simple induction 1.
simple induction 1; intros; auto with arith pts.
elim H2 with x1 y1; auto with arith pts.
simple destruct x0; auto with arith pts.
intros.
apply conv_hn_inv_trans with y0; auto with arith pts.
apply H3.
apply hn_red_hn with x0; auto with arith pts.
Qed.
Hint Resolve inv_red_hn: pts.

(* Specifications: *)
Lemma inv_conv_hn :
forall (e : env) (x y : term),
conv R e x y ->
forall x' y' : term,
red R e x x' ->
head_normal R e x' ->
red R e y y' -> head_normal R e y' -> conv_hn_inv e x' y'.