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_iter_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 <= n /\ be_eq (be_iter1 bef be m) (be_iter1 bef be (S m))) -> lfp_be bef be (be_iter bef be n).