Definition eqo i j :=
   (if i =? j as b return ((b = true -> i = j) -> option (i=j))
    then fun Heq : true = true -> i = j =>
             Some (Heq (eq_refl true))
     else fun _ : false = true -> i = j => None) (eqb_correct i j).

Lemma to_Z_gcd : forall i j, φ (gcd i j) = Zgcdn (2 * size) (φ j) (φ i).