Proof with simpl_env; auto.
unfold transitivity_on.
intros Q E S T SsubQ QsubT.
assert (W : type Q) by auto.
generalize dependent T.
generalize dependent S.
generalize dependent E.
remember Q as Q' in |- *.
generalize dependent Q'.
induction W;
intros Q' EQ E S SsubQ;
induction SsubQ; try discriminate; inversion EQ; subst;
intros T' QsubT;
inversion QsubT; subst; eauto 4 using sub_trans_tvar.
Case "sub_all / sub_top".
assert (sub E (typ_all S1 S2) (typ_all T1 T2)).
SCase "proof of assertion".
pick fresh y and apply sub_all...
auto.
Case "sub_all / sub_all".
pick fresh Y and apply sub_all.
SCase "bounds".
eauto.
SCase "bodies".
lapply (H0 Y); [ intros K | auto ].
apply (K (open_tt T2 Y))...
rewrite_env (empty ++ Y ~ bind_sub T0 ++ E).
apply (sub_narrowing_aux T1)...
unfold transitivity_on.
auto using (IHW T1).
Qed.

(* Specifications: *)
Lemma sub_narrowing : forall Q E F Z P S T,
sub E P Q ->
sub (F ++ Z ~ bind_sub Q ++ E) S T ->
sub (F ++ Z ~ bind_sub P ++ E) S T.