simple induction 1; intros; auto with arith pts.
inversion_clear H2.
apply chi_abs.
apply conv_trans with A'; auto with arith pts.
apply conv_trans with M'; auto with arith pts.
apply stable_conv with (fun (_ : env) (_ x : term) => x = A'0) (Ax A' :: e0);
auto with arith pts.
inversion_clear H2.
apply chi_app.
apply conv_trans with u'; auto with arith pts.
apply conv_trans with v'; auto with arith pts.
inversion_clear H2.
apply chi_prod.
apply conv_trans with A'; auto with arith pts.
apply conv_trans with B'; auto with arith pts.
apply stable_conv with (fun (_ : env) (_ x : term) => x = A'0) (Ax A' :: e0);
auto with arith pts.
Qed.

Lemma inv_red_hn :
forall (e : env) (x y : term),
red R e x y -> head_normal R e x -> conv_hn_inv e x y.