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(* 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 *)
simple induction E.
intros A f HR.
simpl in |- *.
split; intros.
exists x; auto with zfc.
exists y; auto with zfc.
Qed.
simple induction E1; simple induction E2; simple induction E3; simpl in |- *;
intros.
split; (elim X2; intros; elim X3; intros).
elim (a x); intros.
elim (a0 x0); intros.
exists x1.
apply X with (e0 x0); auto with zfc.
elim (b0 y); intros.
elim (b x); intros.
exists x0.
apply X with (e0 x); auto with zfc.
Qed.
simple induction E1; simple induction E2; simpl in |- *; intros.
elim X1; intros; split; intros.
elim (b x); intros.
exists x0; auto with zfc.
elim (a y); intros; exists x; auto with zfc.
Qed.
simple induction E; simple induction E'; simpl in |- *; intros;
unfold CINC in |- *; simpl in |- *.
elim X1; intros.
elim X2; intros.
elim (a x); intros.
exists x0; apply EQC_tran with (e x); auto with zfc.
Qed.
Hint Resolve EQC_sym EQC_refl EQC_INC: zfc.
simple induction E; simple induction E'; unfold CINC in |- *; simpl in |- *;
intros; split; intros.
apply X1.
exists x; auto with zfc.
cut (depprod A (fun x : A => EQC (e0 y) (e x)));
try (simple induction 1; intros x p; exists x; auto with zfc).
apply X2; exists y; auto with zfc.
Qed.
Hint Resolve CINC_EQC: zfc.
simple induction E''; simpl in |- *; intros.
elim X1; intros y p; exists y.
apply EQC_tran with E; auto with zfc.
Qed.
simple induction E'; simple induction E''; simpl in |- *; intros.
elim X1; intros Xl Xr; elim X2; intros y p; elim (Xl y); intros y0 p0;
exists y0; apply EQC_tran with (e y); auto with zfc.
Qed.
unfold CINC in |- *; auto with zfc.
Qed.
unfold CINC in |- *; auto with zfc.
Qed.

Theorem CINC_sound_left :
forall E E' E'' : Ens, EQC E E' -> CINC E E'' -> CINC E' E''.