Proof using.
induction ns; intros; simpl in *; auto.
set (f' := update2 _ _ _ _ _).
set (g' := update2 _ _ _ _ _).
rewrite (IHns f' g'); auto.
unfold f', g', update2.
break_if; auto.
break_and.
subst.
find_rewrite.
trivial.
Qed.

(* Specifications: *)
Lemma collate_ls_neq_update2 :
forall ns (f : A -> A -> list B) n h h' ms mg,
n <> h' ->
collate_ls A_eq_dec ns (update2 A_eq_dec f n h ms) h mg h' h = collate_ls A_eq_dec ns f h mg h' h.