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

(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))))

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

filter₃ > [cons₁, nats₁] > [0, s₁, sieve₁]
zprimes > [cons₁, nats₁] > [0, s₁, sieve₁]

With 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))
zprimes → sieve(nats(s(s(0))))

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

nats(N) → cons(N)

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Recursive Path Order [RPO].
Precedence:

nats₁ > cons₁

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

nats(N) → cons(N)

(4) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(5) RisEmptyProof (EQUIVALENT transformation)

The TRS R is empty. Hence, termination is trivially proven.

(6) TRUE