Section Hurkens.
Variable down : Type -> Type.
Variable up : Type -> Type.
Hypothesis back : forall A, up (down A) -> A.
Hypothesis forth : forall A, A -> up (down A).
Hypothesis backforth : forall (A:Type) (P:A->Type) (a:A),
P (back A (forth A a)) -> P a.
Hypothesis backforth_r : forall (A:Type) (P:A->Type) (a:A),
P a -> P (back A (forth A a)).
(** Proof *)
Definition V : Type := forall A:Type, ((up A -> Type) -> up A -> Type) -> up A -> Type.
Definition U : Type := V -> Type.
Definition sb (z:V) : V := fun A r a => r (z A r) a.
Definition le (i:U -> Type) (x:U) : Type := x (fun A r a => i (fun v => sb v A r a)).
Definition le' (i:up (down U) -> Type) (x:up (down U)) : Type := le (fun a:U => i (forth _ a)) (back _ x).
Definition induct (i:U -> Type) : Type := forall x:U, up (le i x) -> up (i x).
Definition WF : U := fun z => down (induct (fun a => z (down U) le' (forth _ a))).
Definition I (x:U) : Type :=
(forall i:U -> Type, up (le i x) -> up (i (fun v => sb v (down U) le' (forth _ x)))) -> False.

Lemma Omega : forall i:U -> Type, induct i -> up (i WF).