End Be_iter1.

Lemma be_iter1_preserves_eq :
forall (n : nat) (be1 be2 : bool_expr) (f : bool_expr -> bool_expr),
(forall be1 be2 : bool_expr, be_eq be1 be2 -> be_eq (f be1) (f be2)) ->
be_eq be1 be2 -> be_eq (be_iter1 f be1 n) (be_iter1 f be2 n).