(*
  Definition saKey := sa_equiv Key.
  Definition saKeyA := sa_prod saKey saA.
  Definition saKeyAList := sa_list saKeyA.
  Definition saSum := sa_sum saA sa_unit.
  Definition saRange := sa_bijection _ _ (bij_sa_sum_option A) _ saSum.
*)

  Fixpoint lookup k (rho: list (Key*A)) :=
    match rho with
      | nil => None
      | (k', a) :: rho' =>
          if Key_dec_eq k k' then Some a else lookup k rho'
    end.Instance Join_Key : Join Key := @Join_equiv Key.

Lemma join_hom_join_sub {A}{B}`{Join A}`{Join B}: forall (f: A -> B) a1 a2, join_sub a1 a2 -> join_hom f -> join_sub (f a1) (f a2).