927.38/291.46	WORST_CASE(Omega(n^1), ?)
927.38/291.46	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
927.38/291.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
927.38/291.46	
927.38/291.46	
927.38/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
927.38/291.46	
927.38/291.46	(0) CpxTRS
927.38/291.46	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
927.38/291.46	(2) TRS for Loop Detection
927.38/291.46	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
927.38/291.46	(4) BEST
927.38/291.46	    (5) proven lower bound
927.38/291.46	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
927.38/291.46	        (7) BOUNDS(n^1, INF)
927.38/291.46	    (8) TRS for Loop Detection
927.38/291.46	
927.38/291.46	
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(0)
927.38/291.46	Obligation:
927.38/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
927.38/291.46	
927.38/291.46	
927.38/291.46	The TRS R consists of the following rules:
927.38/291.46	
927.38/291.46	   plus(0, Y) -> Y
927.38/291.46	   plus(s(X), Y) -> s(plus(X, Y))
927.38/291.46	   min(X, 0) -> X
927.38/291.46	   min(s(X), s(Y)) -> min(X, Y)
927.38/291.46	   min(min(X, Y), Z) -> min(X, plus(Y, Z))
927.38/291.46	   quot(0, s(Y)) -> 0
927.38/291.46	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
927.38/291.46	
927.38/291.46	S is empty.
927.38/291.46	Rewrite Strategy: FULL
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
927.38/291.46	Transformed a relative TRS into a decreasing-loop problem.
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(2)
927.38/291.46	Obligation:
927.38/291.46	Analyzing the following TRS for decreasing loops:
927.38/291.46	
927.38/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
927.38/291.46	
927.38/291.46	
927.38/291.46	The TRS R consists of the following rules:
927.38/291.46	
927.38/291.46	   plus(0, Y) -> Y
927.38/291.46	   plus(s(X), Y) -> s(plus(X, Y))
927.38/291.46	   min(X, 0) -> X
927.38/291.46	   min(s(X), s(Y)) -> min(X, Y)
927.38/291.46	   min(min(X, Y), Z) -> min(X, plus(Y, Z))
927.38/291.46	   quot(0, s(Y)) -> 0
927.38/291.46	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
927.38/291.46	
927.38/291.46	S is empty.
927.38/291.46	Rewrite Strategy: FULL
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(3) DecreasingLoopProof (LOWER BOUND(ID))
927.38/291.46	The following loop(s) give(s) rise to the lower bound Omega(n^1):
927.38/291.46	
927.38/291.46	The rewrite sequence
927.38/291.46	
927.38/291.46	min(s(X), s(Y)) ->^+ min(X, Y)
927.38/291.46	
927.38/291.46	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
927.38/291.46	
927.38/291.46	The pumping substitution is [X / s(X), Y / s(Y)].
927.38/291.46	
927.38/291.46	The result substitution is [ ].
927.38/291.46	
927.38/291.46	
927.38/291.46	
927.38/291.46	
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(4)
927.38/291.46	Complex Obligation (BEST)
927.38/291.46	
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(5)
927.38/291.46	Obligation:
927.38/291.46	Proved the lower bound n^1 for the following obligation:
927.38/291.46	
927.38/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
927.38/291.46	
927.38/291.46	
927.38/291.46	The TRS R consists of the following rules:
927.38/291.46	
927.38/291.46	   plus(0, Y) -> Y
927.38/291.46	   plus(s(X), Y) -> s(plus(X, Y))
927.38/291.46	   min(X, 0) -> X
927.38/291.46	   min(s(X), s(Y)) -> min(X, Y)
927.38/291.46	   min(min(X, Y), Z) -> min(X, plus(Y, Z))
927.38/291.46	   quot(0, s(Y)) -> 0
927.38/291.46	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
927.38/291.46	
927.38/291.46	S is empty.
927.38/291.46	Rewrite Strategy: FULL
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(6) LowerBoundPropagationProof (FINISHED)
927.38/291.46	Propagated lower bound.
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(7)
927.38/291.46	BOUNDS(n^1, INF)
927.38/291.46	
927.38/291.46	----------------------------------------
927.38/291.46	
927.38/291.46	(8)
927.38/291.46	Obligation:
927.38/291.46	Analyzing the following TRS for decreasing loops:
927.38/291.46	
927.38/291.46	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
927.38/291.46	
927.38/291.46	
927.38/291.46	The TRS R consists of the following rules:
927.38/291.46	
927.38/291.46	   plus(0, Y) -> Y
927.38/291.46	   plus(s(X), Y) -> s(plus(X, Y))
927.38/291.46	   min(X, 0) -> X
927.38/291.46	   min(s(X), s(Y)) -> min(X, Y)
927.38/291.46	   min(min(X, Y), Z) -> min(X, plus(Y, Z))
927.38/291.46	   quot(0, s(Y)) -> 0
927.38/291.46	   quot(s(X), s(Y)) -> s(quot(min(X, Y), s(Y)))
927.38/291.46	
927.38/291.46	S is empty.
927.38/291.46	Rewrite Strategy: FULL
927.48/291.53	EOF