Proof with auto.
intros x u e; induction e; simpl; intro H; f_equal...
Case "exp_fvar".
destruct (a==x)...
contradict H; fsetdec.
Qed.

Lemma subst_ee_open_ee_rec : forall e1 e2 x u k,
expr u ->
subst_ee x u (open_ee_rec k e2 e1) =
open_ee_rec k (subst_ee x u e2) (subst_ee x u e1).