proposition: Lemma div_mod a b : a <> 0 -> (a | b) <-> b mod a = 0.
proof: Proof. intros az; split. - intros (k, ->). apply Nat.mod_mul, az. - intros e. exists (b / a). etransitivity. apply (Nat.div_mod _ a), az. lia. Qed.
imports: Coq(Arith Euclid Compare Lia)