End join_hom_fun.
Arguments lookup [Key A] _ _ _.

(* Some simple properties preserved by join homs *)
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).