Definition bool_fun_mu_all (bft bfg : bool_fun) : bool_fun :=
  bool_fun_univl (bool_fun_impl bft (bool_fun_replacel bfg lx lx')) lx'.Definition bool_fun_mu_ex (bft bfg : bool_fun) : bool_fun :=
  bool_fun_exl (bool_fun_and bft (bool_fun_replacel bfg lx lx')) lx'.Definition mu_all_eval (t be : bool_expr) : bool_expr :=
  univl (Impl t (replacel be lx lx')) lx'.Definition mu_ex_eval (t be : bool_expr) : bool_expr :=
  exl (ANd t (replacel be lx lx')) lx'.Definition BDDmu_all (gc : BDDconfig -> list ad -> BDDconfig)
  (cfg : BDDconfig) (ul : list ad) (nodet nodeg : ad) :=
  match BDDreplacel gc cfg ul nodeg lx lx' with
  | (cfgr, noder) =>
      match BDDimpl gc cfgr (noder :: ul) nodet noder with
      | (cfgi, nodei) => BDDunivl gc cfgi (nodei :: ul) nodei lx'
      end
  end.Definition BDDmu_ex (gc : BDDconfig -> list ad -> BDDconfig)
  (cfg : BDDconfig) (ul : list ad) (nodet nodeg : ad) :=
  match BDDreplacel gc cfg ul nodeg lx lx' with
  | (cfgr, noder) =>
      match BDDand gc cfgr (noder :: ul) nodet noder with
      | (cfga, nodea) => BDDexl gc cfga (nodea :: ul) nodea lx'
      end
  end.

Lemma mu_ex_x_free : forall (t be : bool_expr) (x : BDDvar), be_x_free x (mu_ex_eval t be) = true -> ~ In x lx' /\ (be_x_free x t = true \/ be_x_free x be = true /\ ~ In x lx).