Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 352:e27769992399
next-is-limit
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 14 Jul 2020 11:12:09 +0900 |
| parents | fbabb20f222e |
| children | 17adeeee0c2a |
| 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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188
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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) |
| 274 | 22 : Set (suc (n ⊔ suc 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} _≈_ |
| 247 | 25 -- ∀ x ∀ y ∃ z ∀ t ( z ∋ t → t ≈ x ∨ t ≈ y) |
| 26 pair→ : ( x y t : ZFSet ) → (x , y) ∋ t → ( t ≈ x ) ∨ ( t ≈ y ) | |
| 27 pair← : ( x y t : ZFSet ) → ( t ≈ x ) ∨ ( t ≈ y ) → (x , y) ∋ t | |
| 69 | 28 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) |
| 73 | 29 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z |
| 159 | 30 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 3 | 31 _∈_ : ( A B : ZFSet ) → Set m |
| 32 A ∈ B = B ∋ A | |
| 23 | 33 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
| 34 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 3 | 35 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 36 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 37 _∪_ : ( A B : ZFSet ) → ZFSet |
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disjunction and conjunction
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38 A ∪ B = Union (A , B) |
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infinite and replacement begin
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39 ｛_｝ : ZFSet → ZFSet |
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infinite and replacement begin
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40 ｛ x ｝ = ( x , x ) |
| 3 | 41 infixr 200 _∈_ |
| 42 infixr 230 _∩_ _∪_ | |
| 43 infixr 220 _⊆_ | |
| 44 field | |
| 4 | 45 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 46 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 166 | 47 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} |
| 77 | 48 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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49 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 186 | 50 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) |
| 276 | 51 -- regularity without minimum |
| 324 | 52 ε-induction : { ψ : ZFSet → Set m} |
| 274 | 53 → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) |
| 54 → (x : ZFSet ) → ψ x | |
| 3 | 55 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 56 infinity∅ : ∅ ∈ infinite | |
| 160 | 57 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite |
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remove otrans again. start over
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58 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) |
| 3 | 59 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 130 | 60 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ |
| 138 | 61 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) |
| 3 | 62 |
| 274 | 63 record ZF {n m : Level } : Set (suc (n ⊔ suc m)) where |
| 18 | 64 infixr 210 _,_ |
| 6 | 65 infixl 200 _∋_ |
| 66 infixr 220 _≈_ | |
| 67 field | |
| 68 ZFSet : Set n | |
| 69 _∋_ : ( A x : ZFSet ) → Set m | |
| 70 _≈_ : ( A B : ZFSet ) → Set m | |
| 71 -- ZF Set constructor | |
| 72 ∅ : ZFSet | |
| 18 | 73 _,_ : ( A B : ZFSet ) → ZFSet |
| 6 | 74 Union : ( A : ZFSet ) → ZFSet |
| 75 Power : ( A : ZFSet ) → ZFSet | |
| 115 | 76 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet |
| 18 | 77 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet |
| 6 | 78 infinite : ZFSet |
| 18 | 79 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
| 6 | 80 |