Section Properties.
Context {A: Type}.
Definition finite_captured (P: Ensemble A -> Prop): Prop :=
forall (Phi: Ensemble A),
(forall xs: list A, Forall Phi xs -> P (fun x => In x xs)) ->
P Phi.
Definition finite_witnessed (P: Ensemble A -> Prop): Prop :=
forall (Phi: Ensemble A), P Phi ->
exists xs: list A, Forall Phi xs /\ P (fun x => In x xs).
Definition subset_preserved (P: Ensemble A -> Prop): Prop :=
forall (Phi Psi: Ensemble A),
Included _ Phi Psi -> P Psi -> P Phi.
Definition superset_preserved (P: Ensemble A -> Prop): Prop :=
forall (Phi Psi: Ensemble A),
Included _ Phi Psi -> P Phi -> P Psi.

(* Specifications: *)
Lemma not_finite_witnessed_finite_captured: forall P,
finite_witnessed P ->
finite_captured (fun X => ~ P X).
Lemma not_superset_preserved_subset_preserved: forall P,
superset_preserved P ->
subset_preserved (fun X => ~ P X).
Lemma superset_preserved_same_set_preserved: forall P,
superset_preserved P ->
Proper (Same_set _ ==> iff) P.
Lemma subset_preserved_same_set_preserved: forall P,
subset_preserved P ->
Proper (Same_set _ ==> iff) P.