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-(************************************************************************)
-(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
-(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
-(*   \VV/  **************************************************************)
-(*    //   *      This file is distributed under the terms of the       *)
-(*         *       GNU Lesser General Public License Version 2.1        *)
-(************************************************************************)
-(*                              Hurkens.v                               *)
-(************************************************************************)
-
-(** This is Hurkens paradox [Hurkens] in system U-, adapted by Herman
-    Geuvers [Geuvers] to show the inconsistency in the pure calculus of
-    constructions of a retract from Prop into a small type.
-
-    References:
-
-    - [Hurkens] A. J. Hurkens, "A simplification of Girard's paradox",
-      Proceedings of the 2nd international conference Typed Lambda-Calculi
-      and Applications (TLCA'95), 1995.
-
-    - [Geuvers] "Inconsistency of Classical Logic in Type Theory", 2001
-      (see www.cs.kun.nl/~herman/note.ps.gz).
-*)
-
-Section Paradox.
-
-Variable bool : Prop.
-Variable p2b : Prop -> bool.
-Variable b2p : bool -> Prop.
-Hypothesis p2p1 : (A:Prop)(b2p (p2b A))->A.
-Hypothesis p2p2 : (A:Prop)A->(b2p (p2b A)).
-Variable B:Prop.
-
-Definition V := (A:Prop)((A->bool)->(A->bool))->(A->bool).
-Definition U := V->bool.
-Definition sb : V -> V := [z][A;r;a](r (z A r) a).
-Definition le : (U->bool)->(U->bool) := [i][x](x [A;r;a](i [v](sb v A r a))).
-Definition induct : (U->bool)->Prop := [i](x:U)(b2p (le i x))->(b2p (i x)).
-Definition WF : U := [z](p2b (induct (z U le))).
-Definition I : U->Prop :=
-  [x]((i:U->bool)(b2p (le i x))->(b2p (i [v](sb v U le x))))->B.
-
-Lemma Omega : (i:U->bool)(induct i)->(b2p (i WF)).
-Proof.
-Intros i y.
-Apply y.
-Unfold le WF induct.
-Apply p2p2.
-Intros x H0.
-Apply y.
-Exact H0.
-Qed.
-
-Lemma lemma1 : (induct [u](p2b (I u))).
-Proof.
-Unfold induct.
-Intros x p.
-Apply (p2p2 (I x)).
-Intro q.
-Apply (p2p1 (I [v:V](sb v U le x)) (q [u](p2b (I u)) p)).
-Intro i.
-Apply q with i:=[y:?](i [v:V](sb v U le y)).
-Qed.
-
-Lemma lemma2 : ((i:U->bool)(induct i)->(b2p (i WF)))->B.
-Proof.
-Intro x.
-Apply (p2p1 (I WF) (x [u](p2b (I u)) lemma1)).
-Intros i H0.
-Apply (x [y](i [v](sb v U le y))).
-Apply (p2p1 ? H0).
-Qed.
-
-Theorem paradox : B.
-Proof.
-Exact (lemma2 Omega).
-Qed.
-
-End Paradox.