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.
Fact Qnat_semiring_closed : semiring_closed Qnat.
Canonical Qnat_addrPred := AddrPred Qnat_semiring_closed.
Canonical Qnat_mulrPred := MulrPred Qnat_semiring_closed.
Canonical Qnat_semiringPred := SemiringPred Qnat_semiring_closed.
End QnatPred.

Lemma natq_div m n : n %| m -> (m %/ n)%:R = m%:R / n%:R :> rat.