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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.
simple induction E''; unfold CINC in |- *; simpl in |- *;
intros A f XR e X1 E0 i; apply X1.
apply CIN_sound_right with E'; auto with zfc.
Qed.
simple induction E'; simple induction E''; unfold CINC in |- *; simpl in |- *;
intros.
elim (X2 E0); try assumption; intros.
elim X1; intros XA XB; elim (XA x); intros.
exists x0; apply EQC_tran with (e x); auto with zfc.
Qed.
unfold Vide in |- *; simple induction E; simpl in |- *; intros A e X H;
split.
intros; elim (H (e x)); auto with zfc.
exists x; auto with zfc.
simple induction y.
Qed.
unfold Paire in |- *.
simpl in |- *.
intros; split.
simple induction x.
exists true; auto with zfc.
exists false; auto with zfc.
simple induction y; simpl in |- *.
exists true; auto with zfc.
exists false; auto with zfc.
Qed.
unfold Paire in |- *; simpl in |- *; intros; split.
simple induction x.
exists true; auto with zfc.
exists false; auto with zfc.
simple induction y.
exists true; auto with zfc.
exists false; auto with zfc.
Qed.
unfold Paire in |- *; simpl in |- *; exists true; simpl in |- *;
auto with zfc.
Qed.
unfold Paire in |- *; simpl in |- *; exists false; simpl in |- *;
auto with zfc.
Qed.
Inductive sum_t (A B : Type) : Type :=
| inl_t : A -> sum_t A B
| inr_t : B -> sum_t A B.
Hint Resolve inl_t inr_t: zfc.
unfold Paire in |- *; simpl in |- *; simple induction 1; intros b; elim b;
simpl in |- *; auto with zfc.
Qed.
Hint Resolve CIN_Paire_left CIN_Paire_right: zfc.
(* Singleton *)
unfold Sing in |- *; auto with zfc.
Qed.
unfold Sing in |- *; intros E E' H; elim (Paire_CIN E' E' E);
auto with zfc.
Qed.
simple induction E; intros A f ra; simple induction E'; intros B g rb;
simpl in |- *; simple induction 1; intros H1 H2; split.
intros a; elim (H1 a); intros b; intros; exists b; auto with zfc.
intros b; elim (H2 b); intros a; intros; exists a; auto with zfc.
Qed.
simple induction E; intros A f ra; simple induction E'; intros B g rb;
simple induction 1; intros a; unfold IN in |- *; exists a;
auto with zfc.
apply EQC_EQ; auto with zfc.
Qed.
simple induction E; simple induction E'; simpl in |- *.
intros.
elim X1; intros.
elim (a0 a); intros.
exists x; auto with zfc.
Defined.
simple induction E; simpl in |- *.
intros A f r.
simple induction 1; simpl in |- *.
intros.
exists x; auto with zfc.
Qed.
simple induction E; intros A f r.
intros.
simpl in |- *.
elim (CIN_EXType (sup A f) E' X).
intros x e.
cut (EQC (pi2 (sup A f) x) E'); auto with zfc.
intros e1.
cut (CIN E'' (pi2 (sup A f) x)).
intros i1.
elim (CIN_EXType _ _ i1).
intros x0 e2.
simpl in x0.
exists (dep_i A (fun x : A => pi1 (f x)) x x0).
simpl in |- *.
exact e2.
apply CIN_sound_right with E'; auto with zfc.
Qed.
unfold CINC in |- *; simple induction E; intros A f r.
unfold Union in |- *.
intros.
simpl in |- *.
elim (CIN_EXType (sup A f) E' X).
intro.
simpl in x.
intros.
simpl in p.
elim (CIN_EXType E' E0 X0).
cut (CIN E0 (f x)).
intros.
elim (CIN_EXType _ _ X1).
simpl in |- *.
intros.
exists (dep_i A (fun x : A => pi1 (f x)) x x1); auto with zfc.
apply CIN_sound_right with E'; auto with zfc.
Qed.
(* idem depprod *)
Inductive depprod' (A : Type) (P : A -> Type) : Type :=
dep_i' : forall x : A, P x -> depprod' A P.
simple induction E; unfold Union in |- *; simpl in |- *; intros A f r.
simple induction 1.
simple induction x.
intros a b; simpl in |- *.
intros.
exists (f a).
split.
exists a; auto with zfc.
apply CIN_sound_left with (pi2 (f a) b); auto with zfc.
simpl in |- *.
generalize b; elim (f a); simpl in |- *.
intros.
exists b0; auto with zfc.
Qed.
unfold Union in |- *.
simpl in |- *.
simple induction E; intros A f r; simple induction E'; intros A' f' r'.
simpl in |- *.
intros.
elim X; intros.
split.
simple induction x.
intros.
elim (a x0).
intros.
elim (EQC_EXType (f x0) (f' x1) p0 p).
intros.
exists (dep_i A' (fun x : A' => pi1 (f' x)) x1 x2).
simpl in |- *.
auto with zfc.
simple induction y; intros.
elim (b x); intros.
cut (EQC (f' x) (f x0)); auto with zfc.
intros e.
elim (EQC_EXType (f' x) (f x0) e p); intros.
exists (dep_i A (fun x0 : A => pi1 (f x0)) x0 x1).
simpl in |- *; auto with zfc.
Qed.

Theorem Union_monC :
forall E E' : Ens, CINC E E' -> CINC (Union E) (Union E').