Definition Qplus' (p q : Q) := Qred (Qplus p q).
Definition Qmult' (p q : Q) := Qred (Qmult p q).
Definition Qminus' x y := Qred (Qminus x y).

(* Specifications: *)
Lemma Qplus'_correct : forall p q : Q, (Qplus' p q)==(Qplus p q).
Lemma Qmult'_correct : forall p q : Q, (Qmult' p q)==(Qmult p q).
Lemma Qminus'_correct : forall p q : Q, (Qminus' p q)==(Qminus p q).
Add Morphism Qplus' with signature (Qeq ==> Qeq ==> Qeq) as Qplus'_comp.