Section CofiniteQuantification.
Local Notation "{ k ~> u } t" := (open_rec k u t) (at level 67).
(* With this new definition, we can almost use the same proof for [open_rec_lc], we only need one change. We need to add the line [pick fresh x for L.]  immediately before the use of [apply open_rec_lc_core].  This tactic, defined in [Metatheory], introduces a new atom [x] that is known not to be in the set [L]. *)
(* Take-home Exercise: The next two lemmas have exactly the same proofs as before. *)

Lemma subst_open_var_c : forall (x y : atom) u e,
y <> x ->
lc_c u ->
open ([x ~> u] e) y = [x ~> u] (open e y).