Fixpoint be_iter1 (be : bool_expr) (n : nat) {struct n} : bool_expr :=
    match n with
    | O => be
    | S m => be_iter1 (bef be) m
    end.Fixpoint be_iter2 (be : bool_expr) (n : nat) {struct n} : bool_expr :=
    match n with
    | O => be
    | S m => bef (be_iter2 be m)
    end.

Lemma be_iter2n_false : forall (n : nat) (be : bool_expr) (f : bool_expr -> bool_expr), (forall be1 be2 : bool_expr, be_eq be1 be2 -> be_eq (f be1) (f be2)) -> snd (be_iter2n f be n) = false -> be_eq (fst (be_iter2n f be n)) (be_iter1 f be (two_power n)).