Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 209:e59e682ad120
∀-imply-or
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 01 Aug 2019 10:22:16 +0900 |
| parents | 64ef1db53c49 |
| children | 22d435172d1a |
| 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 | |
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24 open import Data.Empty |
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25 |
| 6 | 26 |
| 138 | 27 contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A |
| 28 contra-position {n} {m} {A} {B} f ¬b a = ¬b ( f a ) | |
| 103 | 29 |
| 166 | 30 double-neg : {n : Level } {A : Set n} → A → ¬ ¬ A |
| 31 double-neg A notnot = notnot A | |
| 32 | |
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33 double-neg2 : {n : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A |
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34 double-neg2 notnot A = notnot ( double-neg A ) |
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35 |
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36 de-morgan : {n : Level } {A B : Set n} → A ∧ B → ¬ ( (¬ A ) ∨ (¬ B ) ) |
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37 de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and )) |
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38 de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and )) |
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39 |
| 208 | 40 dont-or : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ A → B |
| 41 dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a ) | |
| 42 dont-or {A} {B} (case2 b) ¬A = b | |
| 43 | |
| 209 | 44 dont-orb : {n m : Level} {A : Set n} { B : Set m } → A ∨ B → ¬ B → A |
| 45 dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b ) | |
| 46 dont-orb {A} {B} (case1 a) ¬B = a | |
| 47 | |
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48 -- mid-ex-neg : {n : Level } {A : Set n} → ( ¬ ¬ A ) ∨ (¬ A) |
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49 -- mid-ex-neg {n} {A} = {!!} |
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50 |
| 3 | 51 infixr 130 _∧_ |
| 52 infixr 140 _∨_ | |
| 53 infixr 150 _⇔_ | |
| 54 | |
| 6 | 55 record IsZF {n m : Level } |
| 56 (ZFSet : Set n) | |
| 57 (_∋_ : ( A x : ZFSet ) → Set m) | |
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try to fix axiom of replacement
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58 (_≈_ : Rel ZFSet m) |
| 6 | 59 (∅ : ZFSet) |
| 18 | 60 (_,_ : ( A B : ZFSet ) → ZFSet) |
| 6 | 61 (Union : ( A : ZFSet ) → ZFSet) |
| 62 (Power : ( A : ZFSet ) → ZFSet) | |
| 115 | 63 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet ) |
| 18 | 64 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet ) |
| 6 | 65 (infinite : ZFSet) |
| 66 : Set (suc (n ⊔ m)) where | |
| 3 | 67 field |
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posturate OD is isomorphic to Ordinal
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68 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ |
| 3 | 69 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
| 18 | 70 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B ) |
| 69 | 71 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u)) |
| 73 | 72 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z |
| 159 | 73 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z ))) |
| 3 | 74 _∈_ : ( A B : ZFSet ) → Set m |
| 75 A ∈ B = B ∋ A | |
| 23 | 76 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m |
| 77 _⊆_ A B {x} = A ∋ x → B ∋ x | |
| 3 | 78 _∩_ : ( A B : ZFSet ) → ZFSet |
| 115 | 79 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 80 _∪_ : ( A B : ZFSet ) → ZFSet |
| 103 | 81 A ∪ B = Union (A , B) -- Select A ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) is easer |
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82 ｛_｝ : ZFSet → ZFSet |
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83 ｛ x ｝ = ( x , x ) |
| 3 | 84 infixr 200 _∈_ |
| 85 infixr 230 _∩_ _∪_ | |
| 86 infixr 220 _⊆_ | |
| 87 field | |
| 4 | 88 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 89 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 166 | 90 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x} |
| 77 | 91 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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92 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 186 | 93 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w ) |
| 183 | 94 -- This form of regurality forces choice function |
| 3 | 95 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 183 | 96 -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet |
| 97 -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) ) | |
| 98 -- another form of regularity | |
| 99 ε-induction : { ψ : ZFSet → Set m} | |
| 100 → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x ) | |
| 101 → (x : ZFSet ) → ψ x | |
| 3 | 102 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 103 infinity∅ : ∅ ∈ infinite | |
| 160 | 104 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ ｛ x ｝) ∈ infinite |
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105 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ ) |
| 3 | 106 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 130 | 107 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ |
| 138 | 108 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) ) |
| 183 | 109 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] |
| 110 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet | |
| 111 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A | |
| 3 | 112 |
| 6 | 113 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
| 18 | 114 infixr 210 _,_ |
| 6 | 115 infixl 200 _∋_ |
| 116 infixr 220 _≈_ | |
| 117 field | |
| 118 ZFSet : Set n | |
| 119 _∋_ : ( A x : ZFSet ) → Set m | |
| 120 _≈_ : ( A B : ZFSet ) → Set m | |
| 121 -- ZF Set constructor | |
| 122 ∅ : ZFSet | |
| 18 | 123 _,_ : ( A B : ZFSet ) → ZFSet |
| 6 | 124 Union : ( A : ZFSet ) → ZFSet |
| 125 Power : ( A : ZFSet ) → ZFSet | |
| 115 | 126 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet |
| 18 | 127 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet |
| 6 | 128 infinite : ZFSet |
| 18 | 129 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite |
| 6 | 130 |