310.84/291.56	WORST_CASE(Omega(n^1), ?)
310.84/291.57	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
310.84/291.57	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
310.84/291.57	
310.84/291.57	
310.84/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.84/291.57	
310.84/291.57	(0) CpxTRS
310.84/291.57	(1) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
310.84/291.57	(2) TRS for Loop Detection
310.84/291.57	(3) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
310.84/291.57	(4) BEST
310.84/291.57	    (5) proven lower bound
310.84/291.57	        (6) LowerBoundPropagationProof [FINISHED, 0 ms]
310.84/291.57	        (7) BOUNDS(n^1, INF)
310.84/291.57	    (8) TRS for Loop Detection
310.84/291.57	
310.84/291.57	
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(0)
310.84/291.57	Obligation:
310.84/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.84/291.57	
310.84/291.57	
310.84/291.57	The TRS R consists of the following rules:
310.84/291.57	
310.84/291.57	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
310.84/291.57	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
310.84/291.57	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
310.84/291.57	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
310.84/291.57	   nats(N) -> cons(N, n__nats(s(N)))
310.84/291.57	   zprimes -> sieve(nats(s(s(0))))
310.84/291.57	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
310.84/291.57	   sieve(X) -> n__sieve(X)
310.84/291.57	   nats(X) -> n__nats(X)
310.84/291.57	   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
310.84/291.57	   activate(n__sieve(X)) -> sieve(X)
310.84/291.57	   activate(n__nats(X)) -> nats(X)
310.84/291.57	   activate(X) -> X
310.84/291.57	
310.84/291.57	S is empty.
310.84/291.57	Rewrite Strategy: FULL
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(1) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
310.84/291.57	Transformed a relative TRS into a decreasing-loop problem.
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(2)
310.84/291.57	Obligation:
310.84/291.57	Analyzing the following TRS for decreasing loops:
310.84/291.57	
310.84/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.84/291.57	
310.84/291.57	
310.84/291.57	The TRS R consists of the following rules:
310.84/291.57	
310.84/291.57	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
310.84/291.57	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
310.84/291.57	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
310.84/291.57	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
310.84/291.57	   nats(N) -> cons(N, n__nats(s(N)))
310.84/291.57	   zprimes -> sieve(nats(s(s(0))))
310.84/291.57	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
310.84/291.57	   sieve(X) -> n__sieve(X)
310.84/291.57	   nats(X) -> n__nats(X)
310.84/291.57	   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
310.84/291.57	   activate(n__sieve(X)) -> sieve(X)
310.84/291.57	   activate(n__nats(X)) -> nats(X)
310.84/291.57	   activate(X) -> X
310.84/291.57	
310.84/291.57	S is empty.
310.84/291.57	Rewrite Strategy: FULL
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(3) DecreasingLoopProof (LOWER BOUND(ID))
310.84/291.57	The following loop(s) give(s) rise to the lower bound Omega(n^1):
310.84/291.57	
310.84/291.57	The rewrite sequence
310.84/291.57	
310.84/291.57	sieve(cons(s(N), n__sieve(X1_0))) ->^+ cons(s(N), n__sieve(filter(sieve(X1_0), N, N)))
310.84/291.57	
310.84/291.57	gives rise to a decreasing loop by considering the right hand sides subterm at position [1,0,0].
310.84/291.57	
310.84/291.57	The pumping substitution is [X1_0 / cons(s(N), n__sieve(X1_0))].
310.84/291.57	
310.84/291.57	The result substitution is [ ].
310.84/291.57	
310.84/291.57	
310.84/291.57	
310.84/291.57	
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(4)
310.84/291.57	Complex Obligation (BEST)
310.84/291.57	
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(5)
310.84/291.57	Obligation:
310.84/291.57	Proved the lower bound n^1 for the following obligation:
310.84/291.57	
310.84/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.84/291.57	
310.84/291.57	
310.84/291.57	The TRS R consists of the following rules:
310.84/291.57	
310.84/291.57	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
310.84/291.57	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
310.84/291.57	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
310.84/291.57	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
310.84/291.57	   nats(N) -> cons(N, n__nats(s(N)))
310.84/291.57	   zprimes -> sieve(nats(s(s(0))))
310.84/291.57	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
310.84/291.57	   sieve(X) -> n__sieve(X)
310.84/291.57	   nats(X) -> n__nats(X)
310.84/291.57	   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
310.84/291.57	   activate(n__sieve(X)) -> sieve(X)
310.84/291.57	   activate(n__nats(X)) -> nats(X)
310.84/291.57	   activate(X) -> X
310.84/291.57	
310.84/291.57	S is empty.
310.84/291.57	Rewrite Strategy: FULL
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(6) LowerBoundPropagationProof (FINISHED)
310.84/291.57	Propagated lower bound.
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(7)
310.84/291.57	BOUNDS(n^1, INF)
310.84/291.57	
310.84/291.57	----------------------------------------
310.84/291.57	
310.84/291.57	(8)
310.84/291.57	Obligation:
310.84/291.57	Analyzing the following TRS for decreasing loops:
310.84/291.57	
310.84/291.57	The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).
310.84/291.57	
310.84/291.57	
310.84/291.57	The TRS R consists of the following rules:
310.84/291.57	
310.84/291.57	   filter(cons(X, Y), 0, M) -> cons(0, n__filter(activate(Y), M, M))
310.84/291.57	   filter(cons(X, Y), s(N), M) -> cons(X, n__filter(activate(Y), N, M))
310.84/291.57	   sieve(cons(0, Y)) -> cons(0, n__sieve(activate(Y)))
310.84/291.57	   sieve(cons(s(N), Y)) -> cons(s(N), n__sieve(filter(activate(Y), N, N)))
310.84/291.57	   nats(N) -> cons(N, n__nats(s(N)))
310.84/291.57	   zprimes -> sieve(nats(s(s(0))))
310.84/291.57	   filter(X1, X2, X3) -> n__filter(X1, X2, X3)
310.84/291.57	   sieve(X) -> n__sieve(X)
310.84/291.57	   nats(X) -> n__nats(X)
310.84/291.57	   activate(n__filter(X1, X2, X3)) -> filter(X1, X2, X3)
310.84/291.57	   activate(n__sieve(X)) -> sieve(X)
310.84/291.57	   activate(n__nats(X)) -> nats(X)
310.84/291.57	   activate(X) -> X
310.84/291.57	
310.84/291.57	S is empty.
310.84/291.57	Rewrite Strategy: FULL
310.96/291.60	EOF