304.92/291.50	WORST_CASE(Omega(n^1), ?)
304.92/291.50	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
304.92/291.50	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
304.92/291.50	
304.92/291.50	
304.92/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.50	
304.92/291.50	(0) CpxTRS
304.92/291.50	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
304.92/291.50	(2) TRS for Loop Detection
304.92/291.50	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
304.92/291.50	(4) BEST
304.92/291.50	    (5) proven lower bound
304.92/291.50	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
304.92/291.50	        (7) BOUNDS(n^1, INF)
304.92/291.50	    (8) TRS for Loop Detection
304.92/291.50	
304.92/291.50	
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(0)
304.92/291.50	Obligation:
304.92/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.50	
304.92/291.50	
304.92/291.50	The TRS R consists of the following rules:
304.92/291.50	
304.92/291.50	   le(0, Y) -> true
304.92/291.50	   le(s(X), 0) -> false
304.92/291.50	   le(s(X), s(Y)) -> le(X, Y)
304.92/291.50	   minus(0, Y) -> 0
304.92/291.50	   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
304.92/291.50	   ifMinus(true, s(X), Y) -> 0
304.92/291.50	   ifMinus(false, s(X), Y) -> s(minus(X, Y))
304.92/291.50	   quot(0, s(Y)) -> 0
304.92/291.50	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
304.92/291.50	
304.92/291.50	S is empty.
304.92/291.50	Rewrite Strategy: FULL
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
304.92/291.50	Transformed a relative TRS into a decreasing-loop problem.
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(2)
304.92/291.50	Obligation:
304.92/291.50	Analyzing the following TRS for decreasing loops:
304.92/291.50	
304.92/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.50	
304.92/291.50	
304.92/291.50	The TRS R consists of the following rules:
304.92/291.50	
304.92/291.50	   le(0, Y) -> true
304.92/291.50	   le(s(X), 0) -> false
304.92/291.50	   le(s(X), s(Y)) -> le(X, Y)
304.92/291.50	   minus(0, Y) -> 0
304.92/291.50	   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
304.92/291.50	   ifMinus(true, s(X), Y) -> 0
304.92/291.50	   ifMinus(false, s(X), Y) -> s(minus(X, Y))
304.92/291.50	   quot(0, s(Y)) -> 0
304.92/291.50	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
304.92/291.50	
304.92/291.50	S is empty.
304.92/291.50	Rewrite Strategy: FULL
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(3) DecreasingLoopProof (LOWER BOUND(ID))
304.92/291.50	The following loop(s) give(s) rise to the lower bound Omega(n^1):
304.92/291.50	
304.92/291.50	The rewrite sequence
304.92/291.50	
304.92/291.50	le(s(X), s(Y)) ->^+ le(X, Y)
304.92/291.50	
304.92/291.50	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
304.92/291.50	
304.92/291.50	The pumping substitution is [X / s(X), Y / s(Y)].
304.92/291.50	
304.92/291.50	The result substitution is [ ].
304.92/291.50	
304.92/291.50	
304.92/291.50	
304.92/291.50	
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(4)
304.92/291.50	Complex Obligation (BEST)
304.92/291.50	
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(5)
304.92/291.50	Obligation:
304.92/291.50	Proved the lower bound n^1 for the following obligation:
304.92/291.50	
304.92/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.50	
304.92/291.50	
304.92/291.50	The TRS R consists of the following rules:
304.92/291.50	
304.92/291.50	   le(0, Y) -> true
304.92/291.50	   le(s(X), 0) -> false
304.92/291.50	   le(s(X), s(Y)) -> le(X, Y)
304.92/291.50	   minus(0, Y) -> 0
304.92/291.50	   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
304.92/291.50	   ifMinus(true, s(X), Y) -> 0
304.92/291.50	   ifMinus(false, s(X), Y) -> s(minus(X, Y))
304.92/291.50	   quot(0, s(Y)) -> 0
304.92/291.50	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
304.92/291.50	
304.92/291.50	S is empty.
304.92/291.50	Rewrite Strategy: FULL
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(6) LowerBoundPropagationProof (FINISHED)
304.92/291.50	Propagated lower bound.
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(7)
304.92/291.50	BOUNDS(n^1, INF)
304.92/291.50	
304.92/291.50	----------------------------------------
304.92/291.50	
304.92/291.50	(8)
304.92/291.50	Obligation:
304.92/291.50	Analyzing the following TRS for decreasing loops:
304.92/291.50	
304.92/291.50	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
304.92/291.50	
304.92/291.50	
304.92/291.50	The TRS R consists of the following rules:
304.92/291.50	
304.92/291.50	   le(0, Y) -> true
304.92/291.50	   le(s(X), 0) -> false
304.92/291.50	   le(s(X), s(Y)) -> le(X, Y)
304.92/291.50	   minus(0, Y) -> 0
304.92/291.50	   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
304.92/291.50	   ifMinus(true, s(X), Y) -> 0
304.92/291.50	   ifMinus(false, s(X), Y) -> s(minus(X, Y))
304.92/291.50	   quot(0, s(Y)) -> 0
304.92/291.50	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
304.92/291.50	
304.92/291.50	S is empty.
304.92/291.50	Rewrite Strategy: FULL
304.92/291.54	EOF