Definition ad_to_be_eq1 (f : mu_form) (f1 f2 : ad -> bool_expr) :=
forall x : ad, mu_rel_free x f = true -> be_eq (f1 x) (f2 x).

Lemma be_iter2n_is_lfp_be :
forall (bef : bool_expr -> bool_expr) (n : nat) (be : bool_expr),
be_le be (bef be) ->
(forall be1 be2 : bool_expr, be_eq be1 be2 -> be_eq (bef be1) (bef be2)) ->
(forall be1 be2 : bool_expr, be_le be1 be2 -> be_le (bef be1) (bef be2)) ->
(exists m : nat,
m <= two_power n /\ be_eq (be_iter1 bef be m) (be_iter1 bef be (S m))) ->
lfp_be bef be (fst (be_iter2n bef be n)).