Proof with eauto 6 using wf_env_narrowing, wf_typ_narrowing, sub_narrowing.
intros Q E F X P e T PsubQ Typ.
remember (F ++ X ~ bind_sub Q ++ E) as E'.
generalize dependent F.
induction Typ; intros F EQ; subst...
Case "typing_var".
analyze_binds H0...
Case "typing_abs".
pick fresh y and apply typing_abs.
rewrite <- app_assoc.
apply H0...
Case "typing_tabs".
pick fresh Y and apply typing_tabs.
rewrite <- app_assoc.
apply H0...
Case "typing_let".
pick fresh y and apply typing_let...
rewrite <- app_assoc.
apply H0...
Case "typing_case".
pick fresh y and apply typing_case...
SCase "inl branch".
rewrite <- app_assoc.
apply H0...
SCase "inr branch".
rewrite <- app_assoc.
apply H2...
Qed.
(************************************************************************ *)
(** ** Substitution preserves typing (8) *)

Lemma typing_through_subst_ee : forall U E F x T e u,
typing (F ++ x ~ bind_typ U ++ E) e T ->
typing E u U ->
typing (F ++ E) (subst_ee x u e) T.