Mercurial > hg > Members > kono > Proof > ZF-in-agda
view zf.agda @ 210:2c7d45734e3b
axiom of choice from exclusive middle done
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 01 Aug 2019 12:23:07 +0900 |
parents | e59e682ad120 |
children | 22d435172d1a |
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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 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where case1 : A → A ∨ B case2 : B → A ∨ B _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m) _⇔_ A B = ( A → B ) ∧ ( B → A ) open import Relation.Nullary open import Relation.Binary open import Data.Empty contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a ) double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A double-neg A notnot = notnot A double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A double-neg2 notnot A = notnot ( double-neg A ) de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) ) de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and )) de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and )) dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a ) dont-or {A} {B} (case2 b) ¬A = b dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b ) dont-orb {A} {B} (case1 a) ¬B = a -- mid-ex-neg : {n : Level } {A : Set n} → ( ¬ ¬ A ) ∨ (¬ A) -- mid-ex-neg {n} {A} = {!!} infixr 130 _∧_ infixr 140 _∨_ infixr 150 _⇔_ 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 : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) (infinite : ZFSet) : Set (suc (n ⊔ m)) where field isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) _∈_ : ( 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 ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) _∪_ : ( A B : ZFSet ) → ZFSet A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easer {_} : ZFSet → ZFSet { x } = ( x , 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 ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) -- This form of regurality forces choice function -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) -- another form of regularity ε-induction : { ψ : ZFSet → Set m} → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) → (x : ZFSet ) → ψ x -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) infinity∅ : ∅ ∈ infinite infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ { x }) ∈ infinite selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) -- 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 ) → x ∈ X → ψ x ∈ Replace X ψ replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A 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 : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet infinite : ZFSet isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite