simple induction 1; intros; auto with arith pts.
inversion_clear H1.
apply cuhi_srt.
red in |- *.
apply rt_trans with s'; auto with arith pts.
inversion_clear H2.
apply cuhi_abs.
apply conv_trans with A'; auto with arith pts.
apply conv_trans with M'; auto with arith pts.
apply (stable_conv (conv hdr) (Ax A' :: e0)); auto 10 with arith pts.
inversion_clear H2.
apply cuhi_app.
apply conv_trans with u'; auto with arith pts.
apply conv_trans with v'; auto with arith pts.
inversion_clear H2.
apply cuhi_prod.
apply cumul_trans with A'; auto with arith pts.
apply cumul_trans with B'; auto with arith pts.
apply
stable_rt_cumul with (fun e : env => transp _ (rt_cumul e)) (Ax A' :: e0);
auto with arith pts.
Qed.
Section EquivCumulInv.

(* Specifications: *)
Lemma inv_cumul_cumul :
forall (e : env) (x y : term), cumul_hn_inv e x y -> rt_cumul e x y.