Definition augment (vb : var_binding) (x : BDDvar) 
  (b : bool) (y : BDDvar) := if BDDvar_eq x y then b else vb y.Definition bool_fun_restrict (bf : bool_fun) (x : BDDvar) 
  (b : bool) (vb : var_binding) := bool_fun_eval bf (augment vb x b).

(* Specifications: *)
Lemma bool_fun_restrict_zero : forall (x : BDDvar) (b : bool), bool_fun_eq (bool_fun_restrict bool_fun_zero x b) bool_fun_zero.
Lemma bool_fun_restrict_one : forall (x : BDDvar) (b : bool), bool_fun_eq (bool_fun_restrict bool_fun_one x b) bool_fun_one.
Lemma bool_fun_restrict_eq : forall (bf bf' : bool_fun) (x : BDDvar) (b : bool), bool_fun_eq bf bf' -> bool_fun_eq (bool_fun_restrict bf x b) (bool_fun_restrict bf' x b).