Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 65:164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
lemma ox = TransFinite {suc n} {λ ox → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} } { }0 { }1 { }2 ox
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 30 May 2019 01:02:47 +0900 |
| parents | 33fb8228ace9 |
| children | 93abc0133b8a |
| rev | line source |
|---|---|
| 3 | 1 module zf where |
| 2 | |
| 3 open import Level | |
| 4 | |
| 23 | 5 data Bool : Set where |
| 6 true : Bool | |
| 7 false : Bool | |
| 3 | 8 |
| 9 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where | |
| 10 field | |
| 11 proj1 : A | |
| 12 proj2 : B | |
| 13 | |
| 14 open _∧_ | |
| 15 | |
| 16 | |
| 17 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where | |
| 18 case1 : A → A ∨ B | |
| 19 case2 : B → A ∨ B | |
| 20 | |
| 6 | 21 -- open import Relation.Binary.PropositionalEquality |
| 3 | 22 |
| 23 _⇔_ : {n : Level } → ( A B : Set n ) → Set n | |
| 24 _⇔_ A B = ( A → B ) ∧ ( B → A ) | |
| 25 | |
| 6 | 26 open import Data.Empty |
| 27 open import Relation.Nullary | |
| 28 | |
| 29 open import Relation.Binary | |
| 30 open import Relation.Binary.Core | |
| 31 | |
| 3 | 32 infixr 130 _∧_ |
| 33 infixr 140 _∨_ | |
| 34 infixr 150 _⇔_ | |
| 35 | |
| 6 | 36 record IsZF {n m : Level } |
| 37 (ZFSet : Set n) | |
| 38 (_∋_ : ( A x : ZFSet ) → Set m) | |
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try to fix axiom of replacement
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39 (_≈_ : Rel ZFSet m) |
| 6 | 40 (∅ : ZFSet) |
| 18 | 41 (_,_ : ( A B : ZFSet ) → ZFSet) |
| 6 | 42 (Union : ( A : ZFSet ) → ZFSet) |
| 43 (Power : ( A : ZFSet ) → ZFSet) | |
| 18 | 44 (Select : ZFSet → ( ZFSet → Set m ) → ZFSet ) |
| 45 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) | |
| 6 | 46 (infinite : ZFSet) |
| 47 : Set (suc (n ⊔ m)) where | |
| 3 | 48 field |
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posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ |
| 3 | 50 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
| 18 | 51 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) |
| 3 | 52 -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x)) |
| 53 union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y | |
| 54 union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y | |
| 55 _∈_ : ( A B : ZFSet ) → Set m | |
| 56 A ∈ B = B ∋ A | |
| 23 | 57 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
| 58 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 3 | 59 _∩_ : ( A B : ZFSet ) → ZFSet |
| 51 | 60 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 61 _∪_ : ( A B : ZFSet ) → ZFSet |
| 51 | 62 A ∪ B = Union (A , B) |
| 3 | 63 infixr 200 _∈_ |
| 64 infixr 230 _∩_ _∪_ | |
| 65 infixr 220 _⊆_ | |
| 66 field | |
| 4 | 67 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 68 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 23 | 69 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → _⊆_ t A {x} |
| 70 power← : ∀( A t : ZFSet ) → ∀ {x} → _⊆_ t A {x} → Power A ∋ t | |
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164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
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71 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
|
164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
54
diff
changeset
|
72 extensionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B |
| 3 | 73 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 37 | 74 minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
| 75 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | |
| 3 | 76 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 77 infinity∅ : ∅ ∈ infinite | |
| 18 | 78 infinity : ∀( X x : ZFSet ) → x ∈ infinite → ( x ∪ Select X ( λ y → x ≈ y )) ∈ infinite |
| 54 | 79 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) |
| 3 | 80 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 18 | 81 replacement : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( ψ x ∈ Replace X ψ ) |
| 3 | 82 |
| 6 | 83 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
| 18 | 84 infixr 210 _,_ |
| 6 | 85 infixl 200 _∋_ |
| 86 infixr 220 _≈_ | |
| 87 field | |
| 88 ZFSet : Set n | |
| 89 _∋_ : ( A x : ZFSet ) → Set m | |
| 90 _≈_ : ( A B : ZFSet ) → Set m | |
| 91 -- ZF Set constructor | |
| 92 ∅ : ZFSet | |
| 18 | 93 _,_ : ( A B : ZFSet ) → ZFSet |
| 6 | 94 Union : ( A : ZFSet ) → ZFSet |
| 95 Power : ( A : ZFSet ) → ZFSet | |
| 18 | 96 Select : ZFSet → ( ZFSet → Set m ) → ZFSet |
| 97 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet | |
| 6 | 98 infinite : ZFSet |
| 18 | 99 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
| 6 | 100 |
| 10 | 101 module zf-exapmle {n m : Level } ( zf : ZF {m} {n} ) where |
| 7 | 102 |
| 10 | 103 _≈_ = ZF._≈_ zf |
| 104 ZFSet = ZF.ZFSet zf | |
| 105 Select = ZF.Select zf | |
| 106 ∅ = ZF.∅ zf | |
| 107 _∩_ = ( IsZF._∩_ ) (ZF.isZF zf) | |
| 108 _∋_ = ZF._∋_ zf | |
| 109 replacement = IsZF.replacement ( ZF.isZF zf ) | |
| 110 selection = IsZF.selection ( ZF.isZF zf ) | |
| 111 minimul = IsZF.minimul ( ZF.isZF zf ) | |
| 112 regularity = IsZF.regularity ( ZF.isZF zf ) | |
| 7 | 113 |
| 11 | 114 -- russel : Select ( λ x → x ∋ x ) ≈ ∅ |
| 115 -- russel with Select ( λ x → x ∋ x ) | |
| 116 -- ... | s = {!!} | |
| 7 | 117 |