(** *** Exercise  Complete the proof of the substitution lemma stated in the form we need it.  The proof is similar to that of [typing_weakening].  HINT: You'll need to use [rewrite_env] to prepend [nil], and [simpl_env] to simplify nil away. *)

Lemma typing_c_subst_simple : forall (E : env) e u S T (z : atom),
typing_c ((z ~ S) ++ E) e T ->
typing_c E u S ->
typing_c E ([z ~> u] e) T.