z) -> mod(z, y)
304.15/291.51	
304.15/291.51	S is empty.
304.15/291.51	Rewrite Strategy: FULL
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
304.15/291.51	Transformed a relative TRS into a decreasing-loop problem.
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(2)
304.15/291.51	Obligation:
304.15/291.51	Analyzing the following TRS for decreasing loops:
304.15/291.51	
304.15/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.15/291.51	
304.15/291.51	
304.15/291.51	The TRS R consists of the following rules:
304.15/291.51	
304.15/291.51	   le(0, y) -> true
304.15/291.51	   le(s(x), 0) -> false
304.15/291.51	   le(s(x), s(y)) -> le(x, y)
304.15/291.51	   minus(x, x) -> 0
304.15/291.51	   minus(x, 0) -> x
304.15/291.51	   minus(0, x) -> 0
304.15/291.51	   minus(s(x), s(y)) -> minus(x, y)
304.15/291.51	   isZero(0) -> true
304.15/291.51	   isZero(s(x)) -> false
304.15/291.51	   mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y))
304.15/291.51	   if_mod(true, b, x, y, z) -> divByZeroError
304.15/291.51	   if_mod(false, false, x, y, z) -> x
304.15/291.51	   if_mod(false, true, x, y, z) -> mod(z, y)
304.15/291.51	
304.15/291.51	S is empty.
304.15/291.51	Rewrite Strategy: FULL
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
304.15/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
304.15/291.51	
304.15/291.51	The rewrite sequence
304.15/291.51	
304.15/291.51	le(s(x), s(y)) ->^+ le(x, y)
304.15/291.51	
304.15/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
304.15/291.51	
304.15/291.51	The pumping substitution is [x / s(x), y / s(y)].
304.15/291.51	
304.15/291.51	The result substitution is [ ].
304.15/291.51	
304.15/291.51	
304.15/291.51	
304.15/291.51	
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(4)
304.15/291.51	Complex Obligation (BEST)
304.15/291.51	
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(5)
304.15/291.51	Obligation:
304.15/291.51	Proved the lower bound n^1 for the following obligation:
304.15/291.51	
304.15/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.15/291.51	
304.15/291.51	
304.15/291.51	The TRS R consists of the following rules:
304.15/291.51	
304.15/291.51	   le(0, y) -> true
304.15/291.51	   le(s(x), 0) -> false
304.15/291.51	   le(s(x), s(y)) -> le(x, y)
304.15/291.51	   minus(x, x) -> 0
304.15/291.51	   minus(x, 0) -> x
304.15/291.51	   minus(0, x) -> 0
304.15/291.51	   minus(s(x), s(y)) -> minus(x, y)
304.15/291.51	   isZero(0) -> true
304.15/291.51	   isZero(s(x)) -> false
304.15/291.51	   mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y))
304.15/291.51	   if_mod(true, b, x, y, z) -> divByZeroError
304.15/291.51	   if_mod(false, false, x, y, z) -> x
304.15/291.51	   if_mod(false, true, x, y, z) -> mod(z, y)
304.15/291.51	
304.15/291.51	S is empty.
304.15/291.51	Rewrite Strategy: FULL
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(6) LowerBoundPropagationProof (FINISHED)
304.15/291.51	Propagated lower bound.
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(7)
304.15/291.51	BOUNDS(n^1, INF)
304.15/291.51	
304.15/291.51	----------------------------------------
304.15/291.51	
304.15/291.51	(8)
304.15/291.51	Obligation:
304.15/291.51	Analyzing the following TRS for decreasing loops:
304.15/291.51	
304.15/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.15/291.51	
304.15/291.51	
304.15/291.51	The TRS R consists of the following rules:
304.15/291.51	
304.15/291.51	   le(0, y) -> true
304.15/291.51	   le(s(x), 0) -> false
304.15/291.51	   le(s(x), s(y)) -> le(x, y)
304.15/291.51	   minus(x, x) -> 0
304.15/291.51	   minus(x, 0) -> x
304.15/291.51	   minus(0, x) -> 0
304.15/291.51	   minus(s(x), s(y)) -> minus(x, y)
304.15/291.51	   isZero(0) -> true
304.15/291.51	   isZero(s(x)) -> false
304.15/291.51	   mod(x, y) -> if_mod(isZero(y), le(y, x), x, y, minus(x, y))
304.15/291.51	   if_mod(true, b, x, y, z) -> divByZeroError
304.15/291.51	   if_mod(false, false, x, y, z) -> x
304.15/291.51	   if_mod(false, true, x, y, z) -> mod(z, y)
304.15/291.51	
304.15/291.51	S is empty.
304.15/291.51	Rewrite Strategy: FULL
304.15/291.54	EOF