Added a proof that Prop<>Type, for arbitrary fixed Type.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15973 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 128 insertions, 0 deletions

diff --git a/theories/Logic/UniversesFacts.v b/theories/Logic/UniversesFacts.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, *   INRIA - CNRS - LIX - LRI - PPS - Copyright 1999-2012     *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(** A proof of the inconsistency of Prop=Type (for some fixed Type) in
+    Coq using Hurkens' paradox for system Type:Type [Hurkens].
+
+    Adapted from an initial formulation by Herman Geuvers [Geuvers] (this
+    formulation was used to show the inconsistency in the pure
+    calculus of constructions of a retract from Prop into a small
+    type, see Hurkens.v).
+
+    - [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 http://www.cs.kun.nl/~herman/note.ps.gz).
+*)
+
+Section Paradox.
+
+Definition Type2 := Type.
+Definition Type1 := Type : Type2.
+
+(** Preliminary *)
+
+Notation "'rew2' <- H 'in' H'" := (@eq_rect_r Type2 _ (fun X : Type2 => X) H' _ H)
+    (at level 10, H' at level 10).
+Notation "'rew2' H 'in' H'" := (@eq_rect Type2 _ (fun X : Type2 => X) H' _ H)
+    (at level 10, H' at level 10).
+Notation "'rew1' <- H 'in' H'" := (@eq_rect_r Type1 _ (fun X : Type1 => X) H' _ H)
+    (at level 10, H' at level 10).
+Notation "'rew1' H 'in' H'" := (@eq_rect Type1 _ (fun X : Type1 => X) H' _ H)
+    (at level 10, H' at level 10).
+
+Lemma rew_rew : forall (A B:Type1) (H:B=A) (x:A), rew1 H in rew1 <- H in x = x.
+Proof.
+intros.
+destruct H.
+reflexivity.
+Defined. 
+
+(** Main assumption and proof *)
+
+Variable Heq : Prop = Type1 :> Type2.
+
+Definition down : Type1 -> Prop := fun A => rew2 <- Heq in A.
+Definition up : Prop -> Type1 := fun A => rew2 Heq in A.
+
+Lemma up_down : forall (A:Type1), up (down A) = A :> Type1.
+Proof.
+unfold up, down. rewrite Heq. reflexivity.
+Defined.
+
+Definition V : Type1 := forall A:Prop, ((up A -> Prop) -> up A -> Prop) -> up A -> Prop.
+Definition U : Type1 := V -> Prop.
+
+Definition forth (a:U) : up (down U) := rew1 <- (up_down U) in a.
+Definition back (x:up (down U)) : U := rew1 (up_down U) in x. 
+
+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 back_forth (a:U) : back (forth a) = a.
+Proof.
+apply rew_rew.
+Defined.
+
+Lemma Omega : forall i:U -> Prop, induct i -> up (i WF).
+Proof.
+intros i y.
+apply y.
+unfold le, WF, induct.
+rewrite up_down.
+intros x H0.
+apply y.
+unfold sb, le', le.
+rewrite back_forth.
+exact H0.
+Qed.
+
+Lemma lemma1 : induct (fun u => down (I u)).
+Proof.
+unfold induct.
+intros x p.
+rewrite up_down.
+intro q.
+generalize (q (fun u => down (I u)) p).
+rewrite up_down.
+intro r.
+apply r.
+intros i j.
+unfold le, sb, le', le in j |-.
+rewrite back_forth 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).
+rewrite up_down.
+intro 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.
+rewrite up_down in H0.
+exact H0.
+Qed.
+
+Theorem paradox : False.
+Proof.
+exact (lemma2 Omega).
+Qed.
+
+End Paradox.