Add Morphism Qmult' with signature (Qeq ==> Qeq ==> Qeq) as Qmult'_comp.
Add Morphism Qminus' with signature (Qeq ==> Qeq ==> Qeq) as Qminus'_comp.

(* Specifications: *)
Lemma Qred_opp: forall q, Qred (-q) = - (Qred q).
Theorem Qred_compare: forall x y,
Qcompare x y = Qcompare (Qred x) (Qred y).
Lemma Qred_le q q' : Qred q <= Qred q' <-> q <= q'.
Lemma Qred_lt q q' : Qred q < Qred q' <-> q < q'.