303.07/291.50	WORST_CASE(Omega(n^1), ?)
303.07/291.51	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
303.07/291.51	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
303.07/291.51	
303.07/291.51	
303.07/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.07/291.51	
303.07/291.51	(0) CpxTRS
303.07/291.51	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
303.07/291.51	(2) TRS for Loop Detection
303.07/291.51	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
303.07/291.51	(4) BEST
303.07/291.51	    (5) proven lower bound
303.07/291.51	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
303.07/291.51	        (7) BOUNDS(n^1, INF)
303.07/291.51	    (8) TRS for Loop Detection
303.07/291.51	
303.07/291.51	
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(0)
303.07/291.51	Obligation:
303.07/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.07/291.51	
303.07/291.51	
303.07/291.51	The TRS R consists of the following rules:
303.07/291.51	
303.07/291.51	   minus(x, x) -> 0
303.07/291.51	   minus(0, x) -> 0
303.07/291.51	   minus(x, 0) -> x
303.07/291.51	   minus(s(x), s(y)) -> minus(x, y)
303.07/291.51	   le(0, y) -> true
303.07/291.51	   le(s(x), 0) -> false
303.07/291.51	   le(s(x), s(y)) -> le(x, y)
303.07/291.51	   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0), le(y, x))
303.07/291.51	   if_quot(x, y, true, z) -> divByZeroError
303.07/291.51	   if_quot(x, y, false, true) -> s(quot(x, y))
303.07/291.51	   if_quot(x, y, false, false) -> 0
303.07/291.51	
303.07/291.51	S is empty.
303.07/291.51	Rewrite Strategy: FULL
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
303.07/291.51	Transformed a relative TRS into a decreasing-loop problem.
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(2)
303.07/291.51	Obligation:
303.07/291.51	Analyzing the following TRS for decreasing loops:
303.07/291.51	
303.07/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.07/291.51	
303.07/291.51	
303.07/291.51	The TRS R consists of the following rules:
303.07/291.51	
303.07/291.51	   minus(x, x) -> 0
303.07/291.51	   minus(0, x) -> 0
303.07/291.51	   minus(x, 0) -> x
303.07/291.51	   minus(s(x), s(y)) -> minus(x, y)
303.07/291.51	   le(0, y) -> true
303.07/291.51	   le(s(x), 0) -> false
303.07/291.51	   le(s(x), s(y)) -> le(x, y)
303.07/291.51	   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0), le(y, x))
303.07/291.51	   if_quot(x, y, true, z) -> divByZeroError
303.07/291.51	   if_quot(x, y, false, true) -> s(quot(x, y))
303.07/291.51	   if_quot(x, y, false, false) -> 0
303.07/291.51	
303.07/291.51	S is empty.
303.07/291.51	Rewrite Strategy: FULL
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(3) DecreasingLoopProof (LOWER BOUND(ID))
303.07/291.51	The following loop(s) give(s) rise to the lower bound Omega(n^1):
303.07/291.51	
303.07/291.51	The rewrite sequence
303.07/291.51	
303.07/291.51	minus(s(x), s(y)) ->^+ minus(x, y)
303.07/291.51	
303.07/291.51	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
303.07/291.51	
303.07/291.51	The pumping substitution is [x / s(x), y / s(y)].
303.07/291.51	
303.07/291.51	The result substitution is [ ].
303.07/291.51	
303.07/291.51	
303.07/291.51	
303.07/291.51	
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(4)
303.07/291.51	Complex Obligation (BEST)
303.07/291.51	
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(5)
303.07/291.51	Obligation:
303.07/291.51	Proved the lower bound n^1 for the following obligation:
303.07/291.51	
303.07/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.07/291.51	
303.07/291.51	
303.07/291.51	The TRS R consists of the following rules:
303.07/291.51	
303.07/291.51	   minus(x, x) -> 0
303.07/291.51	   minus(0, x) -> 0
303.07/291.51	   minus(x, 0) -> x
303.07/291.51	   minus(s(x), s(y)) -> minus(x, y)
303.07/291.51	   le(0, y) -> true
303.07/291.51	   le(s(x), 0) -> false
303.07/291.51	   le(s(x), s(y)) -> le(x, y)
303.07/291.51	   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0), le(y, x))
303.07/291.51	   if_quot(x, y, true, z) -> divByZeroError
303.07/291.51	   if_quot(x, y, false, true) -> s(quot(x, y))
303.07/291.51	   if_quot(x, y, false, false) -> 0
303.07/291.51	
303.07/291.51	S is empty.
303.07/291.51	Rewrite Strategy: FULL
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(6) LowerBoundPropagationProof (FINISHED)
303.07/291.51	Propagated lower bound.
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(7)
303.07/291.51	BOUNDS(n^1, INF)
303.07/291.51	
303.07/291.51	----------------------------------------
303.07/291.51	
303.07/291.51	(8)
303.07/291.51	Obligation:
303.07/291.51	Analyzing the following TRS for decreasing loops:
303.07/291.51	
303.07/291.51	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
303.07/291.51	
303.07/291.51	
303.07/291.51	The TRS R consists of the following rules:
303.07/291.51	
303.07/291.51	   minus(x, x) -> 0
303.07/291.51	   minus(0, x) -> 0
303.07/291.51	   minus(x, 0) -> x
303.07/291.51	   minus(s(x), s(y)) -> minus(x, y)
303.07/291.51	   le(0, y) -> true
303.07/291.51	   le(s(x), 0) -> false
303.07/291.51	   le(s(x), s(y)) -> le(x, y)
303.07/291.51	   quot(x, y) -> if_quot(minus(x, y), y, le(y, 0), le(y, x))
303.07/291.51	   if_quot(x, y, true, z) -> divByZeroError
303.07/291.51	   if_quot(x, y, false, true) -> s(quot(x, y))
303.07/291.51	   if_quot(x, y, false, false) -> 0
303.07/291.51	
303.07/291.51	S is empty.
303.07/291.51	Rewrite Strategy: FULL
303.14/291.55	EOF