(* 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.(* APPARTENANCE *)

Definition CIN : Ens -> Ens -> Type.(* INCLUSION *)

Definition CINC : Ens -> Ens -> Type.

Theorem Paire_sound_leftC : forall A A' B : Ens, EQC A A' -> EQC (Paire A B) (Paire A' B).