Section Lindenbaum.
Context {A: Type}
(CA: Countable A)
(init: Ensemble A)
(P: Ensemble A -> Prop).
Fixpoint LindenbaumChain (n: nat): Ensemble A :=
match n with
| 0 => init
| S n => fun a => LindenbaumChain n a \/ CA a n /\ P (Union _ (LindenbaumChain n) (Singleton _ a))
end.
Definition LindenbaumConstruction: Ensemble A :=
fun a => exists n, LindenbaumChain n a.
Hypothesis H_init: P init.
End Lindenbaum.
Definition Lindenbaum_preserves {A: Type} (P: Ensemble A -> Prop): Prop :=
forall (CA: Countable A) init, P init -> P (LindenbaumConstruction CA init P).
Definition Lindenbaum_ensures {A: Type} (P cP: Ensemble A -> Prop): Prop :=
forall (CA: Countable A) init, P init -> cP (LindenbaumConstruction CA init P).
Definition Lindenbaum_constructable {A: Type} (P cP: Ensemble A -> Prop): Prop :=
forall Phi, P Phi -> exists Psi: sig cP, Included _ Phi (proj1_sig Psi) /\ P (proj1_sig Psi).

Lemma Lindenbaum_preserves_by_finiteness {A: Type}: forall P: Ensemble A -> Prop,
finite_captured P ->
subset_preserved P ->
Lindenbaum_preserves P.