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ZF.v
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ZF.v
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(** * Definition of ZF-Entailment *)
Require Import Tarski.
(** ** Signature *)
Inductive ZF_Funcs : Type :=
| eset : ZF_Funcs
| pair : ZF_Funcs
| union : ZF_Funcs
| power : ZF_Funcs
| om : ZF_Funcs.
Definition ZF_fun_ar (f : ZF_Funcs) : nat :=
match f with
| eset => 0
| pair => 2
| union => 1
| power => 1
| om => 0
end.
Inductive ZF_Preds : Type :=
| elem : ZF_Preds
| equal : ZF_Preds.
Definition ZF_pred_ar (P : ZF_Preds) : nat :=
match P with _ => 2 end.
Instance ZF_func_sig : funcs_signature :=
{| syms := ZF_Funcs; ar_syms := ZF_fun_ar; |}.
Instance ZF_pred_sig : preds_signature :=
{| preds := ZF_Preds; ar_preds := ZF_pred_ar; |}.
Arguments Vector.nil {_}, _.
Arguments Vector.cons {_} _ {_} _, _ _ _ _.
Notation "¬ A" := (A --> ⊥) (at level 42).
(** ** Axioms *)
Notation "x ∈ y" := (atom ZF_func_sig ZF_pred_sig elem (Vector.cons x (Vector.cons y Vector.nil))) (at level 35).
Notation "x ≡ y" := (atom ZF_func_sig ZF_pred_sig equal (Vector.cons x (Vector.cons y Vector.nil))) (at level 35).
Notation "∅" := (func ZF_func_sig eset Vector.nil).
Notation "'ω'" := (func ZF_func_sig om Vector.nil).
Notation "{ x ; y }" := (func ZF_func_sig pair (Vector.cons x (Vector.cons y Vector.nil))) (at level 31).
Notation "⋃ x" := (func ZF_func_sig union (Vector.cons x Vector.nil)) (at level 32).
Notation "'PP' x" := (func ZF_func_sig power (Vector.cons x Vector.nil)) (at level 31).
Notation "x ∪ y" := (⋃ {x; y}) (at level 32).
Notation "'σ' x" := (x ∪ {x; x}) (at level 32).
Fixpoint shift n x :=
match n with
| O => x
| S n => subst_term ↑ (shift n x)
end.
Definition sub x y :=
∀ $0 ∈ x`[↑] --> $0 ∈ y`[↑].
Notation "x ⊆ y" := (sub x y) (at level 34).
Definition ax_ext :=
∀ ∀ $1 ⊆ $0 --> $0 ⊆ $1 --> $1 ≡ $0.
Definition ax_eset :=
∀ ¬ ($0 ∈ ∅).
Definition ax_pair :=
∀ ∀ ∀ $0 ∈ {$1; $2} <--> $0 ≡ $1 ∨ $0 ≡ $2.
Definition ax_union :=
∀ ∀ $0 ∈ ⋃ $1 <--> ∃ $0 ∈ $2 ∧ $1 ∈ $0.
Definition ax_power :=
∀ ∀ $0 ∈ PP $1 <--> $0 ⊆ $1.
Definition inductive x :=
∅ ∈ x ∧ ∀ $0 ∈ shift 1 x --> σ $0 ∈ shift 1 x.
Definition ax_om1 :=
inductive ω.
Definition ax_om2 :=
∀ inductive $0 --> ω ⊆ $0.
Definition ax_sep phi :=
∀ ∃ ∀ $0 ∈ $1 <--> $0 ∈ $2 ∧ phi[$0.: Nat.add 3 >> var].
Definition fun_rel phi :=
∀ ∀ ∀ phi[$2.:($1)..] --> phi[$2.:($0)..] --> $1 ≡ $0.
Definition ax_rep phi :=
fun_rel phi --> ∀ ∃ ∀ $0 ∈ $1 <--> ∃ $0 ∈ $3 ∧ phi.
(** ** ZF-Models *)
Class ZF_Model :=
{
V :> Type ;
VI : interp V ;
V_ext : forall rho, rho ⊨ ax_ext ;
V_eset : forall rho, rho ⊨ ax_eset ;
V_pair : forall rho, rho ⊨ ax_pair ;
V_union : forall rho, rho ⊨ ax_union ;
V_power : forall rho, rho ⊨ ax_power ;
V_om1 : forall rho, rho ⊨ ax_om1 ;
V_om2 : forall rho, rho ⊨ ax_om2 ;
V_sep : forall phi rho, rho ⊨ ax_sep phi ;
V_rep : forall phi rho, rho ⊨ ax_rep phi ;
}.
Coercion V : ZF_Model >-> Sortclass.
Instance VI_instance (M : ZF_Model) : interp M := @VI M.
Notation "x i∈ y" := (@i_P _ _ _ VI elem (Vector.cons x (Vector.cons y Vector.nil))) (at level 20).
Notation "x i≡ y" := (@i_P _ _ _ VI equal (Vector.cons x (Vector.cons y Vector.nil))) (at level 20).
Notation "x i⊆ y" := (forall z, z i∈ x -> z i∈ y) (at level 20).
Notation "i∅" := (@i_f _ _ _ VI eset Vector.nil).
Notation "'iω'" := (@i_f _ _ _ VI om Vector.nil).
Notation "i{ x ; y }" := (@i_f _ _ _ VI pair (Vector.cons x (Vector.cons y Vector.nil))) (at level 10).
Notation "i⋃ x" := (@i_f _ _ _ VI union (Vector.cons x Vector.nil)) (at level 15).
Notation "'iPP' x" := (@i_f _ _ _ VI power (Vector.cons x Vector.nil)) (at level 15).
Notation "x i∪ y" := (i⋃ i{x; y}) (at level 16).
Notation "'iσ' x" := (x i∪ i{x; x}) (at level 15).
(* Extensional models interpret set equality as equality *)
Notation extensional M :=
(forall x y : M, x i≡ y <-> x = y).
(* Standard models contain only standard natural numbers *)
Fixpoint numeral {M : ZF_Model} n :=
match n with
| O => i∅
| S n => iσ (numeral n)
end.
Definition standard (M : ZF_Model) :=
forall x, x i∈ iω -> exists n, x = numeral n.
(** ** ZF-Entailment *)
Definition ZF_entails phi :=
forall (M : ZF_Model), extensional M -> forall rho, rho ⊨ phi.
(** ** Internal axioms *)
Section ZF.
Context { M : ZF_Model }.
Hypothesis VIEQ : extensional M.
Lemma M_ext (x y : M) :
x i⊆ y -> y i⊆ x -> x = y.
Proof.
rewrite <- VIEQ. apply (V_ext (fun _ => i∅)).
Qed.
Lemma M_eset x :
~ x i∈ i∅.
Proof.
specialize V_eset with (rho:=fun _ => i∅). cbn.
intros H1 H2. now apply (H1 x).
Qed.
Lemma M_pair x y z :
x i∈ i{y; z} <-> x = y \/ x = z.
Proof.
rewrite <- !VIEQ. apply V_pair with (rho:=fun _ => i∅).
Qed.
Lemma M_union x y :
x i∈ i⋃ y <-> exists z, z i∈ y /\ x i∈ z.
Proof.
apply V_union with (rho:=fun _ => i∅).
Qed.
Lemma M_power x y :
x i∈ iPP y <-> x i⊆ y.
Proof.
apply V_power with (rho:=fun _ => i∅).
Qed.
Definition M_inductive x :=
i∅ i∈ x /\ forall y, y i∈ x -> iσ y i∈ x.
Lemma M_om1 :
M_inductive iω.
Proof.
apply V_om1 with (rho:=fun _ => i∅).
Qed.
Lemma M_om2 x :
M_inductive x -> iω i⊆ x.
Proof.
apply V_om2 with (rho:=fun _ => i∅).
Qed.
Definition agrees_fun phi (P : M -> Prop) :=
forall x rho, P x <-> (x.:rho) ⊨ phi.
Definition representable (P : V -> Prop) :=
exists phi rho, forall d, P d <-> (d.:rho) ⊨ phi.
Lemma M_sep P x :
representable P -> exists y, forall z, z i∈ y <-> z i∈ x /\ P z.
Proof.
intros [phi [rho Hp]]. specialize V_sep with (rho:=rho). cbn.
intros H. destruct (H phi x) as [y H']. clear H.
exists y. intros z. specialize (H' z). setoid_rewrite sat_comp in H'.
rewrite (sat_ext _ _ (xi:=z.:rho)) in H'; try now intros [].
firstorder.
Qed.
Definition functional (R : M -> M -> Prop) :=
forall x y y', R x y -> R x y' -> y = y'.
Definition representable__rel phi (R : M -> M -> Prop) :=
forall x y rho, R x y <-> (x.:(y.:rho)) ⊨ phi.
Definition def_rel (R : M -> M -> Prop) :=
exists phi rho, forall x y, R x y <-> (x.:(y.:rho)) ⊨ phi.
Definition M_is_rep R x y :=
forall v, v i∈ y <-> exists u, u i∈ x /\ R u v.
Lemma M_rep R x :
def_rel R -> functional R -> exists y, M_is_rep R x y.
Proof.
Admitted.
Lemma is_rep_unique R x y y' :
M_is_rep R x y -> M_is_rep R x y' -> y = y'.
Proof.
intros H1 H2. apply M_ext; intros v.
- intros H % H1. now apply H2.
- intros H % H2. now apply H1.
Qed.
End ZF.