Mercurial > hg > Members > kono > Proof > ZF-in-agda
view zf.agda @ 11:2df90eb0896c
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 13 May 2019 20:51:45 +0900 |
parents | 8022e14fce74 |
children | b531d2b417ad e11e95d5ddee |
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module zf where open import Level record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where field proj1 : A proj2 : B open _∧_ data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where case1 : A → A ∨ B case2 : B → A ∨ B -- open import Relation.Binary.PropositionalEquality _⇔_ : {n : Level } → ( A B : Set n ) → Set n _⇔_ A B = ( A → B ) ∧ ( B → A ) open import Data.Empty open import Relation.Nullary open import Relation.Binary open import Relation.Binary.Core infixr 130 _∧_ infixr 140 _∨_ infixr 150 _⇔_ record Func {n m : Level } (ZFSet : Set n) (_≈_ : Rel ZFSet m) : Set (n ⊔ suc m) where field rel : Rel ZFSet m dom : ( y : ZFSet ) → ∀ { x : ZFSet } → rel x y fun-eq : { x y z : ZFSet } → ( rel x y ∧ rel x z ) → y ≈ z open Func record IsZF {n m : Level } (ZFSet : Set n) (_∋_ : ( A x : ZFSet ) → Set m) (_≈_ : Rel ZFSet m) (∅ : ZFSet) (_×_ : ( A B : ZFSet ) → ZFSet) (Union : ( A : ZFSet ) → ZFSet) (Power : ( A : ZFSet ) → ZFSet) (Select : ( ZFSet → Set m ) → ZFSet ) (Replace : ( ZFSet → ZFSet ) → ZFSet ) (infinite : ZFSet) : Set (suc (n ⊔ m)) where field isEquivalence : {A B : ZFSet} → IsEquivalence {n} {m} {ZFSet} _≈_ -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) pair : ( A B : ZFSet ) → ( (A × B) ∋ A ) ∧ ( (A × B) ∋ B ) -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x)) union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y _∈_ : ( A B : ZFSet ) → Set m A ∈ B = B ∋ A _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → ∀{ A∋x : Set m } → Set m _⊆_ A B {x} {A∋x} = B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) _∪_ : ( A B : ZFSet ) → ZFSet A ∪ B = Select ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) infixr 200 _∈_ infixr 230 _∩_ _∪_ infixr 220 _⊆_ field empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y} power← : ∀( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) extentionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) minimul : ZFSet → ZFSet regularity : ∀( x : ZFSet ) → ¬ (x ≈ ∅) → ( minimul x ∈ x ∧ ( minimul x ∩ x ≈ ∅ ) ) -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ Select ( λ y → x ≈ y )) ∈ infinite selection : { ψ : ZFSet → Set m } → ∀ ( y : ZFSet ) → ( y ∈ Select ψ ) → ψ y -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) replacement : {ψ : ZFSet → ZFSet} → ∀ ( x : ZFSet ) → ( ψ x ∈ Replace ψ ) record ZF {n m : Level } : Set (suc (n ⊔ m)) where infixr 210 _×_ infixl 200 _∋_ infixr 220 _≈_ field ZFSet : Set n _∋_ : ( A x : ZFSet ) → Set m _≈_ : ( A B : ZFSet ) → Set m -- ZF Set constructor ∅ : ZFSet _×_ : ( A B : ZFSet ) → ZFSet Union : ( A : ZFSet ) → ZFSet Power : ( A : ZFSet ) → ZFSet Select : ( ZFSet → Set m ) → ZFSet Replace : ( ZFSet → ZFSet ) → ZFSet infinite : ZFSet isZF : IsZF ZFSet _∋_ _≈_ ∅ _×_ Union Power Select Replace infinite module zf-exapmle {n m : Level } ( zf : ZF {m} {n} ) where _≈_ = ZF._≈_ zf ZFSet = ZF.ZFSet zf Select = ZF.Select zf ∅ = ZF.∅ zf _∩_ = ( IsZF._∩_ ) (ZF.isZF zf) _∋_ = ZF._∋_ zf replacement = IsZF.replacement ( ZF.isZF zf ) selection = IsZF.selection ( ZF.isZF zf ) minimul = IsZF.minimul ( ZF.isZF zf ) regularity = IsZF.regularity ( ZF.isZF zf ) -- russel : Select ( λ x → x ∋ x ) ≈ ∅ -- russel with Select ( λ x → x ∋ x ) -- ... | s = {!!} module constructible-set where open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ) open import Relation.Binary.PropositionalEquality data Ordinal {n : Level } : Set n where Φ : {lv : Nat} → Ordinal {n} lv T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv ℵ_ : (lv : Nat) → Ordinal (Suc lv) data _o<_ {n : Level } : Ordinal {n} → Ordinal {n} → Set n where l< : {lx ly : Nat } → {x : Ordinal {n} lx } → {y : Ordinal {n} ly } → lx < ly → x o< y Φ< : {lx : Nat} → {x : Ordinal {n} lx} → Φ {n} {lx} o< T-suc {n} {lx} x s< : {lx : Nat} → {x : Ordinal {n} lx} → x o< T-suc {n} {lx} x ℵΦ< : {lx : Nat} → {x : Ordinal {n} lx } → Φ {n} {lx} o< (ℵ lx) ℵ< : {lx : Nat} → {x : Ordinal {n} lx } → T-suc {n} {lx} x o< (ℵ lx) _o≈_ : {n : Level } {lv : Nat } → Rel ( Ordinal {n} lv ) n _o≈_ = {!!} triO : {n : Level } → {lx ly : Nat} → Trichotomous _o≈_ ( _o<_ {n} {lx} {ly} ) triO {n} {lv} Φ y = {!!} triO {n} {lv} (T-suc x) y = {!!} triO {n} {.(Suc lv)} (ℵ lv) y = {!!} max = Data.Nat._⊔_ maxα : {n : Level } → { lx ly : Nat } → Ordinal {n} lx → Ordinal {n} ly → Ordinal {n} (max lx ly) maxα x y with x o< y ... | t = {!!} -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) ' data Constructible {n : Level } {lv : Nat} ( α : Ordinal {n} lv ) : Set (suc n) where fsub : ( ψ : Ordinal {n} lv → Set n ) → Constructible α xself : Ordinal {n} lv → Constructible α record ConstructibleSet {n : Level } : Set (suc n) where field level : Nat α : Ordinal {n} level constructible : Constructible α open ConstructibleSet data _c∋_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } → Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where c> : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } (ta : Constructible {n} {lv} α ) ( tx : Constructible {n} {lv'} α' ) → α' o< α → ta c∋ tx xself-fsub : {lv : Nat} {α : Ordinal {n} lv } (ta : Ordinal {n} lv ) ( ψ : Ordinal {n} lv → Set n ) → _c∋_ {n} {_} {_} {α} {α} (xself ta ) ( fsub ψ) fsub-fsub : {lv lv' : Nat} {α : Ordinal {n} lv } ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) → ( ∀ ( x : Ordinal {n} lv ) → ψ x → ψ₁ x ) → _c∋_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) _∋_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n a ∋ x = constructible a c∋ constructible x data _c≈_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } → Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where crefl : {lv : Nat} {α : Ordinal {n} lv } → _c≈_ {n} {_} {_} {α} {α} (xself α ) (xself α ) feq : {lv : Nat} {α : Ordinal {n} lv } → ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) → (∀ ( x : Ordinal {n} lv ) → ψ x ⇔ ψ₁ x ) → _c≈_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) _≈_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n a ≈ x = constructible a c≈ constructible x ConstructibleSet→ZF : {n : Level } → ZF {suc n} {n} ConstructibleSet→ZF {n} = record { ZFSet = ConstructibleSet ; _∋_ = _∋_ ; _≈_ = _≈_ ; ∅ = record { level = Zero ; α = Φ ; constructible = xself Φ } ; _×_ = {!!} ; Union = {!!} ; Power = {!!} ; Select = {!!} ; Replace = {!!} ; infinite = {!!} ; isZF = {!!} } where