Ltac inj_pair_tac :=
 match goal with H: (@existT ?U ?P ?p ?x = @existT _ _ _ ?y) |- _ =>
   generalize (@inj_pair2 U P p x y H); clear H; intro; try (subst x || subst y)
 end.

Lemma preds_fmap_NoneP: forall f, preds_fmap f NoneP = NoneP.