End NoRetractToImpredicativeUniverse.
(** * Modal fragments of [Prop] are not retracts *)
(** In presence of a a monadic modality on [Prop], we can define a subset of [Prop] of modal propositions which is also a complete Heyting algebra. These cannot be a retract of a modal proposition. This is a case where the universe in system U- are not encoded as Coq universes. *)
Module NoRetractToModalProposition.
(** ** Monadic modality *)
Section Paradox.
Variable M : Prop -> Prop.
Hypothesis incr : forall A B:Prop, (A->B) -> M A -> M B.

(* Specifications: *)
Lemma strength: forall A (P:A->Prop), M(forall x:A,P x) -> forall x:A,M(P x).
Definition MProp := { P:Prop | M P -> P }.