Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 224:afc864169325
recover ε-induction
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Sat, 10 Aug 2019 12:31:25 +0900 |
| parents | 2e1f19c949dc |
| children | 650bdad56729 |
| rev | line source |
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| 3 | 1 module zf where |
| 2 | |
| 3 open import Level | |
| 4 | |
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separate logic and nat
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5 open import logic |
| 123 | 6 |
| 6 | 7 open import Relation.Nullary |
| 8 open import Relation.Binary | |
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axiom of choice → p ∨ ¬ p
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9 open import Data.Empty |
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axiom of choice → p ∨ ¬ p
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10 |
| 6 | 11 record IsZF {n m : Level } |
| 12 (ZFSet : Set n) | |
| 13 (_∋_ : ( A x : ZFSet ) → Set m) | |
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try to fix axiom of replacement
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14 (_≈_ : Rel ZFSet m) |
| 6 | 15 (∅ : ZFSet) |
| 18 | 16 (_,_ : ( A B : ZFSet ) → ZFSet) |
| 6 | 17 (Union : ( A : ZFSet ) → ZFSet) |
| 18 (Power : ( A : ZFSet ) → ZFSet) | |
| 115 | 19 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) |
| 18 | 20 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) |
| 6 | 21 (infinite : ZFSet) |
| 22 : Set (suc (n ⊔ m)) where | |
| 3 | 23 field |
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posturate OD is isomorphic to Ordinal
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24 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ |
| 3 | 25 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
| 18 | 26 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) |
| 69 | 27 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) |
| 73 | 28 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z |
| 159 | 29 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 3 | 30 _∈_ : ( A B : ZFSet ) → Set m |
| 31 A ∈ B = B ∋ A | |
| 23 | 32 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
| 33 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 3 | 34 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 35 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 36 _∪_ : ( A B : ZFSet ) → ZFSet |
| 103 | 37 A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easer |
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38 ｛_｝ : ZFSet → ZFSet |
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infinite and replacement begin
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39 ｛ x ｝ = ( x , x ) |
| 3 | 40 infixr 200 _∈_ |
| 41 infixr 230 _∩_ _∪_ | |
| 42 infixr 220 _⊆_ | |
| 43 field | |
| 4 | 44 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 45 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 166 | 46 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} |
| 77 | 47 power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t |
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¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
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48 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 186 | 49 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) |
| 183 | 50 -- This form of regurality forces choice function |
| 3 | 51 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 183 | 52 -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
| 53 -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | |
| 54 -- another form of regularity | |
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sepration of ordinal from OD
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55 -- ε-induction : { ψ : ZFSet → Set m} |
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56 -- → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) |
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57 -- → (x : ZFSet ) → ψ x |
| 3 | 58 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 59 infinity∅ : ∅ ∈ infinite | |
| 160 | 60 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite |
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61 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) |
| 3 | 62 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 130 | 63 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ |
| 138 | 64 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) |
| 183 | 65 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
| 66 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet | |
| 67 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A | |
| 3 | 68 |
| 6 | 69 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
| 18 | 70 infixr 210 _,_ |
| 6 | 71 infixl 200 _∋_ |
| 72 infixr 220 _≈_ | |
| 73 field | |
| 74 ZFSet : Set n | |
| 75 _∋_ : ( A x : ZFSet ) → Set m | |
| 76 _≈_ : ( A B : ZFSet ) → Set m | |
| 77 -- ZF Set constructor | |
| 78 ∅ : ZFSet | |
| 18 | 79 _,_ : ( A B : ZFSet ) → ZFSet |
| 6 | 80 Union : ( A : ZFSet ) → ZFSet |
| 81 Power : ( A : ZFSet ) → ZFSet | |
| 115 | 82 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet |
| 18 | 83 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet |
| 6 | 84 infinite : ZFSet |
| 18 | 85 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
| 6 | 86 |