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-Section Hurkens.
-
-Definition Type2 := Type.
-Definition Type1 := Type : Type2.
-
-(** Assumption of a retract from Type into Prop *)
-
-Variable down : Type1 -> Prop.
-Variable up : Prop -> Type1.
-
-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 : Type1 := forall A:Prop, ((up A -> Prop) -> up A -> Prop) -> up A -> Prop.
-Definition U : Type1 := V -> Prop.
-
-Definition sb (z:V) : V := fun A r a => r (z A r) a.
-Definition le (i:U -> Prop) (x:U) : Prop := x (fun A r a => i (fun v => sb v A r a)).
-Definition le' (i:up (down U) -> Prop) (x:up (down U)) : Prop := le (fun a:U => i (forth _ a)) (back _ x).
-Definition induct (i:U -> Prop) : Type1 := 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) : Prop :=
-  (forall i:U -> Prop, up (le i x) -> up (i (fun v => sb v (down U) le' (forth _ x)))) -> False.
-
-Lemma Omega : forall i:U -> Prop, induct i -> up (i WF).
-Proof.
-intros i y.
-apply y.
-unfold le, WF, induct.
-apply forth.
-intros x H0.
-apply y.
-unfold sb, le', le.
-compute.
-apply backforth_r.
-exact H0.
-Qed.
-
-Lemma lemma1 : induct (fun u => down (I u)).
-Proof.
-unfold induct.
-intros x p.
-apply forth.
-intro q.
-generalize (q (fun u => down (I u)) p).
-intro r.
-apply back in r.
-apply r.
-intros i j.
-unfold le, sb, le', le in j |-.
-apply backforth in j.
-specialize q with (i := fun y => i (fun v:V => sb v (down U) le' (forth _ y))).
-apply q.
-exact j.
-Qed.
-
-Lemma lemma2 : (forall i:U -> Prop, induct i -> up (i WF)) -> False.
-Proof.
-intro x.
-generalize (x (fun u => down (I u)) lemma1).
-intro r; apply back in r.
-apply r.
-intros i H0.
-apply (x (fun y => i (fun v => sb v (down U) le' (forth _ y)))).
-unfold le, WF in H0.
-apply back in H0.
-exact H0.
-Qed.
-
-Theorem paradox : False.
-Proof.
-exact (lemma2 Omega).
-Qed.
-
-End Hurkens.
-
-Definition informative (x : bool) :=
-  match x with
-  | true => Type
-  | false => Prop
-  end.
-
-Definition depsort (T : Type) (x : bool) : informative x :=
-  match x with
-  | true => T
-  | false => True
-  end.
-
-(* The projection prop should not be definable *)
-Set Primitive Projections.
-Record Box (T : Type) : Prop := wrap {prop : T}.
-
-Definition down (x : Type) : Prop := Box x.
-Definition up (x : Prop) : Type := x.
-
-Definition back A : up (down A) -> A := @prop A.
-
-Definition forth A : A -> up (down A) := wrap A.
-
-Definition backforth (A:Type) (P:A->Type) (a:A) :
-      P (back A (forth A a)) -> P a := fun H => H.
-
-Definition backforth_r (A:Type) (P:A->Type) (a:A) :
-      P a -> P (back A (forth A a)) := fun H => H.
-
-Theorem pandora : False.
-apply (paradox down up back forth backforth backforth_r).
-Qed.
-
-Print Assumptions pandora.
-(* Closed under the global context *)