1036.52/291.52	WORST_CASE(Omega(n^1), ?)
1036.52/291.54	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
1036.52/291.54	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.52/291.54	
1036.52/291.54	(0) CpxTRS
1036.52/291.54	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
1036.52/291.54	(2) TRS for Loop Detection
1036.52/291.54	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
1036.52/291.54	(4) BEST
1036.52/291.54	    (5) proven lower bound
1036.52/291.54	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
1036.52/291.54	        (7) BOUNDS(n^1, INF)
1036.52/291.54	    (8) TRS for Loop Detection
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(0)
1036.52/291.54	Obligation:
1036.52/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	The TRS R consists of the following rules:
1036.52/291.54	
1036.52/291.54	   from(X) -> cons(X, n__from(s(X)))
1036.52/291.54	   sel(0, cons(X, XS)) -> X
1036.52/291.54	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1036.52/291.54	   minus(X, 0) -> 0
1036.52/291.54	   minus(s(X), s(Y)) -> minus(X, Y)
1036.52/291.54	   quot(0, s(Y)) -> 0
1036.52/291.54	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
1036.52/291.54	   zWquot(XS, nil) -> nil
1036.52/291.54	   zWquot(nil, XS) -> nil
1036.52/291.54	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
1036.52/291.54	   from(X) -> n__from(X)
1036.52/291.54	   zWquot(X1, X2) -> n__zWquot(X1, X2)
1036.52/291.54	   activate(n__from(X)) -> from(X)
1036.52/291.54	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
1036.52/291.54	   activate(X) -> X
1036.52/291.54	
1036.52/291.54	S is empty.
1036.52/291.54	Rewrite Strategy: INNERMOST
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
1036.52/291.54	Transformed a relative TRS into a decreasing-loop problem.
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(2)
1036.52/291.54	Obligation:
1036.52/291.54	Analyzing the following TRS for decreasing loops:
1036.52/291.54	
1036.52/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	The TRS R consists of the following rules:
1036.52/291.54	
1036.52/291.54	   from(X) -> cons(X, n__from(s(X)))
1036.52/291.54	   sel(0, cons(X, XS)) -> X
1036.52/291.54	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1036.52/291.54	   minus(X, 0) -> 0
1036.52/291.54	   minus(s(X), s(Y)) -> minus(X, Y)
1036.52/291.54	   quot(0, s(Y)) -> 0
1036.52/291.54	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
1036.52/291.54	   zWquot(XS, nil) -> nil
1036.52/291.54	   zWquot(nil, XS) -> nil
1036.52/291.54	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
1036.52/291.54	   from(X) -> n__from(X)
1036.52/291.54	   zWquot(X1, X2) -> n__zWquot(X1, X2)
1036.52/291.54	   activate(n__from(X)) -> from(X)
1036.52/291.54	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
1036.52/291.54	   activate(X) -> X
1036.52/291.54	
1036.52/291.54	S is empty.
1036.52/291.54	Rewrite Strategy: INNERMOST
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(3) DecreasingLoopProof (LOWER BOUND(ID))
1036.52/291.54	The following loop(s) give(s) rise to the lower bound Omega(n^1):
1036.52/291.54	
1036.52/291.54	The rewrite sequence
1036.52/291.54	
1036.52/291.54	minus(s(X), s(Y)) ->^+ minus(X, Y)
1036.52/291.54	
1036.52/291.54	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
1036.52/291.54	
1036.52/291.54	The pumping substitution is [X / s(X), Y / s(Y)].
1036.52/291.54	
1036.52/291.54	The result substitution is [ ].
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(4)
1036.52/291.54	Complex Obligation (BEST)
1036.52/291.54	
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(5)
1036.52/291.54	Obligation:
1036.52/291.54	Proved the lower bound n^1 for the following obligation:
1036.52/291.54	
1036.52/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	The TRS R consists of the following rules:
1036.52/291.54	
1036.52/291.54	   from(X) -> cons(X, n__from(s(X)))
1036.52/291.54	   sel(0, cons(X, XS)) -> X
1036.52/291.54	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1036.52/291.54	   minus(X, 0) -> 0
1036.52/291.54	   minus(s(X), s(Y)) -> minus(X, Y)
1036.52/291.54	   quot(0, s(Y)) -> 0
1036.52/291.54	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
1036.52/291.54	   zWquot(XS, nil) -> nil
1036.52/291.54	   zWquot(nil, XS) -> nil
1036.52/291.54	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
1036.52/291.54	   from(X) -> n__from(X)
1036.52/291.54	   zWquot(X1, X2) -> n__zWquot(X1, X2)
1036.52/291.54	   activate(n__from(X)) -> from(X)
1036.52/291.54	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
1036.52/291.54	   activate(X) -> X
1036.52/291.54	
1036.52/291.54	S is empty.
1036.52/291.54	Rewrite Strategy: INNERMOST
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(6) LowerBoundPropagationProof (FINISHED)
1036.52/291.54	Propagated lower bound.
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(7)
1036.52/291.54	BOUNDS(n^1, INF)
1036.52/291.54	
1036.52/291.54	----------------------------------------
1036.52/291.54	
1036.52/291.54	(8)
1036.52/291.54	Obligation:
1036.52/291.54	Analyzing the following TRS for decreasing loops:
1036.52/291.54	
1036.52/291.54	The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
1036.52/291.54	
1036.52/291.54	
1036.52/291.54	The TRS R consists of the following rules:
1036.52/291.54	
1036.52/291.54	   from(X) -> cons(X, n__from(s(X)))
1036.52/291.54	   sel(0, cons(X, XS)) -> X
1036.52/291.54	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
1036.52/291.54	   minus(X, 0) -> 0
1036.52/291.54	   minus(s(X), s(Y)) -> minus(X, Y)
1036.52/291.54	   quot(0, s(Y)) -> 0
1036.52/291.54	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
1036.52/291.54	   zWquot(XS, nil) -> nil
1036.52/291.54	   zWquot(nil, XS) -> nil
1036.52/291.54	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
1036.52/291.54	   from(X) -> n__from(X)
1036.52/291.54	   zWquot(X1, X2) -> n__zWquot(X1, X2)
1036.52/291.54	   activate(n__from(X)) -> from(X)
1036.52/291.54	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
1036.52/291.54	   activate(X) -> X
1036.52/291.54	
1036.52/291.54	S is empty.
1036.52/291.54	Rewrite Strategy: INNERMOST
1036.95/291.64	EOF