3.31/1.59	YES
3.31/1.60	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.31/1.60	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.31/1.60	
3.31/1.60	
3.31/1.60	Termination w.r.t. Q of the given QTRS could be proven:
3.31/1.60	
3.31/1.60	(0) QTRS
3.31/1.60	(1) QTRSRRRProof [EQUIVALENT, 0 ms]
3.31/1.60	(2) QTRS
3.31/1.60	(3) RisEmptyProof [EQUIVALENT, 0 ms]
3.31/1.60	(4) YES
3.31/1.60	
3.31/1.60	
3.31/1.60	----------------------------------------
3.31/1.60	
3.31/1.60	(0)
3.31/1.60	Obligation:
3.31/1.60	Q restricted rewrite system:
3.31/1.60	The TRS R consists of the following rules:
3.31/1.60	
3.31/1.60	   filter(cons(X), 0, M) -> cons(0)
3.31/1.60	   filter(cons(X), s(N), M) -> cons(X)
3.31/1.60	   sieve(cons(0)) -> cons(0)
3.31/1.60	   sieve(cons(s(N))) -> cons(s(N))
3.31/1.60	   nats(N) -> cons(N)
3.31/1.60	   zprimes -> sieve(nats(s(s(0))))
3.31/1.60	
3.31/1.60	The set Q consists of the following terms:
3.31/1.60	
3.31/1.60	   filter(cons(x0), 0, x1)
3.31/1.60	   filter(cons(x0), s(x1), x2)
3.31/1.60	   sieve(cons(0))
3.31/1.60	   sieve(cons(s(x0)))
3.31/1.60	   nats(x0)
3.31/1.60	   zprimes
3.31/1.60	
3.31/1.60	
3.31/1.60	----------------------------------------
3.31/1.60	
3.31/1.60	(1) QTRSRRRProof (EQUIVALENT)
3.31/1.60	Used ordering:
3.31/1.60	Knuth-Bendix order [KBO] with precedence:sieve_1 > zprimes > s_1 > cons_1 > nats_1 > 0 > filter_3
3.31/1.60	
3.31/1.60	and weight map:
3.31/1.60	
3.31/1.60	   0=3
3.31/1.60	   zprimes=8
3.31/1.60	   cons_1=1
3.31/1.60	   s_1=1
3.31/1.60	   sieve_1=0
3.31/1.60	   nats_1=2
3.31/1.60	   filter_3=0
3.31/1.60	
3.31/1.60	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.31/1.60	
3.31/1.60	   filter(cons(X), 0, M) -> cons(0)
3.31/1.60	   filter(cons(X), s(N), M) -> cons(X)
3.31/1.60	   sieve(cons(0)) -> cons(0)
3.31/1.60	   sieve(cons(s(N))) -> cons(s(N))
3.31/1.60	   nats(N) -> cons(N)
3.31/1.60	   zprimes -> sieve(nats(s(s(0))))
3.31/1.60	
3.31/1.60	
3.31/1.60	
3.31/1.60	
3.31/1.60	----------------------------------------
3.31/1.60	
3.31/1.60	(2)
3.31/1.60	Obligation:
3.31/1.60	Q restricted rewrite system:
3.31/1.60	R is empty.
3.31/1.60	The set Q consists of the following terms:
3.31/1.60	
3.31/1.60	   filter(cons(x0), 0, x1)
3.31/1.60	   filter(cons(x0), s(x1), x2)
3.31/1.60	   sieve(cons(0))
3.31/1.60	   sieve(cons(s(x0)))
3.31/1.60	   nats(x0)
3.31/1.60	   zprimes
3.31/1.60	
3.31/1.60	
3.31/1.60	----------------------------------------
3.31/1.60	
3.31/1.60	(3) RisEmptyProof (EQUIVALENT)
3.31/1.60	The TRS R is empty. Hence, termination is trivially proven.
3.31/1.60	----------------------------------------
3.31/1.60	
3.31/1.60	(4)
3.31/1.60	YES
3.31/1.62	EOF