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]. *)

Lemma open_rec_lc_c : forall k u e,
lc_c e ->
e = {k ~> u} e.