Definition sub_silhouette {A} `{Perm_alg A} (a b: A) : Prop :=
forall c, joins c b -> joins c a.
Definition same_silhouette {A} `{Perm_alg A} (a b: A) : Prop :=
forall c, joins c a <-> joins c b.

Lemma sub_silhouette_refl {A} `{Perm_alg A}: forall a, sub_silhouette a a.