Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 129:2a5519dcc167
ord power set
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 02 Jul 2019 09:28:26 +0900 |
| parents | 0c2cbf37e002 |
| children | 3849614bef18 |
| 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 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where | |
| 15 case1 : A → A ∨ B | |
| 16 case2 : B → A ∨ B | |
| 17 | |
| 116 | 18 _⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m ) → Set (n ⊔ m) |
| 77 | 19 _⇔_ A B = ( A → B ) ∧ ( B → A ) |
| 3 | 20 |
| 123 | 21 |
| 6 | 22 open import Relation.Nullary |
| 23 open import Relation.Binary | |
| 24 | |
| 103 | 25 contra-position : {n : Level } {A B : Set n} → (A → B) → ¬ B → ¬ A |
| 26 contra-position {n} {A} {B} f ¬b a = ¬b ( f a ) | |
| 27 | |
| 3 | 28 infixr 130 _∧_ |
| 29 infixr 140 _∨_ | |
| 30 infixr 150 _⇔_ | |
| 31 | |
| 6 | 32 record IsZF {n m : Level } |
| 33 (ZFSet : Set n) | |
| 34 (_∋_ : ( A x : ZFSet ) → Set m) | |
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9
5ed16e2d8eb7
try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
changeset
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35 (_≈_ : Rel ZFSet m) |
| 6 | 36 (∅ : ZFSet) |
| 18 | 37 (_,_ : ( A B : ZFSet ) → ZFSet) |
| 6 | 38 (Union : ( A : ZFSet ) → ZFSet) |
| 39 (Power : ( A : ZFSet ) → ZFSet) | |
| 115 | 40 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) |
| 18 | 41 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) |
| 6 | 42 (infinite : ZFSet) |
| 43 : Set (suc (n ⊔ m)) where | |
| 3 | 44 field |
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29
fce60b99dc55
posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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45 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ |
| 3 | 46 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
| 18 | 47 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) |
| 69 | 48 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) |
| 70 | 49 union-u : ( X z : ZFSet ) → Union X ∋ z → ZFSet |
| 73 | 50 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z |
| 72 | 51 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z ) |
| 3 | 52 _∈_ : ( A B : ZFSet ) → Set m |
| 53 A ∈ B = B ∋ A | |
| 23 | 54 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
| 55 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 3 | 56 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 57 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 58 _∪_ : ( A B : ZFSet ) → ZFSet |
| 103 | 59 A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easer |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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diff
changeset
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60 ｛_｝ : ZFSet → ZFSet |
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9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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61 ｛ x ｝ = ( x , x ) |
| 3 | 62 infixr 200 _∈_ |
| 63 infixr 230 _∩_ _∪_ | |
| 64 infixr 220 _⊆_ | |
| 65 field | |
| 4 | 66 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 67 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 23 | 68 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → _⊆_ t A {x} |
| 77 | 69 power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t |
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65
164ad5a703d8
¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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70 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 76 | 71 extensionality : { A B : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B |
| 3 | 72 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 37 | 73 minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
| 74 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | |
| 3 | 75 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 76 infinity∅ : ∅ ∈ infinite | |
|
78
9a7a64b2388c
infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
77
diff
changeset
|
77 infinity : ∀( X x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite |
| 116 | 78 selection : ∀ { X : ZFSet } → { ψ : (x : ZFSet ) → Set m } → ∀ { y : ZFSet } → (((y : ZFSet) → y ∈ X → ψ y ) ∧ ( y ∈ X ) ) ⇔ (y ∈ Select X ψ ) |
| 3 | 79 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 18 | 80 replacement : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( ψ x ∈ Replace X ψ ) |
| 103 | 81 -- -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ] |
| 82 -- axiom-of-choice : Set (suc n) | |
| 83 -- axiom-of-choice = ? | |
| 3 | 84 |
| 6 | 85 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
| 18 | 86 infixr 210 _,_ |
| 6 | 87 infixl 200 _∋_ |
| 88 infixr 220 _≈_ | |
| 89 field | |
| 90 ZFSet : Set n | |
| 91 _∋_ : ( A x : ZFSet ) → Set m | |
| 92 _≈_ : ( A B : ZFSet ) → Set m | |
| 93 -- ZF Set constructor | |
| 94 ∅ : ZFSet | |
| 18 | 95 _,_ : ( A B : ZFSet ) → ZFSet |
| 6 | 96 Union : ( A : ZFSet ) → ZFSet |
| 97 Power : ( A : ZFSet ) → ZFSet | |
| 115 | 98 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet |
| 18 | 99 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet |
| 6 | 100 infinite : ZFSet |
| 18 | 101 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
| 6 | 102 |