proposition: Lemma nat_to_B_lt x y: x < y <-> nat_to_B x <c nat_to_B y.
proof: Proof. case:y; first by split => // /clt0. move => y; rewrite ltnS /=; apply: (iff_trans ( nat_to_B_le x y)). apply:iff_sym;apply:(cltSleP (nat_to_B_Nat y)). Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)