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 ZOmod_plus_pow_digit : forall u v n, (0 <= u * v)%Z -> (forall k, (0 <= k < n)%Z -> Zdigit u k = Z0 \/ Zdigit v k = Z0) -> Z.rem (u + v) (Zpower beta n) = (Z.rem u (Zpower beta n) + Z.rem v (Zpower beta n))%Z.