(* ---------------------------------------------------------------------------- *)
(* Working with closedness. *)
(* Again, we prove the following lemmas only in order to illustrate the use of the tactic [inversion_closed]. *)

(* Specifications: *)
Lemma inversion_closed_TVar:
forall k x,
x >= k ->
closed k (TVar x) ->
False.
Lemma inversion_closed_TApp_1:
forall t1 t2 k,
closed k (TApp t1 t2) ->
closed k t1.
Lemma inversion_closed_TApp_2:
forall t1 t2 k,
closed k (TApp t1 t2) ->
closed k t2.
Lemma inversion_closed_TAbs:
forall t k,
closed k (TAbs t) ->
closed (1 + k) t.