Proof with
simpl_env;
eauto 4 using wf_typ_subst_tb, wf_env_subst_tb, wf_typ_weaken_head.
intros Q E F Z S T P SsubT PsubQ.
remember (F ++ Z ~ bind_sub Q ++ E) as G.
generalize dependent F.
induction SsubT; intros G EQ; subst; simpl subst_tt...
Case "sub_top".
apply sub_top...
Case "sub_refl_tvar".
destruct (X == Z); subst.
SCase "X = Z".
apply sub_reflexivity...
SCase "X <> Z".
apply sub_reflexivity...
inversion H0; subst.
analyze_binds H3...
apply (wf_typ_var (subst_tt Z P U))...
Case "sub_trans_tvar".
destruct (X == Z); subst.
SCase "X = Z".
apply (sub_transitivity Q).
SSCase "left branch".
rewrite_env (empty ++ map (subst_tb Z P) G ++ E).
apply sub_weakening...
SSCase "right branch".
rewrite (subst_tt_fresh Z P Q).
analyze_binds_uniq H.
inversion BindsTacVal; subst...
apply (notin_fv_wf E); eauto using fresh_mid_tail.
SCase "X <> Z".
apply (sub_trans_tvar (subst_tt Z P U))...
rewrite (map_subst_tb_id E Z P);
[ | auto | eapply fresh_mid_tail; eauto ].
analyze_binds H...
Case "sub_all".
pick fresh X and apply sub_all...
rewrite subst_tt_open_tt_var...
rewrite subst_tt_open_tt_var...
rewrite_env (map (subst_tb Z P) (X ~ bind_sub T1 ++ G) ++ E).
apply H0...
Qed.
(* ********************************************************************** *)
(** * #<a name="typing"></a># Properties of typing *)
(* ********************************************************************** *)
(** ** Weakening (5) *)

(* Specifications: *)
Lemma typing_weakening : forall E F G e T,
typing (G ++ E) e T ->
wf_env (G ++ F ++ E) ->
typing (G ++ F ++ E) e T.