Fixpoint Zsum_digit f k :=
  match k with
  | O => Z0
  | S k => (Zsum_digit f k + f (Z_of_nat k) * Zpower beta (Z_of_nat k))%Z
  end.

Theorem Zdigit_ext : forall n1 n2, (forall k, (0 <= k)%Z -> Zdigit n1 k = Zdigit n2 k) -> n1 = n2.