Definition same_constructive_silhouette {A} {JOIN: Join A} (a b: A) :=
forall c, (constructive_joins c a -> constructive_joins c b) *
(constructive_joins c b -> constructive_joins c a).
Definition sub_constructive_silhouette {A} {JOIN: Join A}  (a b: A) :=
forall c, constructive_joins c b -> constructive_joins c a.

Lemma sub_constructive_silhouette_refl {A} {JOIN: Join A} : forall a, sub_constructive_silhouette a a.