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_plus : forall u v, (0 <= u * v)%Z -> (forall k, (0 <= k)%Z -> Zdigit u k = Z0 \/ Zdigit v k = Z0) -> forall k, Zdigit (u + v) k = (Zdigit u k + Zdigit v k)%Z.