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 mu_eval_lemma1 :
forall f : mu_form,
f_ok f ->
mu_form_ap_ok (var_lu 0 N) f ->
(forall re : rel_env,
ad_to_be_ok (var_lu 0 N) re -> be_ok (var_lu 0 N) (mu_eval f re)) /\
(forall P : ad, f_P_even P f true -> re_to_be_inc (mu_eval f) P) /\
(forall P : ad, f_P_even P f false -> re_to_be_dec (mu_eval f) P) /\
(forall re1 re2 : rel_env,
ad_to_be_eq re1 re2 -> be_eq (mu_eval f re1) (mu_eval f re2)).