352.69/291.51	WORST_CASE(Omega(n^1), ?)
352.69/291.52	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
352.69/291.52	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
352.69/291.52	
352.69/291.52	
352.69/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.69/291.52	
352.69/291.52	(0) CpxTRS
352.69/291.52	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
352.69/291.52	(2) TRS for Loop Detection
352.69/291.52	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
352.69/291.52	(4) BEST
352.69/291.52	    (5) proven lower bound
352.69/291.52	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
352.69/291.52	        (7) BOUNDS(n^1, INF)
352.69/291.52	    (8) TRS for Loop Detection
352.69/291.52	
352.69/291.52	
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(0)
352.69/291.52	Obligation:
352.69/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.69/291.52	
352.69/291.52	
352.69/291.52	The TRS R consists of the following rules:
352.69/291.52	
352.69/291.52	   from(X) -> cons(X, n__from(s(X)))
352.69/291.52	   sel(0, cons(X, XS)) -> X
352.69/291.52	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
352.69/291.52	   minus(X, 0) -> 0
352.69/291.52	   minus(s(X), s(Y)) -> minus(X, Y)
352.69/291.52	   quot(0, s(Y)) -> 0
352.69/291.52	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
352.69/291.52	   zWquot(XS, nil) -> nil
352.69/291.52	   zWquot(nil, XS) -> nil
352.69/291.52	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
352.69/291.52	   from(X) -> n__from(X)
352.69/291.52	   zWquot(X1, X2) -> n__zWquot(X1, X2)
352.69/291.52	   activate(n__from(X)) -> from(X)
352.69/291.52	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
352.69/291.52	   activate(X) -> X
352.69/291.52	
352.69/291.52	S is empty.
352.69/291.52	Rewrite Strategy: FULL
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
352.69/291.52	Transformed a relative TRS into a decreasing-loop problem.
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(2)
352.69/291.52	Obligation:
352.69/291.52	Analyzing the following TRS for decreasing loops:
352.69/291.52	
352.69/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.69/291.52	
352.69/291.52	
352.69/291.52	The TRS R consists of the following rules:
352.69/291.52	
352.69/291.52	   from(X) -> cons(X, n__from(s(X)))
352.69/291.52	   sel(0, cons(X, XS)) -> X
352.69/291.52	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
352.69/291.52	   minus(X, 0) -> 0
352.69/291.52	   minus(s(X), s(Y)) -> minus(X, Y)
352.69/291.52	   quot(0, s(Y)) -> 0
352.69/291.52	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
352.69/291.52	   zWquot(XS, nil) -> nil
352.69/291.52	   zWquot(nil, XS) -> nil
352.69/291.52	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
352.69/291.52	   from(X) -> n__from(X)
352.69/291.52	   zWquot(X1, X2) -> n__zWquot(X1, X2)
352.69/291.52	   activate(n__from(X)) -> from(X)
352.69/291.52	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
352.69/291.52	   activate(X) -> X
352.69/291.52	
352.69/291.52	S is empty.
352.69/291.52	Rewrite Strategy: FULL
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(3) DecreasingLoopProof (LOWER BOUND(ID))
352.69/291.52	The following loop(s) give(s) rise to the lower bound Omega(n^1):
352.69/291.52	
352.69/291.52	The rewrite sequence
352.69/291.52	
352.69/291.52	minus(s(X), s(Y)) ->^+ minus(X, Y)
352.69/291.52	
352.69/291.52	gives rise to a decreasing loop by considering the right hand sides subterm at position [].
352.69/291.52	
352.69/291.52	The pumping substitution is [X / s(X), Y / s(Y)].
352.69/291.52	
352.69/291.52	The result substitution is [ ].
352.69/291.52	
352.69/291.52	
352.69/291.52	
352.69/291.52	
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(4)
352.69/291.52	Complex Obligation (BEST)
352.69/291.52	
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(5)
352.69/291.52	Obligation:
352.69/291.52	Proved the lower bound n^1 for the following obligation:
352.69/291.52	
352.69/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.69/291.52	
352.69/291.52	
352.69/291.52	The TRS R consists of the following rules:
352.69/291.52	
352.69/291.52	   from(X) -> cons(X, n__from(s(X)))
352.69/291.52	   sel(0, cons(X, XS)) -> X
352.69/291.52	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
352.69/291.52	   minus(X, 0) -> 0
352.69/291.52	   minus(s(X), s(Y)) -> minus(X, Y)
352.69/291.52	   quot(0, s(Y)) -> 0
352.69/291.52	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
352.69/291.52	   zWquot(XS, nil) -> nil
352.69/291.52	   zWquot(nil, XS) -> nil
352.69/291.52	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
352.69/291.52	   from(X) -> n__from(X)
352.69/291.52	   zWquot(X1, X2) -> n__zWquot(X1, X2)
352.69/291.52	   activate(n__from(X)) -> from(X)
352.69/291.52	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
352.69/291.52	   activate(X) -> X
352.69/291.52	
352.69/291.52	S is empty.
352.69/291.52	Rewrite Strategy: FULL
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(6) LowerBoundPropagationProof (FINISHED)
352.69/291.52	Propagated lower bound.
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(7)
352.69/291.52	BOUNDS(n^1, INF)
352.69/291.52	
352.69/291.52	----------------------------------------
352.69/291.52	
352.69/291.52	(8)
352.69/291.52	Obligation:
352.69/291.52	Analyzing the following TRS for decreasing loops:
352.69/291.52	
352.69/291.52	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
352.69/291.52	
352.69/291.52	
352.69/291.52	The TRS R consists of the following rules:
352.69/291.52	
352.69/291.52	   from(X) -> cons(X, n__from(s(X)))
352.69/291.52	   sel(0, cons(X, XS)) -> X
352.69/291.52	   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
352.69/291.52	   minus(X, 0) -> 0
352.69/291.52	   minus(s(X), s(Y)) -> minus(X, Y)
352.69/291.52	   quot(0, s(Y)) -> 0
352.69/291.52	   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))
352.69/291.52	   zWquot(XS, nil) -> nil
352.69/291.52	   zWquot(nil, XS) -> nil
352.69/291.52	   zWquot(cons(X, XS), cons(Y, YS)) -> cons(quot(X, Y), n__zWquot(activate(XS), activate(YS)))
352.69/291.52	   from(X) -> n__from(X)
352.69/291.52	   zWquot(X1, X2) -> n__zWquot(X1, X2)
352.69/291.52	   activate(n__from(X)) -> from(X)
352.69/291.52	   activate(n__zWquot(X1, X2)) -> zWquot(X1, X2)
352.69/291.52	   activate(X) -> X
352.69/291.52	
352.69/291.52	S is empty.
352.69/291.52	Rewrite Strategy: FULL
352.76/291.55	EOF