(* 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_flatten_eq : forall (A B C : Set)(eqda : EqDec A)(eqdb : EqDec B)(ls : list (list (A * B * C))) keys a b, list_pred (fun a b => forall x, In x b -> fst (fst x) = a) keys ls -> NoDup keys -> arrayLookup _ (flatten ls) (a, b) = arrayLookup _ (arrayLookupList _ (combine keys ls) a) (a, b).