End Be_iter1.

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)).