Fixpoint subst_te (Z : atom) (U : typ) (e : exp) {struct e} : exp :=
  match e with
  | exp_bvar i => exp_bvar i
  | exp_fvar x => exp_fvar x
  | exp_abs V e1 => exp_abs  (subst_tt Z U V)  (subst_te Z U e1)
  | exp_app e1 e2 => exp_app  (subst_te Z U e1) (subst_te Z U e2)
  | exp_tabs V e1 => exp_tabs (subst_tt Z U V)  (subst_te Z U e1)
  | exp_tapp e1 V => exp_tapp (subst_te Z U e1) (subst_tt Z U V)
  | exp_let e1 e2 => exp_let (subst_te Z U e1) (subst_te Z U e2)
  | exp_inl e1 => exp_inl (subst_te Z U e1)
  | exp_inr e1 => exp_inr (subst_te Z U e1)
  | exp_case e1 e2 e3 => exp_case (subst_te Z U e1)
                                  (subst_te Z U e2) (subst_te Z U e3)
  end.Definition subst_tb (Z : atom) (P : typ) (b : binding) : binding :=
  match b with
  | bind_sub T => bind_sub (subst_tt Z P T)
  | bind_typ T => bind_typ (subst_tt Z P T)
  end.

Lemma subst_ee_open_te_rec : forall e P z u k, expr u -> subst_ee z u (open_te_rec k P e) = open_te_rec k P (subst_ee z u e).