YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


Termination w.r.t. Q of the given QTRS could be proven:

(0) QTRS
(1) QTRSRRRProof [EQUIVALENT, 0 ms]
(2) QTRS
(3) RisEmptyProof [EQUIVALENT, 0 ms]
(4) YES


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(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))

The set Q consists of the following terms:

   filter(cons(x0), 0, x1)
   filter(cons(x0), s(x1), x2)
   sieve(cons(0))
   sieve(cons(s(x0)))
   nats(x0)
   zprimes


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(1) QTRSRRRProof (EQUIVALENT)
Used ordering:
Knuth-Bendix order [KBO] with precedence:sieve_1 > zprimes > s_1 > cons_1 > nats_1 > 0 > filter_3

and weight map:

   0=3
   zprimes=8
   cons_1=1
   s_1=1
   sieve_1=0
   nats_1=2
   filter_3=0

The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   filter(cons(X), 0, M) -> cons(0)
   filter(cons(X), s(N), M) -> cons(X)
   sieve(cons(0)) -> cons(0)
   sieve(cons(s(N))) -> cons(s(N))
   nats(N) -> cons(N)
   zprimes -> sieve(nats(s(s(0))))




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(2)
Obligation:
Q restricted rewrite system:
R is empty.
The set Q consists of the following terms:

   filter(cons(x0), 0, x1)
   filter(cons(x0), s(x1), x2)
   sieve(cons(0))
   sieve(cons(s(x0)))
   nats(x0)
   zprimes


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(3) RisEmptyProof (EQUIVALENT)
The TRS R is empty. Hence, termination is trivially proven.
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(4)
YES