Definition iota (P : T -> Prop) := lim (fun A => (forall x, P x -> A x)).

(* Specifications: *)
Lemma iota_correct_weak :
forall P : T -> Prop,
(forall x y, P x -> P y -> close x y) ->
forall x, P x -> close (iota P) x.
Lemma close_iota :
forall P Q : T -> Prop,
(forall x, P x <-> Q x) ->
close (iota P) (iota Q).