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.

(* Specifications: *)
Lemma preds_fmap_NoneP: forall f, preds_fmap f NoneP = NoneP.
Lemma empty_NO: forall r, identity r -> r = NO \/ exists k, exists pds, r = PURE k pds.
Lemma level_age_fash: forall m m': rmap, level m = S (level m') -> exists m1, age m m1.
Lemma level_later_fash: forall m m': rmap, (level m > level m')%nat -> exists m1, laterR m m1 /\ level m1 = level m'.