Canonical prodv_muloid := Monoid.MulLaw prod0v prodv0.

(* Specifications: *)
Lemma prod1v : left_id 1%VS prodv.
Lemma prodv1 : right_id 1%VS prodv.
Lemma prodvS U1 U2 V1 V2 : (U1 <= U2 -> V1 <= V2 -> U1 * V1 <= U2 * V2)%VS.
Lemma prodvSl U1 U2 V : (U1 <= U2 -> U1 * V <= U2 * V)%VS.
Lemma prodvSr U V1 V2 : (V1 <= V2 -> U * V1 <= U * V2)%VS.
Lemma prodvDl : left_distributive prodv addv.
Lemma prodvDr : right_distributive prodv addv.