(* * Definition of semantic and deductive ZF-Entailment in binary signature *)
Require Import Undecidability.FOL.Util.Syntax.
Require Import Undecidability.FOL.Util.sig_bin.
Require Import Undecidability.FOL.Util.FullTarski.
Require Import Undecidability.FOL.Util.FullDeduction.
Import Vector.VectorNotations.
Require Import List.
(* ** Minimal binary signature only containing membership, no function symbols or equality *)
Existing Instance falsity_on.
Notation term' := (term sig_empty).
Notation form' := (form sig_empty sig_binary _ falsity_on).
Arguments Vector.nil {_}, _.
Arguments Vector.cons {_} _ {_} _, _ _ _ _.
Declare Scope syn'.
Open Scope syn'.
Notation "x ∈' y" := (atom sig_empty sig_binary tt ([x; y])) (at level 35) : syn'.
Definition eq' (x y : term') :=
∀ x`[↑] ∈' $0 <~> y`[↑] ∈' $0.
Notation "x ≡' y" := (eq' x y) (at level 35) : syn'.
(* ** Characterisations of set operations *)
Fixpoint shift `{funcs_signature} `{preds_signature} n (t : term) :=
match n with
| O => t
| S n => subst_term ↑ (shift n t)
end.
Definition is_eset (t : term') :=
∀ ¬ ($0 ∈' t`[↑]).
Definition is_pair (x y t : term') :=
∀ $0 ∈' t`[↑] <~> $0 ≡' x`[↑] ∨ $0 ≡' y`[↑].
Definition is_union (x t : term') :=
∀ $0 ∈' t`[↑] <~> ∃ $0 ∈' shift 2 x ∧ $1 ∈' $0.
Definition sub' (x y : term') :=
∀ $0 ∈' x`[↑] ~> $0 ∈' y`[↑].
Definition is_power (x t : term') :=
∀ $0 ∈' t`[↑] <~> sub' $0 x`[↑].
Definition is_sigma (x t : term') :=
∀ $0 ∈' t`[↑] <~> $0 ∈' x`[↑] ∨ $0 ≡' x`[↑].
Definition is_inductive (t : term') :=
(∃ is_eset $0 ∧ $0 ∈' t`[↑]) ∧ ∀ $0 ∈' t`[↑] ~> (∃ is_sigma $1 $0 ∧ $0 ∈' shift 2 t).
Definition is_om (t : term') :=
is_inductive t ∧ ∀ is_inductive $0 ~> sub' t`[↑] $0.
(* ** Symbol-free axiomatisation *)
Definition ax_ext' :=
∀ ∀ sub' $1 $0 ~> sub' $0 $1 ~> $1 ≡' $0.
Definition ax_eq_elem' :=
∀ ∀ ∀ ∀ $3 ≡' $1 ~> $2 ≡' $0 ~> $3 ∈' $2 ~> $1 ∈' $0.
Definition ax_eset' :=
∃ is_eset $0.
Definition ax_pair' :=
∀ ∀ ∃ is_pair $2 $1 $0.
Definition ax_union' :=
∀ ∃ is_union $1 $0.
Definition ax_power' :=
∀ ∃ is_power $1 $0.
Definition ax_om' :=
∃ is_om $0.
Definition binZF :=
ax_ext' :: ax_eq_elem' :: ax_eset' :: ax_pair' :: ax_union' :: ax_power' :: ax_om' :: nil.
(* ** Problems *)
(* Semantic entailment restricted to extensional models and core axioms (without sep and rep). *)
Definition entailment_binZF phi :=
forall D (M : @interp sig_empty _ D) (rho : nat -> D), (forall psi, In psi binZF -> rho ⊨ psi) -> rho ⊨ phi.
(* Deductive entailment restricted to intuitionistic rules and core axioms (without sep and rep). *)
Definition deduction_binZF phi :=
binZF ⊢I phi.
Require Import Undecidability.FOL.Util.Syntax.
Require Import Undecidability.FOL.Util.sig_bin.
Require Import Undecidability.FOL.Util.FullTarski.
Require Import Undecidability.FOL.Util.FullDeduction.
Import Vector.VectorNotations.
Require Import List.
(* ** Minimal binary signature only containing membership, no function symbols or equality *)
Existing Instance falsity_on.
Notation term' := (term sig_empty).
Notation form' := (form sig_empty sig_binary _ falsity_on).
Arguments Vector.nil {_}, _.
Arguments Vector.cons {_} _ {_} _, _ _ _ _.
Declare Scope syn'.
Open Scope syn'.
Notation "x ∈' y" := (atom sig_empty sig_binary tt ([x; y])) (at level 35) : syn'.
Definition eq' (x y : term') :=
∀ x`[↑] ∈' $0 <~> y`[↑] ∈' $0.
Notation "x ≡' y" := (eq' x y) (at level 35) : syn'.
(* ** Characterisations of set operations *)
Fixpoint shift `{funcs_signature} `{preds_signature} n (t : term) :=
match n with
| O => t
| S n => subst_term ↑ (shift n t)
end.
Definition is_eset (t : term') :=
∀ ¬ ($0 ∈' t`[↑]).
Definition is_pair (x y t : term') :=
∀ $0 ∈' t`[↑] <~> $0 ≡' x`[↑] ∨ $0 ≡' y`[↑].
Definition is_union (x t : term') :=
∀ $0 ∈' t`[↑] <~> ∃ $0 ∈' shift 2 x ∧ $1 ∈' $0.
Definition sub' (x y : term') :=
∀ $0 ∈' x`[↑] ~> $0 ∈' y`[↑].
Definition is_power (x t : term') :=
∀ $0 ∈' t`[↑] <~> sub' $0 x`[↑].
Definition is_sigma (x t : term') :=
∀ $0 ∈' t`[↑] <~> $0 ∈' x`[↑] ∨ $0 ≡' x`[↑].
Definition is_inductive (t : term') :=
(∃ is_eset $0 ∧ $0 ∈' t`[↑]) ∧ ∀ $0 ∈' t`[↑] ~> (∃ is_sigma $1 $0 ∧ $0 ∈' shift 2 t).
Definition is_om (t : term') :=
is_inductive t ∧ ∀ is_inductive $0 ~> sub' t`[↑] $0.
(* ** Symbol-free axiomatisation *)
Definition ax_ext' :=
∀ ∀ sub' $1 $0 ~> sub' $0 $1 ~> $1 ≡' $0.
Definition ax_eq_elem' :=
∀ ∀ ∀ ∀ $3 ≡' $1 ~> $2 ≡' $0 ~> $3 ∈' $2 ~> $1 ∈' $0.
Definition ax_eset' :=
∃ is_eset $0.
Definition ax_pair' :=
∀ ∀ ∃ is_pair $2 $1 $0.
Definition ax_union' :=
∀ ∃ is_union $1 $0.
Definition ax_power' :=
∀ ∃ is_power $1 $0.
Definition ax_om' :=
∃ is_om $0.
Definition binZF :=
ax_ext' :: ax_eq_elem' :: ax_eset' :: ax_pair' :: ax_union' :: ax_power' :: ax_om' :: nil.
(* ** Problems *)
(* Semantic entailment restricted to extensional models and core axioms (without sep and rep). *)
Definition entailment_binZF phi :=
forall D (M : @interp sig_empty _ D) (rho : nat -> D), (forall psi, In psi binZF -> rho ⊨ psi) -> rho ⊨ phi.
(* Deductive entailment restricted to intuitionistic rules and core axioms (without sep and rep). *)
Definition deduction_binZF phi :=
binZF ⊢I phi.