Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate zf.agda @ 11:2df90eb0896c
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 13 May 2019 20:51:45 +0900 |
| parents | 8022e14fce74 |
| children | b531d2b417ad e11e95d5ddee |
| rev | line source |
|---|---|
| 3 | 1 module zf where |
| 2 | |
| 3 open import Level | |
| 4 | |
| 5 | |
| 6 record _∧_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where | |
| 7 field | |
| 8 proj1 : A | |
| 9 proj2 : B | |
| 10 | |
| 11 open _∧_ | |
| 12 | |
| 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 | |
| 6 | 18 -- open import Relation.Binary.PropositionalEquality |
| 3 | 19 |
| 20 _⇔_ : {n : Level } → ( A B : Set n ) → Set n | |
| 21 _⇔_ A B = ( A → B ) ∧ ( B → A ) | |
| 22 | |
| 6 | 23 open import Data.Empty |
| 24 open import Relation.Nullary | |
| 25 | |
| 26 open import Relation.Binary | |
| 27 open import Relation.Binary.Core | |
| 28 | |
| 3 | 29 infixr 130 _∧_ |
| 30 infixr 140 _∨_ | |
| 31 infixr 150 _⇔_ | |
| 32 | |
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33 record Func {n m : Level } (ZFSet : Set n) (_≈_ : Rel ZFSet m) : Set (n ⊔ suc m) where |
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parents:
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34 field |
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parents:
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35 rel : Rel ZFSet m |
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parents:
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36 dom : ( y : ZFSet ) → ∀ { x : ZFSet } → rel x y |
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5ed16e2d8eb7
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parents:
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changeset
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37 fun-eq : { x y z : ZFSet } → ( rel x y ∧ rel x z ) → y ≈ z |
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5ed16e2d8eb7
try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
8
diff
changeset
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38 |
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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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39 open Func |
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5ed16e2d8eb7
try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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40 |
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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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41 |
| 6 | 42 record IsZF {n m : Level } |
| 43 (ZFSet : Set n) | |
| 44 (_∋_ : ( A x : ZFSet ) → Set m) | |
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5ed16e2d8eb7
try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
8
diff
changeset
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45 (_≈_ : Rel ZFSet m) |
| 6 | 46 (∅ : ZFSet) |
| 47 (_×_ : ( A B : ZFSet ) → ZFSet) | |
| 48 (Union : ( A : ZFSet ) → ZFSet) | |
| 49 (Power : ( A : ZFSet ) → ZFSet) | |
| 10 | 50 (Select : ( ZFSet → Set m ) → ZFSet ) |
| 51 (Replace : ( ZFSet → ZFSet ) → ZFSet ) | |
| 6 | 52 (infinite : ZFSet) |
| 53 : Set (suc (n ⊔ m)) where | |
| 3 | 54 field |
| 6 | 55 isEquivalence : {A B : ZFSet} → IsEquivalence {n} {m} {ZFSet} _≈_ |
| 3 | 56 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) |
| 6 | 57 pair : ( A B : ZFSet ) → ( (A × B) ∋ A ) ∧ ( (A × B) ∋ B ) |
| 3 | 58 -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x)) |
| 59 union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y | |
| 60 union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y | |
| 61 _∈_ : ( A B : ZFSet ) → Set m | |
| 62 A ∈ B = B ∋ A | |
| 7 | 63 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → ∀{ A∋x : Set m } → Set m |
| 3 | 64 _⊆_ A B {x} {A∋x} = B ∋ x |
| 65 _∩_ : ( A B : ZFSet ) → ZFSet | |
| 10 | 66 A ∩ B = Select ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) ) |
| 3 | 67 _∪_ : ( A B : ZFSet ) → ZFSet |
| 10 | 68 A ∪ B = Select ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) ) |
| 3 | 69 infixr 200 _∈_ |
| 70 infixr 230 _∩_ _∪_ | |
| 71 infixr 220 _⊆_ | |
| 72 field | |
| 4 | 73 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) |
| 3 | 74 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) |
| 8 | 75 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y} |
| 76 power← : ∀( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t | |
| 3 | 77 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) |
| 6 | 78 extentionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B |
| 3 | 79 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) |
| 10 | 80 minimul : ZFSet → ZFSet |
| 81 regularity : ∀( x : ZFSet ) → ¬ (x ≈ ∅) → ( minimul x ∈ x ∧ ( minimul x ∩ x ≈ ∅ ) ) | |
| 3 | 82 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) |
| 83 infinity∅ : ∅ ∈ infinite | |
| 10 | 84 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ Select ( λ y → x ≈ y )) ∈ infinite |
| 85 selection : { ψ : ZFSet → Set m } → ∀ ( y : ZFSet ) → ( y ∈ Select ψ ) → ψ y | |
| 3 | 86 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) |
| 10 | 87 replacement : {ψ : ZFSet → ZFSet} → ∀ ( x : ZFSet ) → ( ψ x ∈ Replace ψ ) |
| 3 | 88 |
| 6 | 89 record ZF {n m : Level } : Set (suc (n ⊔ m)) where |
| 90 infixr 210 _×_ | |
| 91 infixl 200 _∋_ | |
| 92 infixr 220 _≈_ | |
| 93 field | |
| 94 ZFSet : Set n | |
| 95 _∋_ : ( A x : ZFSet ) → Set m | |
| 96 _≈_ : ( A B : ZFSet ) → Set m | |
| 97 -- ZF Set constructor | |
| 98 ∅ : ZFSet | |
| 99 _×_ : ( A B : ZFSet ) → ZFSet | |
| 100 Union : ( A : ZFSet ) → ZFSet | |
| 101 Power : ( A : ZFSet ) → ZFSet | |
| 10 | 102 Select : ( ZFSet → Set m ) → ZFSet |
| 103 Replace : ( ZFSet → ZFSet ) → ZFSet | |
| 6 | 104 infinite : ZFSet |
| 10 | 105 isZF : IsZF ZFSet _∋_ _≈_ ∅ _×_ Union Power Select Replace infinite |
| 6 | 106 |
| 10 | 107 module zf-exapmle {n m : Level } ( zf : ZF {m} {n} ) where |
| 7 | 108 |
| 10 | 109 _≈_ = ZF._≈_ zf |
| 110 ZFSet = ZF.ZFSet zf | |
| 111 Select = ZF.Select zf | |
| 112 ∅ = ZF.∅ zf | |
| 113 _∩_ = ( IsZF._∩_ ) (ZF.isZF zf) | |
| 114 _∋_ = ZF._∋_ zf | |
| 115 replacement = IsZF.replacement ( ZF.isZF zf ) | |
| 116 selection = IsZF.selection ( ZF.isZF zf ) | |
| 117 minimul = IsZF.minimul ( ZF.isZF zf ) | |
| 118 regularity = IsZF.regularity ( ZF.isZF zf ) | |
| 7 | 119 |
| 11 | 120 -- russel : Select ( λ x → x ∋ x ) ≈ ∅ |
| 121 -- russel with Select ( λ x → x ∋ x ) | |
| 122 -- ... | s = {!!} | |
| 7 | 123 |
| 10 | 124 module constructible-set where |
| 7 | 125 |
| 11 | 126 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ) |
| 10 | 127 |
| 128 open import Relation.Binary.PropositionalEquality | |
| 129 | |
| 11 | 130 data Ordinal {n : Level } : Set n where |
| 131 Φ : {lv : Nat} → Ordinal {n} lv | |
| 132 T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv | |
| 133 ℵ_ : (lv : Nat) → Ordinal (Suc lv) | |
| 134 | |
| 135 data _o<_ {n : Level } : Ordinal {n} → Ordinal {n} → Set n where | |
| 136 l< : {lx ly : Nat } → {x : Ordinal {n} lx } → {y : Ordinal {n} ly } → lx < ly → x o< y | |
| 137 Φ< : {lx : Nat} → {x : Ordinal {n} lx} → Φ {n} {lx} o< T-suc {n} {lx} x | |
| 138 s< : {lx : Nat} → {x : Ordinal {n} lx} → x o< T-suc {n} {lx} x | |
| 139 ℵΦ< : {lx : Nat} → {x : Ordinal {n} lx } → Φ {n} {lx} o< (ℵ lx) | |
| 140 ℵ< : {lx : Nat} → {x : Ordinal {n} lx } → T-suc {n} {lx} x o< (ℵ lx) | |
| 141 | |
| 142 _o≈_ : {n : Level } {lv : Nat } → Rel ( Ordinal {n} lv ) n | |
| 143 _o≈_ = {!!} | |
| 144 | |
| 145 triO : {n : Level } → {lx ly : Nat} → Trichotomous _o≈_ ( _o<_ {n} {lx} {ly} ) | |
| 146 triO {n} {lv} Φ y = {!!} | |
| 147 triO {n} {lv} (T-suc x) y = {!!} | |
| 148 triO {n} {.(Suc lv)} (ℵ lv) y = {!!} | |
| 149 | |
| 150 | |
| 151 max = Data.Nat._⊔_ | |
| 10 | 152 |
| 11 | 153 maxα : {n : Level } → { lx ly : Nat } → Ordinal {n} lx → Ordinal {n} ly → Ordinal {n} (max lx ly) |
| 154 maxα x y with x o< y | |
| 155 ... | t = {!!} | |
| 156 | |
| 157 -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) ' | |
| 158 | |
| 159 data Constructible {n : Level } {lv : Nat} ( α : Ordinal {n} lv ) : Set (suc n) where | |
| 160 fsub : ( ψ : Ordinal {n} lv → Set n ) → Constructible α | |
| 161 xself : Ordinal {n} lv → Constructible α | |
| 10 | 162 |
| 163 record ConstructibleSet {n : Level } : Set (suc n) where | |
| 164 field | |
| 165 level : Nat | |
| 11 | 166 α : Ordinal {n} level |
| 10 | 167 constructible : Constructible α |
| 168 | |
| 169 open ConstructibleSet | |
| 170 | |
| 11 | 171 data _c∋_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } → |
| 10 | 172 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where |
| 11 | 173 c> : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } |
| 174 (ta : Constructible {n} {lv} α ) ( tx : Constructible {n} {lv'} α' ) → α' o< α → ta c∋ tx | |
| 175 xself-fsub : {lv : Nat} {α : Ordinal {n} lv } | |
| 176 (ta : Ordinal {n} lv ) ( ψ : Ordinal {n} lv → Set n ) → _c∋_ {n} {_} {_} {α} {α} (xself ta ) ( fsub ψ) | |
| 177 fsub-fsub : {lv lv' : Nat} {α : Ordinal {n} lv } | |
| 178 ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) → | |
| 179 ( ∀ ( x : Ordinal {n} lv ) → ψ x → ψ₁ x ) → _c∋_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) | |
| 7 | 180 |
| 11 | 181 _∋_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n |
| 182 a ∋ x = constructible a c∋ constructible x | |
| 183 | |
| 184 data _c≈_ {n : Level } : {lv lv' : Nat} {α : Ordinal {n} lv } {α' : Ordinal {n} lv' } → | |
| 185 Constructible {n} {lv} α → Constructible {n} {lv'} α' → Set n where | |
| 186 crefl : {lv : Nat} {α : Ordinal {n} lv } → _c≈_ {n} {_} {_} {α} {α} (xself α ) (xself α ) | |
| 187 feq : {lv : Nat} {α : Ordinal {n} lv } | |
| 188 → ( ψ : Ordinal {n} lv → Set n ) ( ψ₁ : Ordinal {n} lv → Set n ) | |
| 189 → (∀ ( x : Ordinal {n} lv ) → ψ x ⇔ ψ₁ x ) → _c≈_ {n} {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) | |
| 190 | |
| 191 _≈_ : {n : Level} → (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set n | |
| 192 a ≈ x = constructible a c≈ constructible x | |
| 10 | 193 |
| 194 | |
| 11 | 195 ConstructibleSet→ZF : {n : Level } → ZF {suc n} {n} |
| 196 ConstructibleSet→ZF {n} = record { | |
| 10 | 197 ZFSet = ConstructibleSet |
| 198 ; _∋_ = _∋_ | |
| 11 | 199 ; _≈_ = _≈_ |
| 10 | 200 ; ∅ = record { level = Zero ; α = Φ ; constructible = xself Φ } |
| 201 ; _×_ = {!!} | |
| 202 ; Union = {!!} | |
| 203 ; Power = {!!} | |
| 204 ; Select = {!!} | |
| 205 ; Replace = {!!} | |
| 206 ; infinite = {!!} | |
| 207 ; isZF = {!!} | |
| 208 } where |