Proof with congruence || eauto.
induction e; intros j v u i Neq H; simpl in *; inversion H; f_equal...
Case "exp_bvar".
destruct (j===n)... destruct (i===n)...
Qed.

(* Specifications: *)
Lemma open_ee_rec_type_aux : forall e j V u i,
open_te_rec j V e = open_ee_rec i u (open_te_rec j V e) ->
e = open_ee_rec i u e.
Lemma open_ee_rec_expr : forall u e k,
expr e ->
e = open_ee_rec k u e.