Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 104:d92411bed18c
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sun, 16 Jun 2019 02:06:09 +0900 |
parents | c8b79d303867 |
children | 1daa1d24348c |
rev | line source |
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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 | |
18 _⇔_ : {n : Level } → ( A B : Set n ) → Set n | |
77 | 19 _⇔_ A B = ( A → B ) ∧ ( B → A ) |
3 | 20 |
6 | 21 open import Relation.Nullary |
22 open import Relation.Binary | |
23 | |
103 | 24 contra-position : {n : Level } {A B : Set n} → (A → B) → ¬ B → ¬ A |
25 contra-position {n} {A} {B} f ¬b a = ¬b ( f a ) | |
26 | |
3 | 27 infixr 130 _∧_ |
28 infixr 140 _∨_ | |
29 infixr 150 _⇔_ | |
30 | |
6 | 31 record IsZF {n m : Level } |
32 (ZFSet : Set n) | |
33 (_∋_ : ( A x : ZFSet ) → Set m) | |
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try to fix axiom of replacement
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34 (_≈_ : Rel ZFSet m) |
6 | 35 (∅ : ZFSet) |
18 | 36 (_,_ : ( A B : ZFSet ) → ZFSet) |
6 | 37 (Union : ( A : ZFSet ) → ZFSet) |
38 (Power : ( A : ZFSet ) → ZFSet) | |
18 | 39 (Select : ZFSet → ( ZFSet → Set m ) → ZFSet ) |
40 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) | |
6 | 41 (infinite : ZFSet) |
42 : Set (suc (n ⊔ m)) where | |
3 | 43 field |
29
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posturate OD is isomorphic to Ordinal
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parents:
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44 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ |
3 | 45 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
18 | 46 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) |
69 | 47 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) |
70 | 48 union-u : ( X z : ZFSet ) → Union X ∋ z → ZFSet |
73 | 49 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z |
72 | 50 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → (X ∋ union-u X z X∋z) ∧ (union-u X z X∋z ∋ z ) |
3 | 51 _∈_ : ( A B : ZFSet ) → Set m |
52 A ∈ B = B ∋ A | |
23 | 53 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
54 _⊆_ A B {x} = A ∋ x → B ∋ x | |
3 | 55 _∩_ : ( A B : ZFSet ) → ZFSet |
51 | 56 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
3 | 57 _∪_ : ( A B : ZFSet ) → ZFSet |
103 | 58 A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easer |
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infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 {_} : ZFSet → ZFSet |
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infinite and replacement begin
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60 { x } = ( x , x ) |
3 | 61 infixr 200 _∈_ |
62 infixr 230 _∩_ _∪_ | |
63 infixr 220 _⊆_ | |
64 field | |
4 | 65 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
3 | 66 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
23 | 67 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → _⊆_ t A {x} |
77 | 68 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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69 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
76 | 70 extensionality : { A B : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B |
3 | 71 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
37 | 72 minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
73 regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | |
3 | 74 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
75 infinity∅ : ∅ ∈ infinite | |
78
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infinite and replacement begin
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parents:
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76 infinity : ∀( X x : ZFSet ) → x ∈ infinite → ( x ∪ { x }) ∈ infinite |
54 | 77 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) |
3 | 78 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
18 | 79 replacement : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( ψ x ∈ Replace X ψ ) |
103 | 80 -- -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ { t }) ] |
81 -- axiom-of-choice : Set (suc n) | |
82 -- axiom-of-choice = ? | |
3 | 83 |
6 | 84 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
18 | 85 infixr 210 _,_ |
6 | 86 infixl 200 _∋_ |
87 infixr 220 _≈_ | |
88 field | |
89 ZFSet : Set n | |
90 _∋_ : ( A x : ZFSet ) → Set m | |
91 _≈_ : ( A B : ZFSet ) → Set m | |
92 -- ZF Set constructor | |
93 ∅ : ZFSet | |
18 | 94 _,_ : ( A B : ZFSet ) → ZFSet |
6 | 95 Union : ( A : ZFSet ) → ZFSet |
96 Power : ( A : ZFSet ) → ZFSet | |
18 | 97 Select : ZFSet → ( ZFSet → Set m ) → ZFSet |
98 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet | |
6 | 99 infinite : ZFSet |
18 | 100 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
6 | 101 |