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1 module zf where
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2
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3 open import Level
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4
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5
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6 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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7 field
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8 proj1 : A
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9 proj2 : B
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10
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11 open _∧_
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12
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13
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14 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
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15 case1 : A → A ∨ B
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16 case2 : B → A ∨ B
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17
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18 -- open import Relation.Binary.PropositionalEquality
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19
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20 _⇔_ : {n : Level } → ( A B : Set n ) → Set n
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21 _⇔_ A B = ( A → B ) ∧ ( B → A )
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22
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23 open import Data.Empty
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24 open import Relation.Nullary
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25
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26 open import Relation.Binary
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27 open import Relation.Binary.Core
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28
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29 infixr 130 _∧_
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30 infixr 140 _∨_
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31 infixr 150 _⇔_
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32
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33 record IsZF {n m : Level }
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34 (ZFSet : Set n)
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35 (_∋_ : ( A x : ZFSet ) → Set m)
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36 (_≈_ : ( A B : ZFSet ) → Set m)
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37 (∅ : ZFSet)
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38 (_×_ : ( A B : ZFSet ) → ZFSet)
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39 (Union : ( A : ZFSet ) → ZFSet)
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40 (Power : ( A : ZFSet ) → ZFSet)
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41 (Restrict : ( ZFSet → Set m ) → ZFSet)
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42 (infinite : ZFSet)
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43 : Set (suc (n ⊔ m)) where
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44 field
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45 isEquivalence : {A B : ZFSet} → IsEquivalence {n} {m} {ZFSet} _≈_
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46 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
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47 pair : ( A B : ZFSet ) → ( (A × B) ∋ A ) ∧ ( (A × B) ∋ B )
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48 -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x))
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49 union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y
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50 union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y
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51 _∈_ : ( A B : ZFSet ) → Set m
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52 A ∈ B = B ∋ A
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53 _⊆_ : ( A B : ZFSet ) → { x : ZFSet } → { A∋x : Set m } → Set m
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54 _⊆_ A B {x} {A∋x} = B ∋ x
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55 _∩_ : ( A B : ZFSet ) → ZFSet
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56 A ∩ B = Restrict ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) )
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57 _∪_ : ( A B : ZFSet ) → ZFSet
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58 A ∪ B = Restrict ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) )
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59 infixr 200 _∈_
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60 infixr 230 _∩_ _∪_
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61 infixr 220 _⊆_
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62 field
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63 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
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64 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
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65 power→ : ( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y}
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66 power← : ( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t
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67 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
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68 extentionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B
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69 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
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70 -- smaller : ZFSet → ZFSet
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71 -- regularity : ( x : ZFSet ) → ¬ (x ≈ ∅) → smaller x ∩ x ≈ ∅
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72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
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73 infinity∅ : ∅ ∈ infinite
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74 infinity : ( x : ZFSet ) → x ∈ infinite → ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite
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75 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
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76 replacement : ( ψ : ZFSet → Set m ) → ( y : ZFSet ) → y ∈ Restrict ψ → ψ y
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77
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78 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
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79 coinductive
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80 infixr 210 _×_
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81 infixl 200 _∋_
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82 infixr 220 _≈_
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83 field
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84 ZFSet : Set n
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85 _∋_ : ( A x : ZFSet ) → Set m
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86 _≈_ : ( A B : ZFSet ) → Set m
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87 -- ZF Set constructor
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88 ∅ : ZFSet
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89 _×_ : ( A B : ZFSet ) → ZFSet
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90 Union : ( A : ZFSet ) → ZFSet
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91 Power : ( A : ZFSet ) → ZFSet
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92 Restrict : ( ZFSet → Set m ) → ZFSet
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93 infinite : ZFSet
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94 field
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95 isZF : IsZF ZFSet _∋_ _≈_ ∅ _×_ Union Power Restrict infinite
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96
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