(* Exercise [subst_lc_c]:  Once we have changed the definition of lc, we can complete the proof of subst_lc.  HINT: apply lc_abs_c with cofinite set (L `union` singleton x). This gives us an atom x0, and a hypothesis that x0 is fresh for both L and x. *)

Lemma subst_lc_c : forall (x : atom) u e,
lc_c e ->
lc_c u ->
lc_c ([x ~> u] e).