Termination w.r.t. Q proof of TRS/TRCSREx9_BLR02

Termination w.r.t. Q of the following Term Rewriting System could be proven:

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.

↳ QTRS

  ↳ DirectTerminationProof

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.

We use [23] with the following order to prove termination.

Lexicographic path order with status [19].
Quasi-Precedence:

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

Status:

filter₃: [3,2,1]
sieve₁: [1]
zprimes: multiset
s₁: [1]
0: multiset
cons₁: [1]
nats₁: [1]