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.