simple induction 1; intros; auto 10 with arith pts.
Qed.

Lemma equiv_inv_cumul_hn :
forall (e : env) (x y : term),
rt_cumul e x y ->
forall x' y' : term,
red hdr e x x' ->
head_normal hdr e x' ->
red hdr e y y' -> head_normal hdr e y' -> cumul_hn_inv e x' y'.