Definition be_le2 (be1 be2 : bool_expr) :=
  forall ve : var_env,
  bool_fun_of_bool_expr be1 ve = true -> bool_fun_of_bool_expr be2 ve = true.

Lemma mu_ex_le : forall t be1 be2 : bool_expr, be_le be1 be2 -> be_le (mu_ex_eval t be1) (mu_ex_eval t be2).