Mercurial > hg > Members > kono > Proof > ZF-in-agda
view zf.agda @ 24:3186bbee99dd
separte level
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 18 May 2019 16:03:10 +0900 |
parents | 7293a151d949 |
children | fce60b99dc55 |
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module zf where open import Level data Bool : Set where true : Bool false : Bool 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 → ( ZFSet → Set m ) → ZFSet ) (Replace : ZFSet → ( 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 } → Set m _⊆_ A B {x} = A ∋ x → B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) _∪_ : ( A B : ZFSet ) → ZFSet A ∪ B = Select (A , B) ( 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} → _⊆_ t A {x} power← : ∀( A t : ZFSet ) → ∀ {x} → _⊆_ t A {x} → 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 x : ZFSet ) → x ∈ infinite → ( x ∪ Select X ( λ y → x ≈ y )) ∈ infinite selection : { ψ : ZFSet → Set m } → ∀ ( X y : ZFSet ) → ( y ∈ Select X 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 x : ZFSet ) → ( ψ x ∈ Replace X 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 → ( ZFSet → Set m ) → ZFSet Replace : ZFSet → ( 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 = {!!}