proposition: Lemma addsmxS m1 m2 m3 m4 n (A : 'M_(m1, n)) (B : 'M_(m2, n)) (C : 'M_(m3, n)) (D : 'M_(m4, n)) : (A <= C -> B <= D -> A + B <= C + D)%MS.
proof: Proof. move=> sAC sBD. by rewrite addsmx_sub {1}addsmxC !(submx_trans _ (addsmxSr _ _)). Qed.
imports: HB(structures) mathcomp(ssreflect ssrbool ssrfun eqtype ssrnat seq choice) mathcomp(fintype finfun bigop finset fingroup perm order) mathcomp(div prime binomial ssralg finalg zmodp matrix)