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 rat_lrmorphism A B (f : A -> B) : rmorphism f -> lrmorphism f.