red in |- *; intros H'; inversion H'.
intros a l H'; red in |- *; intros H'0; simple inversion H'0.
discriminate H.
injection H1.
injection H0.
intros H'1 H'2 H'3 H'4; rewrite H'2; rewrite H'4; intros H'5.
apply (ltT_not_refl a); auto.
injection H1; injection H2.
intros H'1 H'2 H'3 H'4; rewrite H'1; rewrite H'3; auto.
Qed.
Hint Resolve ltPO.

Theorem ltP_trans : forall x y z, ltP x y -> ltP y z -> ltP x z.
intros x y z H'; generalize z; clear z; elim H'.