End join_hom_list.
(* FPMs This section proves a join hom lemma specialized to finite partial maps producing [option]s. *)
Section join_hom_fun.
Variables (Key A: Type)
(Key_dec_eq: forall k1 k2: Key, {k1=k2}+{~k1=k2})
(JA: Join A).
(* 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.

(* Specifications: *)
Lemma join_hom_fun
: join_hom (fun env k => lookup k env).