(* ---------------------------------------------------------------------------- *)
(* [map] composes with itself. *)

Lemma map_lift_map_lift:
forall T k s wk ws (e : env T),
forall `{Lift T},
@LiftLift T _ ->
k <= s ->
map (lift wk k) (map (lift ws s) e) = map (lift ws (wk + s)) (map (lift wk k) e).