Definition dnorm (d:hexadecimal) : hexadecimal :=
  let norm_i i f :=
    match i with
    | Pos i => Pos (unorm i)
    | Neg i => match nzhead (app i f) with Nil => Pos zero | _ => Neg (unorm i) end
    end in
  match d with
  | Hexadecimal i f => Hexadecimal (norm_i i f) f
  | HexadecimalExp i f e =>
    match Decimal.norm e with
    | Decimal.Pos Decimal.zero => Hexadecimal (norm_i i f) f
    | e => HexadecimalExp (norm_i i f) f e
    end
  end.

Lemma dnorm_i_exact i f : (nb_digits f < nb_digits (unorm (app (abs i) f)))%nat -> match i with | Pos i => Pos (unorm i) | Neg i => match nzhead (app i f) with | Nil => Pos zero | _ => Neg (unorm i) end end = norm i.