intros P z H H0 x; induction 1 as [|x y z]; auto.
apply H0 with y; auto.
Qed.

(* Specifications: *)
Lemma clos_refl_trans_ind_right : forall (P : A -> Prop) (z:A),
P z ->
(forall x y, R x y -> P y -> clos_refl_trans R y z -> P x) ->
forall x, clos_refl_trans R x z -> P x.