annotate zf.agda @ 213:22d435172d1a

separate logic and nat
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 02 Aug 2019 12:17:10 +0900
parents e59e682ad120
children 2e1f19c949dc
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e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1 module zf where
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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2
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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3 open import Level
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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4
213
22d435172d1a separate logic and nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 209
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5 open import logic
123
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 116
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6
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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7 open import Relation.Nullary
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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8 open import Relation.Binary
188
1f2c8b094908 axiom of choice → p ∨ ¬ p
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
9 open import Data.Empty
1f2c8b094908 axiom of choice → p ∨ ¬ p
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
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10
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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11 record IsZF {n m : Level }
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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12 (ZFSet : Set n)
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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13 (_∋_ : ( A x : ZFSet ) → Set m)
9
5ed16e2d8eb7 try to fix axiom of replacement
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 8
diff changeset
14 (_≈_ : Rel ZFSet m)
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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15 (∅ : ZFSet)
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
16 (_,_ : ( A B : ZFSet ) → ZFSet)
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
17 (Union : ( A : ZFSet ) → ZFSet)
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
18 (Power : ( A : ZFSet ) → ZFSet)
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
19 (Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet )
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
20 (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet )
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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21 (infinite : ZFSet)
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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22 : Set (suc (n ⊔ m)) where
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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23 field
29
fce60b99dc55 posturate OD is isomorphic to Ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 23
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24 isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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25 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
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26 pair : ( A B : ZFSet ) → ( (A , B) ∋ A ) ∧ ( (A , B) ∋ B )
69
93abc0133b8a union continue
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 65
diff changeset
27 -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x ∧ (z ∈ u))
73
dd430a95610f fix ordinal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 72
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28 union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
159
3675bd617ac8 infinite continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 140
diff changeset
29 union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → ¬ ( (u : ZFSet ) → ¬ ((X ∋ u) ∧ (u ∋ z )))
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 _∈_ : ( A B : ZFSet ) → Set m
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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31 A ∈ B = B ∋ A
23
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 18
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32 _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set m
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 18
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33 _⊆_ A B {x} = A ∋ x → B ∋ x
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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34 _∩_ : ( A B : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
35 A ∩ B = Select A ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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36 _∪_ : ( A B : ZFSet ) → ZFSet
103
c8b79d303867 starting over HOD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 78
diff changeset
37 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: 77
diff changeset
38 {_} : ZFSet → ZFSet
9a7a64b2388c infinite and replacement begin
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 77
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39 { x } = ( x , x )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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40 infixr 200 _∈_
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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41 infixr 230 _∩_ _∪_
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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42 infixr 220 _⊆_
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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43 field
4
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 3
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44 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
45 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
166
ea0e7927637a use double negation
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 160
diff changeset
46 power→ : ∀( A t : ZFSet ) → Power A ∋ t → ∀ {x} → t ∋ x → ¬ ¬ ( A ∋ x ) -- _⊆_ t A {x}
77
75ba8cf64707 Power Set on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 76
diff changeset
47 power← : ∀( A t : ZFSet ) → ( ∀ {x} → _⊆_ t A {x}) → Power A ∋ t
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: 54
diff changeset
48 -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
186
914cc522c53a fix extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 183
diff changeset
49 extensionality : { A B w : ZFSet } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z) ) → ( A ∈ w ⇔ B ∈ w )
183
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
50 -- This form of regurality forces choice function
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
183
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
52 -- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
53 -- regularity : ∀( x : ZFSet ) → (not : ¬ (x ≈ ∅)) → ( minimul x not ∈ x ∧ ( minimul x not ∩ x ≈ ∅ ) )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
54 -- another form of regularity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
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55 ε-induction : { ψ : ZFSet → Set m}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
56 → ( {x : ZFSet } → ({ y : ZFSet } → x ∋ y → ψ y ) → ψ x )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
57 → (x : ZFSet ) → ψ x
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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58 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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59 infinity∅ : ∅ ∈ infinite
160
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 159
diff changeset
60 infinity : ∀( x : ZFSet ) → x ∈ infinite → ( x ∪ { x }) ∈ infinite
140
312e27aa3cb5 remove otrans again. start over
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 138
diff changeset
61 selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet } → ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈ Select X ψ )
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
62 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
130
3849614bef18 new replacement axiom
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 123
diff changeset
63 replacement← : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → x ∈ X → ψ x ∈ Replace X ψ
138
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 133
diff changeset
64 replacement→ : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet ) → ( lt : x ∈ Replace X ψ ) → ¬ ( ∀ (y : ZFSet) → ¬ ( x ≈ ψ y ) )
183
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
65 -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ]
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
66 choice-func : (X : ZFSet ) → {x : ZFSet } → ¬ ( x ≈ ∅ ) → ( X ∋ x ) → ZFSet
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 166
diff changeset
67 choice : (X : ZFSet ) → {A : ZFSet } → ( X∋A : X ∋ A ) → (not : ¬ ( A ≈ ∅ )) → A ∋ choice-func X not X∋A
3
e7990ff544bf reocrd ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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68
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
69 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
70 infixr 210 _,_
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
71 infixl 200 _∋_
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
72 infixr 220 _≈_
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
73 field
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
74 ZFSet : Set n
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
75 _∋_ : ( A x : ZFSet ) → Set m
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
76 _≈_ : ( A B : ZFSet ) → Set m
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
77 -- ZF Set constructor
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
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78 ∅ : ZFSet
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
79 _,_ : ( A B : ZFSet ) → ZFSet
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
80 Union : ( A : ZFSet ) → ZFSet
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
81 Power : ( A : ZFSet ) → ZFSet
115
277c2f3b8acb Select declaration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 111
diff changeset
82 Select : (X : ZFSet ) → ( ψ : (x : ZFSet ) → Set m ) → ZFSet
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
83 Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
84 infinite : ZFSet
18
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 14
diff changeset
85 isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite
6
d9b704508281 isEquiv and isZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
86