Library ZFskol

Require Import basic.
Require Export ZFdef.
Require Import Sublogic.

Module Skolem (Z : IZF_R_Ex_sig CoqSublogicThms) <: IZF_R_sig CoqSublogicThms.

Instance Zsetoid: Equivalence Z.eq_set.
Proof.
split; red; intros; rewrite Z.eq_set_ax in *; intros.
 reflexivity.
 symmetry; trivial.
 transitivity (Z.in_set x0 y); trivial.
Qed.

Instance Zin_morph : Proper (Z.eq_set ==> Z.eq_set ==> iff) Z.in_set.
Proof.
do 3 red; intros.
rewrite Z.eq_set_ax in H0.
split; intros.
 rewrite <- H0.
 apply Z.in_reg with x; trivial.

 rewrite H0.
 symmetry in H.
 apply Z.in_reg with y; trivial.
Qed.

Definition set :=
  { f : Z.set -> Prop |
    (exists u, f u) /\
    (∀ a a', f a -> f a' -> Z.eq_set a a') }.

Lemma set_intro : ∀ (f:Z.set->Prop) (P:set->Prop),
  (exists u, f u) ->
  (∀ a a', f a -> f a' -> Z.eq_set a a') ->
  (∀ Hex Huniq, P (exist _ f (conj Hex Huniq))) ->
  sig P.
intros.
exists (exist (fun _ => _ /\ _) f (conj H H0)); trivial. Qed.

Inductive in_set_ (x y:set) : Prop :=
 InSet
  (_:exists2 x', proj1_sig x x' &
     exists2 y', proj1_sig y y' & Z.in_set x' y').

Definition in_set := in_set_.

Lemma in_set_elim : ∀ x y, in_set x y <->
  exists2 x', proj1_sig x x' &
  exists2 y', proj1_sig y y' & Z.in_set x' y'.
split; intros.
 destruct H; trivial.
 constructor; trivial.
Qed.

Notation "x ∈ y" := (in_set x y).

Definition eq_set a b := ∀ x, x ∈ a <-> x ∈ b.

Notation "x == y" := (eq_set x y).

Lemma eq_set_ax : ∀ a b, a == b <-> (∀ x, x ∈ a <-> x ∈ b).
reflexivity.
Qed.

Instance eq_setoid: Equivalence eq_set.
Proof.
split; do 2 red; intros.
 reflexivity.
 symmetry; trivial.
 transitivity (x0 ∈ y); trivial.
Qed.

Lemma In_intro: ∀ x y: set,
  (∀ x' y', proj1_sig x x' -> proj1_sig y y' -> Z.in_set x' y') ->
  x ∈ y.
intros.
destruct (proj2_sig x).
destruct H0.
destruct (proj2_sig y).
destruct H2.
constructor.
exists x0; trivial.
exists x1; auto.
Qed.

Definition Z2set (x:Z.set) : set.
exists (fun a => Z.eq_set a x).
split.
 exists x; reflexivity.
 intros; transitivity x; trivial.
 symmetry; trivial.
Defined.

Lemma Z2set_surj : ∀ x, exists y, x == Z2set y.
intros.
destruct (proj2_sig x).
destruct H.
exists x0.
split; intros.
 rewrite in_set_elim in H1.
 destruct H1.
 destruct H2.
 constructor.
 exists x2; trivial.
 exists x3; trivial.
 simpl; auto.

 rewrite in_set_elim in H1.
 destruct H1.
 destruct H2.
 constructor.
 exists x2; trivial.
 simpl in H2.
 exists x0; trivial.
 rewrite <- H2; trivial.
Qed.

Lemma Eq_proj : ∀ (x:set) x', proj1_sig x x' -> x == Z2set x'.
intros.
destruct (proj2_sig x).
clear H0.
split; intros.
 rewrite in_set_elim in H0.
 destruct H0.
 destruct H2.
 constructor.
 exists x1; simpl; trivial.
 exists x2; auto.

 rewrite in_set_elim in H0.
 destruct H0; simpl in *.
 destruct H2.
 constructor.
 exists x1; simpl; trivial.
 exists x'; trivial.
 rewrite <- H2; trivial.
Qed.

Lemma inZ_in : ∀ a b, Z.in_set a b -> Z2set a ∈ Z2set b.
unfold Z2set, in_set; simpl.
intros.
constructor; simpl.
exists a; try reflexivity.
exists b; try reflexivity; trivial.
Qed.

Lemma in_inZ : ∀ a b, Z2set a ∈ Z2set b -> Z.in_set a b.
intros.
rewrite in_set_elim in H.
destruct H.
destruct H0.
unfold Z2set in *; simpl in *.
apply Z.in_reg with x; trivial.
rewrite <- H0; auto.
Qed.

Lemma in_equiv a b : in_set (Z2set a) (Z2set b) <-> Z.in_set a b.
split; intros.
 apply in_inZ; trivial.
 apply inZ_in; trivial.
Qed.

Lemma eq_Zeq : ∀ x y, Z2set x == Z2set y -> Z.eq_set x y.
intros.
rewrite Z.eq_set_ax; split; intros.
 apply in_inZ.
 apply (proj1 (H (Z2set x0))).
 apply inZ_in; trivial.

 apply in_inZ.
 apply (proj2 (H (Z2set x0))).
 apply inZ_in; trivial.
Qed.

Lemma Zeq_eq : ∀ x y, Z.eq_set x y -> Z2set x == Z2set y.
intros.
split; intros.
 rewrite in_set_elim in H0.
 destruct H0.
 destruct H1.
 simpl in H1.
 constructor.
 exists x1; trivial.
 exists x2; trivial.
 simpl.
 transitivity x; trivial.

 rewrite in_set_elim in H0.
 destruct H0.
 destruct H1.
 simpl in H1.
 constructor.
 exists x1; trivial.
 exists x2; trivial.
 simpl.
 transitivity y; trivial.
 symmetry; trivial.
Qed.

Lemma eq_equiv x y : Z2set x == Z2set y <-> Z.eq_set x y.
split; intros.
 apply eq_Zeq; trivial.
 apply Zeq_eq; trivial.
Qed.

Instance Z2set_morph : Proper (Z.eq_set ==> eq_set) Z2set.
exact Zeq_eq.
Qed.

Lemma in_reg : ∀ a a' b, a == a' -> a ∈ b -> a' ∈ b.
intros.
rewrite in_set_elim in H0.
destruct H0 as (a0,eq_a,(b0,eq_b,in_ab)).
destruct (proj2_sig a') as ((a'0, eq_a'),_).
constructor.
exists a'0; trivial.
exists b0; trivial.
apply Z.in_reg with a0; trivial.
apply eq_Zeq.
rewrite <- (Eq_proj _ _ eq_a).
rewrite <- (Eq_proj _ _ eq_a').
trivial.
Qed.

Instance in_morph : Proper (eq_set ==> eq_set ==> iff) in_set.
Proof.
unfold eq_set in |- *; split; intros.
 apply in_reg with x; auto.
 elim (H0 x); auto.

 apply in_reg with y; auto.
  symmetry in |- *; auto.
  elim (H0 y); auto.
Qed.


Lemma empty_sig : { empty | ∀ x, ~ x ∈ empty }.
apply set_intro with (fun x:Z.set => ∀ y, ~ Z.in_set y x); intros.
 apply Z.empty_ex.

 rewrite Z.eq_set_ax; split; intros.
  elim H with x; trivial.
  elim H0 with x; trivial.

 red; intros.
 rewrite in_set_elim in H.
 destruct H; simpl in *.
 destruct H0.
 elim H0 with x0; trivial.
Qed.

Definition empty := proj1_sig empty_sig.
Lemma empty_ax: ∀ x, ~ x ∈ empty.
Proof proj2_sig empty_sig.


Lemma pair_sig : ∀ a b,
  { pair | ∀ x, x ∈ pair <-> (x == a \/ x == b) }.
intros a b.
apply set_intro with
  (fun x => ∀ y, y ∈ Z2set x <-> y == a \/ y == b); intros.
 elim (Z2set_surj a); intros.
 elim (Z2set_surj b); intros.
 elim Z.pair_ex with x x0; intros.
 exists x1; intros.
 elim (Z2set_surj y); intros.
 rewrite H2.
 transitivity (Z.in_set x2 x1).
  split; intros; first [apply inZ_in|apply in_inZ]; trivial.
  apply iff_trans with (1:=H1 x2).
  rewrite H.
  rewrite H0.
  setoid_replace (Z.eq_set x2 x) with (Z2set x2 == Z2set x).
  setoid_replace (Z.eq_set x2 x0) with (Z2set x2 == Z2set x0).
  reflexivity.
  split; intros; first [apply Zeq_eq|apply eq_Zeq]; trivial.
  split; intros; first [apply Zeq_eq|apply eq_Zeq]; trivial.

 intros; rewrite Z.eq_set_ax; intros.
 transitivity (Z2set x ∈ Z2set a0).
  split; intros; first [apply inZ_in|apply in_inZ]; trivial.
  transitivity (Z2set x == a \/ Z2set x == b); auto.
  symmetry.
  transitivity (Z2set x ∈ Z2set a'); trivial.
  split; intros; first [apply inZ_in|apply in_inZ]; trivial.

 split; intros.
  rewrite in_set_elim in H.
  destruct H; simpl in *.
  destruct H0.
  apply (proj1 (H0 x)).
  constructor.
  exists x0; trivial; simpl.
  exists x1; trivial; reflexivity.

  apply In_intro; simpl; intros.
  apply in_inZ.
  rewrite <- (Eq_proj _ _ H0).
  apply (proj2 (H1 x)); trivial.
Qed.

Definition pair a b := proj1_sig (pair_sig a b).
Lemma pair_ax: ∀ a b x, x ∈ pair a b <-> (x == a \/ x == b).
Proof fun a b => proj2_sig (pair_sig a b).


Lemma union_sig: ∀ a,
  { union | ∀ x, x ∈ union <-> (exists2 y, x ∈ y & y ∈ a) }.
intro a.
apply set_intro with
 (fun x => ∀ z,
  Z.in_set z x <-> exists2 y, Z.in_set z y & Z2set y ∈ a); intros.
 elim (Z2set_surj a); intros.
 elim (Z.union_ex x); intros.
 exists x0.
 intros.
 apply iff_trans with (1:=H0 z).
 split; intros.
  destruct H1.
  exists x1; trivial.
  rewrite H; apply inZ_in; trivial.

  destruct H1.
  exists x1; trivial.
  apply in_inZ; rewrite <- H; trivial.

 rewrite Z.eq_set_ax; intros.
 apply iff_trans with (1:=H x).
 symmetry; trivial.

split; intros.
 rewrite in_set_elim in H.
 destruct H; simpl in *.
 destruct H0.
 elim (proj1 (H0 _) H1); intros.
 exists (Z2set x2); trivial.
 rewrite (Eq_proj _ _ H).
 apply inZ_in; trivial.

 destruct H.
 apply In_intro; simpl; intros.
 apply (proj2 (H2 x')).
 elim (Z2set_surj x0); intros.
 exists x1.
  apply in_inZ.
  rewrite <- H3; rewrite <- (Eq_proj _ _ H1); trivial.

  rewrite <- H3; trivial.
Qed.

Definition union a := proj1_sig (union_sig a).
Lemma union_ax: ∀ a x,
  x ∈ union a <-> (exists2 y, x ∈ y & y ∈ a).
Proof fun a => proj2_sig (union_sig a).


Lemma subset_sig: ∀ a P,
  { subset | ∀ x, x ∈ subset <->
             (x ∈ a /\ exists2 x', x==x' & P x') }.
intros a P.
apply set_intro with
  (fun x => ∀ y, y ∈ Z2set x <->
    (y ∈ a /\ exists2 y', y==y' & P y')); intros.
 elim (Z2set_surj a); intros.
 elim Z.subset_ex with
   x (fun z => exists2 z', z' == Z2set z & P z'); intros.
 exists x0; intros.
 elim (Z2set_surj y); intros.
 rewrite H1.
 transitivity (Z.in_set x1 x0).
  split; intros; first [apply inZ_in|apply in_inZ]; trivial.
  apply iff_trans with (1:=H0 x1).
  rewrite H.
  apply and_iff_morphism.
   split; intros; first [apply inZ_in|apply in_inZ]; trivial.

   split; destruct 1.
    destruct H3.
    exists x3; trivial.
    rewrite H1; rewrite H3; apply Zeq_eq; trivial.

    exists x1;[reflexivity|].
    exists x2; trivial.
    rewrite <- H1; symmetry; trivial.

 apply eq_Zeq.
 rewrite eq_set_ax; intros.
 rewrite H.
 rewrite H0.
 reflexivity.

 split; intros.
  rewrite in_set_elim in H.
  destruct H; simpl in *.
  destruct H0.
  apply (proj1 (H0 x)).
  constructor.
  exists x0; trivial; simpl.
  exists x1; trivial; reflexivity.

  apply In_intro; simpl; intros.
  apply in_inZ.
  rewrite <- (Eq_proj _ _ H0).
  apply (proj2 (H1 x)); trivial.
Qed.

Definition subset a P := proj1_sig (subset_sig a P).
Lemma subset_ax : ∀ a P x,
    x ∈ subset a P <-> (x ∈ a /\ exists2 x', x==x' & P x').
Proof fun a P => proj2_sig (subset_sig a P).


Lemma power_sig: ∀ a,
  { power | ∀ x, x ∈ power <-> (∀ y, y ∈ x -> y ∈ a) }.
intro a.
apply set_intro with
 (fun x => ∀ z,
  Z.in_set z x <-> (∀ y, Z.in_set y z -> Z2set y ∈ a)); intros.
 elim (Z2set_surj a); intros.
 elim (Z.power_ex x); intros.
 exists x0; intros.
 apply iff_trans with (1:= H0 z).
 split; intros.
  rewrite H; apply inZ_in; auto.
  apply in_inZ; rewrite <- H; auto.

 rewrite Z.eq_set_ax; intros.
 apply iff_trans with (1:=H x).
 symmetry; trivial.

split; intros.
 rewrite in_set_elim in H.
 destruct H; simpl in *.
 destruct H1.
 elim (Z2set_surj y); intros.
 rewrite H3.
 apply (proj1 (H1 _) H2).
 apply in_inZ; rewrite <- H3; rewrite <- (Eq_proj _ _ H); trivial.

 apply In_intro; simpl; intros.
 apply (proj2 (H1 x')).
 intros.
 apply H.
 rewrite (Eq_proj _ _ H0); apply inZ_in; trivial.
Qed.

Definition power a := proj1_sig (power_sig a).
Lemma power_ax:
  ∀ a x, x ∈ power a <-> (∀ y, y ∈ x -> y ∈ a).
Proof fun a => proj2_sig (power_sig a).


Definition uchoice_pred (P:set->Prop) :=
  (∀ x x', x == x' -> P x -> P x') /\
  (exists x, P x) /\
  (∀ x x', P x -> P x' -> x == x').

Definition uchoice (P : set -> Prop) (Hp : uchoice_pred P) : set.
exists (fun z => P (Z2set z)).
destruct Hp.
destruct H0.
split; intros.
 destruct H0.
 destruct (Z2set_surj x).
 exists x0; apply H with x; trivial.

 apply eq_Zeq.
 auto.
Defined.

Lemma uchoice_ax : ∀ P h x,
  (x ∈ uchoice P h <-> exists2 z, P z & x ∈ z).
split; intros.
 destruct H.
 destruct H.
 destruct H0.
 simpl in H0.
 exists (Z2set x1); trivial.
 apply In_intro; intros.
 simpl in H3.
 rewrite H3.
 destruct (proj2_sig x).
 rewrite (H5 x' x0); trivial.

 destruct H.
 destruct H0.
 destruct H0.
 destruct H1.
 constructor.
 exists x1; trivial.
 simpl.
 exists x2; trivial.
 apply (proj1 h x0); trivial.
 apply Eq_proj; trivial.
Qed.

Lemma uchoice_morph_raw : ∀ (P1 P2:set->Prop) h1 h2,
  (∀ x x', x == x' -> (P1 x <-> P2 x')) ->
  uchoice P1 h1 == uchoice P2 h2.
intros.
apply eq_set_ax; intros.
rewrite uchoice_ax.
rewrite uchoice_ax.
apply ex2_morph; red; intros.
 apply H; reflexivity.

 reflexivity.
Qed.

Instance uchoice_pred_morph : Proper ((eq_set ==> iff) ==> iff) uchoice_pred.
apply morph_impl_iff1; auto with *.
do 3 red; intros.
destruct H0 as (?,(?,?)); split;[|split]; intros.
 assert (x x0).
  revert H4; apply H; auto with *.
 revert H5; apply H; trivial.

 destruct H1; exists x0.
 revert H1; apply H; auto with *.

 apply H2; [revert H3|revert H4]; apply H; auto with *.
Qed.

Lemma uchoice_ext : ∀ (P:set->Prop) h x, P x -> x == uchoice P h.
intros.
apply eq_set_ax;intros.
rewrite uchoice_ax.
split; intros.
 exists x; trivial.

 destruct H0.
 destruct h.
 destruct H3.
 rewrite (H4 x x1); auto.
Qed.

Lemma uchoice_def : ∀ P h, P (uchoice P h).
intros.
destruct h.
destruct a.
destruct e.
apply p with x; trivial.
apply uchoice_ext; trivial.
Qed.


Definition funDom (R:set -> set -> Prop) x :=
  ∀ x' y y', R x y -> R x' y' -> x == x' -> y == y'.
Definition downR (R:set -> set -> Prop) x' y' :=
  exists2 x, x == Z2set x' /\ funDom R x & exists2 y, y == Z2set y' & R x y.

Lemma downR_morph : Proper
  ((eq_set ==> eq_set ==> iff) ==> Z.eq_set ==> Z.eq_set ==> iff) downR.
do 4 red; intros.
unfold downR.
apply Zeq_eq in H0.
apply Zeq_eq in H1.
apply ex2_morph; red; intros.
 apply and_iff_morphism.
  rewrite H0; reflexivity.

  unfold funDom.
  split; intros.
   rewrite <- (fun e1 => H a a e1 y2 y2) in H3; auto with *.
   rewrite <- (fun e1 => H x' x' e1 y' y') in H4; eauto with *.

   rewrite (fun e1 => H a a e1 y2 y2) in H3; auto with *.
   rewrite (fun e1 => H x' x' e1 y' y') in H4; eauto with *.

 apply ex2_morph; red; intros.
  rewrite H1; reflexivity.
  apply H; reflexivity.
Qed.

Lemma downRm : ∀ R x x' y y',
  Z.eq_set x x' ->
  Z.eq_set y y' ->
  downR R x y ->
  downR R x' y'.
intros.
destruct H1 as (xx,(eqx,fdomx), (yy,eqy,rel)).
exists xx.
 split; trivial.
 rewrite eqx.
 rewrite eq_equiv; trivial.
exists yy; trivial.
rewrite eqy.
rewrite eq_equiv; trivial.
Qed.

Lemma downR_fun : ∀ R x y y',
  downR R x y ->
  downR R x y' ->
  Z.eq_set y y'.
intros.
destruct H as (xx,(eqx,fdomx), (yy,eqy,rel)).
destruct H0 as (xx',(eqx',_), (yy',eqy',rel')).
apply eq_Zeq.
rewrite <- eqy; rewrite <- eqy'.
red in fdomx.
apply fdomx with xx'; trivial.
rewrite eqx; rewrite eqx'.
reflexivity.
Qed.

Lemma repl0 : ∀ (a:set) (R:set->set->Prop), set.
intros a R.
exists
 (fun a' => ∀ x,
  Z.in_set x a' <-> exists2 y, Z2set y ∈ a & downR R y x).
split; intros.
 destruct (Z2set_surj a).
 assert (R'm := fun x0 x' y y' (_:Z.in_set x0 x) => downRm R x0 x' y y').
 assert (R'fun := fun x0 y y' (_:Z.in_set x0 x) => downR_fun R x0 y y').
 destruct (Z.repl_ex x (downR R) R'm R'fun); intros.
 exists x0; intros.
 rewrite H0.
 apply ex2_morph; red; intros.
 2:reflexivity.
 split; intros.
  rewrite H.
  apply inZ_in; trivial.

  apply in_inZ.
  rewrite <- H; trivial.

 rewrite Z.eq_set_ax; intros.
 rewrite H.
 rewrite H0.
 reflexivity.
Defined.

Definition incl_set x y := ∀ z, z ∈ x -> z ∈ y.

Lemma repl0_mono :
  Proper (incl_set ==> (eq_set ==> eq_set ==> iff) ==> incl_set) repl0.
do 4 red; simpl; intros.
rewrite in_set_elim in *.
destruct H1.
destruct H2.
simpl in *; intros.
destruct (Z2set_surj y).
assert (R'm := fun x x' y y' (_:Z.in_set x x3) => downRm y0 x x' y y').
assert (R'fun := fun x y y' (_:Z.in_set x x3) => downR_fun y0 x y y').
destruct (Z.repl_ex x3 (downR y0) R'm R'fun).
exists x1; trivial.
exists x4; trivial.
 intro; rewrite H5.
 apply ex2_morph; red; intros; auto with *.
 rewrite H4.
 symmetry; apply in_equiv.

 rewrite H5.
 rewrite H2 in H3.
 clear x2 H1 H2 x4 H5.
 destruct H3.
 exists x2.
  apply H in H1.
  rewrite H4 in H1; rewrite in_equiv in H1; trivial.

  revert H2; apply iff_impl; apply downR_morph; auto with *.
Qed.

Lemma repl_sig :
  { repl |
    Proper (incl_set ==> (eq_set ==> eq_set ==> iff) ==> incl_set) repl /\
    ∀ a (R:set->set->Prop),
    (∀ x x' y y', x ∈ a -> x == x' -> y == y' -> R x y -> R x' y') ->
    (∀ x y y', x ∈ a -> R x y -> R x y' -> y == y') ->
    ∀ x, x ∈ repl a R <-> exists2 y, y ∈ a & R y x }.
exists repl0; split.
 exact repl0_mono.
split; intros.
 rewrite in_set_elim in H1.
 destruct H1.
 destruct H2; simpl in *.
 rewrite H2 in H3.
 destruct H3.
 destruct H4.
 destruct H5.
 destruct H4.
 exists x3.
  rewrite H4; trivial.
 revert H6; apply H; auto with *.
  rewrite H4; trivial.

  apply Eq_proj in H1.
  rewrite H1; trivial.

 destruct H1.
 apply In_intro; simpl; intros.
 rewrite H4; clear H4.
 destruct (Z2set_surj x0).
 exists x1.
  rewrite <- H4; trivial.
 exists x0.
  split; intros; eauto.
  red; intros.
  apply H0 with x0; trivial.
  revert H6; apply H; auto with *.
  rewrite <- H7; trivial.

  exists x; trivial.
  apply Eq_proj; trivial.
Defined.

Definition repl := proj1_sig repl_sig.
Lemma repl_mono :
  Proper (incl_set ==> (eq_set ==> eq_set ==> iff) ==> incl_set) repl.
Proof (proj1 (proj2_sig repl_sig)).
Lemma repl_ax:
    ∀ a (R:set->set->Prop),
    (∀ x x' y y', x ∈ a -> x == x' -> y == y' -> R x y -> R x' y') ->
    (∀ x y y', x ∈ a -> R x y -> R x y' -> y == y') ->
    ∀ x, x ∈ repl a R <-> exists2 y, y ∈ a & R y x.
Proof proj2 (proj2_sig repl_sig).


Definition Nat x :=
  ∀ (P:set),
  empty ∈ P ->
  (∀ x, x ∈ P -> union (pair x (pair x x)) ∈ P) ->
  x ∈ P.

Instance Nat_morph : Proper (eq_set ==> iff) Nat.
unfold Nat; split; intros.
 rewrite <- H; apply H0; trivial.
 rewrite H; apply H0; trivial.
Qed.

Lemma Nat_S : ∀ x,
  Nat x -> Nat (union (pair x (pair x x))).
unfold Nat; intros; auto.
Qed.

Definition infinite_sig :
  { infty | empty ∈ infty /\
      ∀ x, x ∈ infty -> union (pair x (pair x x)) ∈ infty }.
apply set_intro with
  (fun x => ∀ y, Z.in_set y x <-> Nat (Z2set y)); intros.
 elim Z.infinity_ex; intros.
 elim Z.repl_ex with x (fun x' y' => Z.eq_set x' y' /\ Nat (Z2set y')); intros.
  exists x0; intros.
  rewrite H1.
  split; intros.
   destruct H2.
   destruct H.
   destruct H3; trivial.

   exists y.
    red in H2.
    apply in_inZ.
    apply H2; intros.
     destruct H.
     apply inZ_in in H3.
     apply in_reg with (Z2set x1); trivial.
     split; intros.
      destruct (Z2set_surj x2).
      rewrite H5 in H4; apply in_inZ in H4.
      elim H with x3; trivial.

      elim empty_ax with x2; trivial.

     destruct (Z2set_surj x1).
     rewrite H4 in H3; apply in_inZ in H3.
     apply H0 in H3.
     destruct H3.
     apply in_reg with (Z2set x3).
     2:apply inZ_in; trivial.
     split; simpl; intros.
      rewrite union_ax.
      destruct (Z2set_surj x4).
      rewrite H7 in H6; apply in_inZ in H6; rewrite H3 in H6.
      destruct H6.
       exists (pair x1 x1).
        rewrite pair_ax; left.
        rewrite H7; rewrite H4; apply Zeq_eq; trivial.

        rewrite pair_ax; right; reflexivity.

       exists x1.
        rewrite H7; rewrite H4; apply inZ_in; trivial.

        rewrite pair_ax; left; reflexivity.

      destruct (Z2set_surj x4).
      rewrite H7; apply inZ_in.
      rewrite H3.
      rewrite H7 in H6; clear x4 H7.
      rewrite union_ax in H6.
      destruct H6.
      rewrite pair_ax in H7; destruct H7.
       right; apply in_inZ.
       rewrite <- H4; rewrite <- H7; trivial.

       rewrite H7 in H6; clear x4 H7.
       left; apply eq_Zeq.
       rewrite <- H4.
       rewrite pair_ax in H6; destruct H6; trivial.

    split; auto with *.

  rewrite <- H2; rewrite <- H3; trivial.

  destruct H2; destruct H3.
  rewrite <- H2; trivial.

 rewrite Z.eq_set_ax; split; intros.
  rewrite H0; rewrite H in H1; trivial.

  rewrite H; rewrite H0 in H1; trivial.

split; intros.
 apply In_intro; simpl; unfold Nat; intros.
 apply (proj2 (H0 x')); intros.
 apply in_reg with empty; trivial.
 split; intros.
  elim empty_ax with x; trivial.

  elim (Z2set_surj x); intros.
 apply Eq_proj in H.
 rewrite <- H in H3.
 elim empty_ax with (1:=H3).

 rewrite in_set_elim in H.
 destruct H; simpl in *.
 destruct H0.
 apply In_intro; simpl; intros.
 apply (proj2 (H3 x')).
 setoid_replace (Z2set x') with (union (pair x (pair x x))).
  apply Nat_S.
  rewrite (Eq_proj _ _ H).
  apply (proj1 (H0 x0)); trivial.

  split; intros.
   elim (Z2set_surj x2); intros.
   rewrite H5 in H4 |- *; clear H5 x2.
   apply Eq_proj in H2.
   rewrite <- H2 in H4; trivial.

   elim (Z2set_surj x2); intros.
   rewrite H5 in H4 |- *; clear H5 x2.
   apply Eq_proj in H2.
   rewrite <- H2; trivial.
Qed.
Definition infinite := proj1_sig infinite_sig.
Lemma infinity_ax1: empty ∈ infinite.
Proof proj1 (proj2_sig infinite_sig).

Lemma infinity_ax2: ∀ x,
  x ∈ infinite -> union (pair x (pair x x)) ∈ infinite.
Proof proj2 (proj2_sig infinite_sig).


Lemma wf_ax : ∀ (P:set->Prop),
  (∀ x, P x -> P x) ->
  (∀ x, (∀ y, y ∈ x -> P y) -> P x) ->
  ∀ x, P x.
intros P _ H x.
cut (∀ xs (x:set), x == Z2set xs -> P x).
 intros.
 destruct (Z2set_surj x).
 eauto.
clear x.
intros xs; elim xs using Z.wf_ax; intros; trivial.
apply H; intros.
elim (Z2set_surj y); intros.
rewrite H1,H3 in H2.
apply in_inZ in H2; eauto.
Qed.


Section Collection.

Hypothesis lst_rk : (Z.set->Prop) -> Z.set -> Prop.
Hypothesis lst_rk_morph : ∀ (P P':Z.set->Prop),
  (∀ x x', Z.eq_set x x' -> (P x <-> P' x')) ->
  ∀ y y', Z.eq_set y y' -> lst_rk P y -> lst_rk P' y'.
Hypothesis lst_incl : ∀ P y, lst_rk P y -> P y.
Hypothesis lst_fun : ∀ P y y', lst_rk P y -> lst_rk P y' -> Z.eq_set y y'.

Hypothesis coll_ax_uniq : ∀ A (R:Z.set->Z.set->Prop),
    (∀ x x' y y', Z.in_set x A -> Z.eq_set x x' -> Z.eq_set y y' ->
     R x y -> R x' y') ->
    exists B, lst_rk(fun B =>
      ∀ x, Z.in_set x A ->
      (exists y, R x y) ->
      exists2 y, Z.in_set y B & R x y) B.

Lemma coll_sig : ∀ A (R:set->set->Prop),
  {coll| Proper (eq_set==>eq_set==>iff) R ->
     ∀ x, x ∈ A -> (exists y, R x y) ->
     exists2 y, y ∈ coll & R x y }.
intros A R.
pose (R' x y := exists2 x', Z2set x == x' & exists2 y', Z2set y == y' & R x' y').
assert (R'm : ∀ x x' y y', Z.eq_set x x' -> Z.eq_set y y' ->
     R' x y -> R' x' y').
 destruct 3 as (x'',?,(y'',?,?)).
 exists x'';[|exists y'';trivial].
  transitivity (Z2set x); trivial.
  apply Zeq_eq.
  symmetry; trivial.

  transitivity (Z2set y); trivial.
  apply Zeq_eq.
  symmetry; trivial.
apply set_intro with
  (lst_rk(fun B => exists2 A', Z2set A' == A & ∀ x, Z.in_set x A' ->
      (exists y, R' x y) ->
      exists2 y, Z.in_set y B & R' x y)); intros.
 destruct (Z2set_surj A) as (A',e).
 destruct coll_ax_uniq with A' R' as (B,HB); eauto.
 exists B.
 revert HB; apply lst_rk_morph; auto with *.
 intros.
 split; intros.
  exists A'; intros; auto with *.
  destruct H0 with x0 as (y,?,?); trivial.
  exists y; trivial.
  revert H3; apply Zin_morph; auto with *.

  destruct H0 as (A'',e',?).
  rewrite e in e'.
  apply eq_Zeq in e'.
  rewrite <- e' in H1.
  destruct H0 with x0 as (y,?,?); trivial.
  exists y; trivial.
  revert H3; apply Zin_morph; auto with *.

 apply lst_fun with (1:=H) (2:=H0).

 destruct Hex as (B,HB).
 assert (Bok := lst_incl _ _ HB).
 destruct Bok as (A',eA,Bok).
 destruct (Z2set_surj x) as (x',ex).
 destruct Bok with x' as (y,?,?).
  apply in_inZ.
  rewrite eA; rewrite <- ex; trivial.

  destruct H1 as (y,Rxy).
  destruct (Z2set_surj y) as (y',ey).
  exists y'; exists x; [|exists y]; auto with *.

  exists (Z2set y).
   apply In_intro; simpl; intros.
   specialize Huniq with (1:=HB) (2:=H5).
   rewrite <- Huniq.
   rewrite H4; trivial.

   destruct H3 as (x'',?,(y',?,?)).
   revert H5; apply H; trivial.
   rewrite ex; rewrite <- H3; auto with *.
Qed.

Definition coll A R := proj1_sig (coll_sig A R).
Lemma coll_ax : ∀ A (R:set->set->Prop),
  Proper (eq_set==>eq_set==>iff) R ->
  ∀ x, x ∈ A -> (exists y, R x y) ->
  exists2 y, y ∈ coll A R & R x y.
Proof (fun A R => proj2_sig (coll_sig A R)).

End Collection.

End Skolem.