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 to_hexadecimal_inj q q' : to_hexadecimal q <> None -> to_hexadecimal q = to_hexadecimal q' -> q = q'.