(** Elements [a] and [b] join. *)
  Definition joins {A} {J: Join A} (a b : A) : Prop :=
    exists c, join a b c.Definition overlap {A}{J: Join A} (a b: A) := ~(joins a b).

Lemma joins_sym {A} {J: Join A} {PA: Perm_alg A}: forall a b, joins a b <-> joins b a.