(* Take-home Exercise: We've proven the above lemma very explicitly, so that you can step through it slowly to see how it works. However, with automation, it is possible to give a *much* shorter proof. Reprove this lemma on your own to see how compact you can make it. *)
(** With the help of this lemma, we can complete the proof. *)

Lemma open_rec_lc : forall k u e,
lc e -> e = {k ~> u} e.