simple induction 1; intros; auto with arith pts.
apply chi_abs; auto with arith pts.
apply stable_conv with (fun (_ : env) (_ x : term) => x = A) (Ax A' :: e0);
auto with arith pts.
apply chi_prod; auto with arith pts.
apply stable_conv with (fun (_ : env) (_ x : term) => x = A) (Ax A' :: e0);
auto with arith pts.
Qed.

Lemma conv_hn_inv_trans :
forall (e : env) (x y z : term),
conv_hn_inv e x y -> conv_hn_inv e y z -> conv_hn_inv e x z.