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.