(* Some notation for array lookups *)
Notation "f '#' i " := (arrayLookup _ f i) (at level 70) : array_scope.(* A special case for array lookup when the range is lists -- returns nil instead of None when not found.  This is useful for multidimensional arrays. *)
Definition arrayLookupList(A B : Set)(eqd : EqDec A)(ls : list (A * (list B)))(a : A) :=
    match (arrayLookup _ ls a) with
      | None => nil
      | Some ls => ls
    end.Fixpoint arrayLookupOpt(A B : Set)(eqd : EqDec A)(ls : list (option (A * B)))(a : A) :=
  match ls with
    | nil => None
      | o :: ls' =>
        match o with
          | None => arrayLookupOpt _ ls' a
          | Some (a', b') =>
            if (eqb a a') then (Some b') else (arrayLookupOpt _ ls' a)
        end
  end.

Theorem arrayLookup_pred_2 : forall (A B C: Set)(eqda : EqDec A) (P : list B -> list C -> Prop) (ls1 : list (A * list B))(ls2 : list (A * list C)) a, list_pred (fun x y => (fst x = a \/ fst y = a) -> (fst x = a /\ fst y = a /\ P (snd x) (snd y))) ls1 ls2 -> P nil nil -> P (arrayLookupList _ ls1 a) (arrayLookupList _ ls2 a).