Canonical Qint_opprPred := OpprPred Qint_subring_closed.
Canonical Qint_addrPred := AddrPred Qint_subring_closed.
Canonical Qint_mulrPred := MulrPred Qint_subring_closed.
Canonical Qint_zmodPred := ZmodPred Qint_subring_closed.
Canonical Qint_semiringPred := SemiringPred Qint_subring_closed.
Canonical Qint_smulrPred := SmulrPred Qint_subring_closed.
Canonical Qint_subringPred := SubringPred Qint_subring_closed.
End QintPred.
Section QnatPred.
Definition Qnat := [qualify a x : rat | (x \is a Qint) && (0 <= x)].
Fact Qnat_key : pred_key Qnat. Proof. by []. Qed.
Canonical Qnat_keyed := KeyedQualifier Qnat_key.

Lemma Qnat_def x : (x \is a Qnat) = (x \is a Qint) && (0 <= x).
Fact Qnat_semiring_closed : semiring_closed Qnat.