Proof with simpl_env;
eauto using wf_typ_weakening,
wf_typ_from_wf_env_typ,
wf_typ_from_wf_env_sub,
sub_weakening.
intros E F G e T Typ.
remember (G ++ E) as H.
generalize dependent G.
induction Typ; intros G EQ Ok; subst...
Case "typing_abs".
pick fresh x and apply typing_abs.
lapply (H x); [intros K | auto].
rewrite <- app_assoc.
apply (H0 x)...
Case "typing_tabs".
pick fresh X and apply typing_tabs.
lapply (H X); [intros K | auto].
rewrite <- app_assoc.
apply (H0 X)...
Case "typing_let".
pick fresh x and apply typing_let...
lapply (H0 x); [intros K | auto].
rewrite <- app_assoc.
apply H0...
Case "typing_case".
pick fresh x and apply typing_case...
SCase "inl branch".
lapply (H0 x); [intros K | auto].
rewrite <- app_assoc.
apply H0...
assert (J : wf_typ (G ++ F ++ E) (typ_sum T1 T2))...
inversion J...
SCase "inr branch".
lapply (H2 x); [intros K | auto].
rewrite <- app_assoc.
apply H2...
assert (J : wf_typ (G ++ F ++ E) (typ_sum T1 T2))...
inversion J...
Qed.
(* ********************************************************************** *)
(** ** Strengthening (6) *)

(* Specifications: *)
Lemma sub_strengthening : forall x U E F S T,
sub (F ++ x ~ bind_typ U ++ E) S T ->
sub (F ++ E) S T.