induction 1; intuition; simpl in *.
apply rat0_le_all.
rewrite <- IHdat_better1.
rewrite <- IHdat_better2.
eapply leRat_refl.
Qed.

(* Specifications: *)
Lemma datMap_better : forall (A B : Set)(t : DistApproxTree B) n (rel : nat -> B -> DistApproxTree A -> Prop) t1 t2,
(forall n b t1 t2, rel n b t1 -> rel (pred n) b t2 -> dat_better t1 t2) ->
datMap (rel n) t t1 ->
datMap (rel (pred n)) t t2 ->
dat_better t1 t2.