Fixpoint upd (i : nat) {A} new (l : list A) : option (list A) :=
  match i, l with
  | O, _ :: l => Some (new :: l)
  | _, nil => None
  | S i, x :: l =>
    match upd i new l with
    | Some l' => Some (x :: l')
    | None => None
    end
  end.

Lemma upd_app_None {A} i x (l1 l2 : list A) : upd i x l1 = None -> upd i x (l1 ++ l2) = option_map (app l1) (upd (i - length l1) x l2).