Proof with auto.
intros E S1 e1 T Typ.
remember (exp_abs S1 e1) as e.
generalize dependent e1.
generalize dependent S1.
induction Typ; intros S1 b1 EQ U1 U2 Sub; inversion EQ; subst.
Case "typing_abs".
inversion Sub; subst.
split...
exists T1. exists L...
Case "typing_sub".
auto using (sub_transitivity T).
Qed.

(* Specifications: *)
Lemma typing_inv_tabs : forall E S1 e1 T,
typing E (exp_tabs S1 e1) T ->
forall U1 U2, sub E T (typ_all U1 U2) ->
sub E U1 S1
/\ exists S2, exists L, forall X, X `notin` L ->
typing (X ~ bind_sub U1 ++ E) (open_te e1 X) (open_tt S2 X)
/\ sub (X ~ bind_sub U1 ++ E) (open_tt S2 X) (open_tt U2 X).