(* This program is free software; you can redistribute it and/or      *)
(* modify it under the terms of the GNU Lesser General Public License *)
(* as published by the Free Software Foundation; either version 2.1   *)
(* of the License, or (at your option) any later version.             *)
(*                                                                    *)
(* This program is distributed in the hope that it will be useful,    *)
(* but WITHOUT ANY WARRANTY; without even the implied warranty of     *)
(* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the      *)
(* GNU General Public License for more details.                       *)
(*                                                                    *)
(* You should have received a copy of the GNU Lesser General Public   *)
(* License along with this program; if not, write to the Free         *)
(* Software Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA *)
(* 02110-1301 USA                                                     *)
(* Contribution to the Coq Library   V6.3 (July 1999)                    *)
(* Peter Aczel's Encoding of CZF *)
(* Using the same definition "Ens" of sets, we can developp Peter Aczel's   *)
(* encoding of "Constructive Type Theory" (CZF).                            *)
(* It is basically a simillar developement, but this time, the propositions *)
(* are objects of type "Type", i.e. are on the same level (resp. above) the *)
(* sets. The advantage is that we can extract the constructive witness of an*)
(* existential proof. The drawbacks are:                                    *)
(*  - no definition of the powerset                                         *)
(*  - complicated difference between bounded and unbounded quantification   *)
(*  - excluded middle is now much more "dangerous"                          *)
Definition EQC : Ens -> Ens -> Type.
simple induction 1; intros A f eq1.
simple induction 1; intros B g eq2.
refine (prod_t _ _).
exact (forall x : A, depprod _ (fun y : B => eq1 x (g y))).
exact (forall y : B, depprod _ (fun x : A => eq1 x (g y))).
Defined.
(* APPARTENANCE *)
Definition CIN : Ens -> Ens -> Type.
simple induction 2.
intros.
exact (depprod _ (fun y : A => EQC X (e y))).
Defined.
(* INCLUSION *)
Definition CINC : Ens -> Ens -> Type.
intros E1 E2.
exact (forall E : Ens, CIN E E1 -> CIN E E2).
Defined.
(* EQ EST UNE RELATION D'EQUIVALENCE *)

Theorem EQC_refl : forall E : Ens, EQC E E.