308.47/291.50	WORST_CASE(Omega(n^1), ?)
308.52/291.51	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
308.52/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
308.52/291.51	
308.52/291.51	
308.52/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.52/291.51	
308.52/291.51	(0) CpxTRS
308.52/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
308.52/291.51	(2) TRS for Loop Detection
308.52/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
308.52/291.51	(4) BEST
308.52/291.51	    (5) proven lower bound
308.52/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
308.52/291.51	        (7) BOUNDS(n^1, INF)
308.52/291.51	    (8) TRS for Loop Detection
308.52/291.51	
308.52/291.51	
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(0)
308.52/291.51	Obligation:
308.52/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.52/291.51	
308.52/291.51	
308.52/291.51	The TRS R consists of the following rules:
308.52/291.51	
308.52/291.51	   le(0, y) -> true
308.52/291.51	   le(s(x), 0) -> false
308.52/291.51	   le(s(x), s(y)) -> le(x, y)
308.52/291.51	   pred(s(x)) -> x
308.52/291.51	   minus(x, 0) -> x
308.52/291.51	   minus(x, s(y)) -> pred(minus(x, y))
308.52/291.51	   mod(0, y) -> 0
308.52/291.51	   mod(s(x), 0) -> 0
308.52/291.51	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
308.52/291.51	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
308.52/291.51	   if_mod(false, s(x), s(y)) -> s(x)
308.52/291.51	
308.52/291.51	S is empty.
308.52/291.51	Rewrite Strategy: FULL
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
308.52/291.51	Transformed a relative TRS into a decreasing-loop problem.
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(2)
308.52/291.51	Obligation:
308.52/291.51	Analyzing the following TRS for decreasing loops:
308.52/291.51	
308.52/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.52/291.51	
308.52/291.51	
308.52/291.51	The TRS R consists of the following rules:
308.52/291.51	
308.52/291.51	   le(0, y) -> true
308.52/291.51	   le(s(x), 0) -> false
308.52/291.51	   le(s(x), s(y)) -> le(x, y)
308.52/291.51	   pred(s(x)) -> x
308.52/291.51	   minus(x, 0) -> x
308.52/291.51	   minus(x, s(y)) -> pred(minus(x, y))
308.52/291.51	   mod(0, y) -> 0
308.52/291.51	   mod(s(x), 0) -> 0
308.52/291.51	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
308.52/291.51	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
308.52/291.51	   if_mod(false, s(x), s(y)) -> s(x)
308.52/291.51	
308.52/291.51	S is empty.
308.52/291.51	Rewrite Strategy: FULL
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
308.52/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
308.52/291.51	
308.52/291.51	The rewrite sequence
308.52/291.51	
308.52/291.51	minus(x, s(y)) ->^+ pred(minus(x, y))
308.52/291.51	
308.52/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
308.52/291.51	
308.52/291.51	The pumping substitution is [y / s(y)].
308.52/291.51	
308.52/291.51	The result substitution is [ ].
308.52/291.51	
308.52/291.51	
308.52/291.51	
308.52/291.51	
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(4)
308.52/291.51	Complex Obligation (BEST)
308.52/291.51	
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(5)
308.52/291.51	Obligation:
308.52/291.51	Proved the lower bound n^1 for the following obligation:
308.52/291.51	
308.52/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.52/291.51	
308.52/291.51	
308.52/291.51	The TRS R consists of the following rules:
308.52/291.51	
308.52/291.51	   le(0, y) -> true
308.52/291.51	   le(s(x), 0) -> false
308.52/291.51	   le(s(x), s(y)) -> le(x, y)
308.52/291.51	   pred(s(x)) -> x
308.52/291.51	   minus(x, 0) -> x
308.52/291.51	   minus(x, s(y)) -> pred(minus(x, y))
308.52/291.51	   mod(0, y) -> 0
308.52/291.51	   mod(s(x), 0) -> 0
308.52/291.51	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
308.52/291.51	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
308.52/291.51	   if_mod(false, s(x), s(y)) -> s(x)
308.52/291.51	
308.52/291.51	S is empty.
308.52/291.51	Rewrite Strategy: FULL
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(6) LowerBoundPropagationProof (FINISHED)
308.52/291.51	Propagated lower bound.
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(7)
308.52/291.51	BOUNDS(n^1, INF)
308.52/291.51	
308.52/291.51	----------------------------------------
308.52/291.51	
308.52/291.51	(8)
308.52/291.51	Obligation:
308.52/291.51	Analyzing the following TRS for decreasing loops:
308.52/291.51	
308.52/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
308.52/291.51	
308.52/291.51	
308.52/291.51	The TRS R consists of the following rules:
308.52/291.51	
308.52/291.51	   le(0, y) -> true
308.52/291.51	   le(s(x), 0) -> false
308.52/291.51	   le(s(x), s(y)) -> le(x, y)
308.52/291.51	   pred(s(x)) -> x
308.52/291.51	   minus(x, 0) -> x
308.52/291.51	   minus(x, s(y)) -> pred(minus(x, y))
308.52/291.51	   mod(0, y) -> 0
308.52/291.51	   mod(s(x), 0) -> 0
308.52/291.51	   mod(s(x), s(y)) -> if_mod(le(y, x), s(x), s(y))
308.52/291.51	   if_mod(true, s(x), s(y)) -> mod(minus(x, y), s(y))
308.52/291.51	   if_mod(false, s(x), s(y)) -> s(x)
308.52/291.51	
308.52/291.51	S is empty.
308.52/291.51	Rewrite Strategy: FULL
308.52/291.53	EOF