Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison zf.agda @ 6:d9b704508281

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isEquiv and isZF
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 11 May 2019 11:40:31 +0900
parents c12d964a04c0
children 813f1b3b000b
comparison
equal deleted inserted replaced
4:c12d964a04c0 6:d9b704508281
13 13
14 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where 14 data _∨_ {n m : Level} (A : Set n) ( B : Set m ) : Set (n ⊔ m) where
15 case1 : A → A ∨ B 15 case1 : A → A ∨ B
16 case2 : B → A ∨ B 16 case2 : B → A ∨ B
17 17
18 open import Relation.Binary.PropositionalEquality 18 -- open import Relation.Binary.PropositionalEquality
19 19
20 _⇔_ : {n : Level } → ( A B : Set n ) → Set n 20 _⇔_ : {n : Level } → ( A B : Set n ) → Set n
21 _⇔_ A B = ( A → B ) ∧ ( B → A ) 21 _⇔_ A B = ( A → B ) ∧ ( B → A )
22
23 open import Data.Empty
24 open import Relation.Nullary
25
26 open import Relation.Binary
27 open import Relation.Binary.Core
22 28
23 infixr 130 _∧_ 29 infixr 130 _∧_
24 infixr 140 _∨_ 30 infixr 140 _∨_
25 infixr 150 _⇔_ 31 infixr 150 _⇔_
26 32
27 open import Data.Empty 33 record IsZF {n m : Level }
28 open import Relation.Nullary 34 (ZFSet : Set n)
29 35 (_∋_ : ( A x : ZFSet ) → Set m)
30 record ZF (n m : Level ) : Set (suc (n ⊔ m)) where 36 (_≈_ : ( A B : ZFSet ) → Set m)
31 coinductive 37 (∅ : ZFSet)
38 (_×_ : ( A B : ZFSet ) → ZFSet)
39 (Union : ( A : ZFSet ) → ZFSet)
40 (Power : ( A : ZFSet ) → ZFSet)
41 (Restrict : ( ZFSet → Set m ) → ZFSet)
42 (infinite : ZFSet)
43 : Set (suc (n ⊔ m)) where
32 field 44 field
33 ZFSet : Set n 45 isEquivalence : {A B : ZFSet} → IsEquivalence {n} {m} {ZFSet} _≈_
34 _∋_ : ( A x : ZFSet ) → Set m
35 _≈_ : ( A B : ZFSet ) → Set m
36 -- ZF Set constructor
37 ∅ : ZFSet
38 _×_ : ( A B : ZFSet ) → ZFSet
39 Union : ( A : ZFSet ) → ZFSet
40 Power : ( A : ZFSet ) → ZFSet
41 Restrict : ( ZFSet → Set m ) → ZFSet
42 infixl 200 _∋_
43 infixr 210 _×_
44 infixr 220 _≈_
45 field
46 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z) 46 -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
47 pair : ( A B : ZFSet ) → A × B ∋ A ∧ A × B ∋ B 47 pair : ( A B : ZFSet ) → ( (A × B) ∋ A ) ∧ ( (A × B) ∋ B )
48 -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x)) 48 -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t ∈ x))
49 union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y 49 union→ : ( X x y : ZFSet ) → X ∋ x → x ∋ y → Union X ∋ y
50 union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y 50 union← : ( X x y : ZFSet ) → Union X ∋ y → X ∋ x → x ∋ y
51 _∈_ : ( A B : ZFSet ) → Set m 51 _∈_ : ( A B : ZFSet ) → Set m
52 A ∈ B = B ∋ A 52 A ∈ B = B ∋ A
63 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x ) 63 empty : ∀( x : ZFSet ) → ¬ ( ∅ ∋ x )
64 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) 64 -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
65 power→ : ( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y} 65 power→ : ( A t : ZFSet ) → Power A ∋ t → ∀ {x} {y} → _⊆_ t A {x} {y}
66 power← : ( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t 66 power← : ( A t : ZFSet ) → ∀ {x} {y} → _⊆_ t A {x} {y} → Power A ∋ t
67 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) 67 -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
68 extentionality : ( A B z : ZFSet ) → A ∋ z ⇔ B ∋ z → A ≈ B 68 extentionality : ( A B z : ZFSet ) → (( A ∋ z ) ⇔ (B ∋ z) ) → A ≈ B
69 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) ) 69 -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
70 -- smaller : ZFSet → ZFSet 70 -- smaller : ZFSet → ZFSet
71 -- regularity : ( x : ZFSet ) → ¬ (x ≈ ∅) → smaller x ∩ x ≈ ∅ 71 -- regularity : ( x : ZFSet ) → ¬ (x ≈ ∅) → smaller x ∩ x ≈ ∅
72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) 72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
73 infinite : ZFSet
74 infinity∅ : ∅ ∈ infinite 73 infinity∅ : ∅ ∈ infinite
75 infinity : ( x : ZFSet ) → x ∈ infinite → ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite 74 infinity : ( x : ZFSet ) → x ∈ infinite → ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite
76 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) ) 75 -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
77 replacement : ( ψ : ZFSet → Set m ) → ( y : ZFSet ) → y ∈ Restrict ψ → ψ y 76 replacement : ( ψ : ZFSet → Set m ) → ( y : ZFSet ) → y ∈ Restrict ψ → ψ y
78 77
78 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
79 coinductive
80 infixr 210 _×_
81 infixl 200 _∋_
82 infixr 220 _≈_
83 field
84 ZFSet : Set n
85 _∋_ : ( A x : ZFSet ) → Set m
86 _≈_ : ( A B : ZFSet ) → Set m
87 -- ZF Set constructor
88 ∅ : ZFSet
89 _×_ : ( A B : ZFSet ) → ZFSet
90 Union : ( A : ZFSet ) → ZFSet
91 Power : ( A : ZFSet ) → ZFSet
92 Restrict : ( ZFSet → Set m ) → ZFSet
93 infinite : ZFSet
94 field
95 isZF : IsZF ZFSet _∋_ _≈_ ∅ _×_ Union Power Restrict infinite
96