Proof with auto.
intros u e k Hexpr. revert k.
induction Hexpr; intro k; simpl; f_equal; auto*;
try solve [
unfold open_ee in *;
pick fresh x;
eapply open_ee_rec_expr_aux with (j := 0) (v := exp_fvar x);
auto
| unfold open_te in *;
pick fresh X;
eapply open_ee_rec_type_aux with (j := 0) (V := typ_fvar X);
auto
].
Qed.

Lemma subst_ee_fresh : forall (x: atom) u e,
x `notin` fv_ee e ->
e = subst_ee x u e.