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

comparison zf.agda @ 8:cb014a103467

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fix
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 12 May 2019 11:08:17 +0900
parents 813f1b3b000b
children 5ed16e2d8eb7
comparison
equal deleted inserted replaced
7:813f1b3b000b 8:cb014a103467
60 infixr 230 _∩_ _∪_ 60 infixr 230 _∩_ _∪_
61 infixr 220 _⊆_ 61 infixr 220 _⊆_
62 field 62 field
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 ∧ ( smaller x ∩ x ≈ ∅ ) )
72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) ) 72 -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
73 infinity∅ : ∅ ∈ infinite 73 infinity∅ : ∅ ∈ infinite
74 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
75 -- 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 ) )
76 replacement : ( ψ : ZFSet → Set m ) → ( y : ZFSet ) → y ∈ Restrict ψ → ψ y 76 -- this form looks like specification
77 replacement : ( ψ : ZFSet → Set m ) → ( x : ZFSet ) → x ∈ Restrict ψ → ψ x
77 78
78 record ZF {n m : Level } : Set (suc (n ⊔ m)) where 79 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
79 coinductive
80 infixr 210 _×_ 80 infixr 210 _×_
81 infixl 200 _∋_ 81 infixl 200 _∋_
82 infixr 220 _≈_ 82 infixr 220 _≈_
83 field 83 field
84 ZFSet : Set n 84 ZFSet : Set n
124 prev Zero = Zero 124 prev Zero = Zero
125 125
126 open import Relation.Binary.PropositionalEquality 126 open import Relation.Binary.PropositionalEquality
127 127
128 128
129 -- data Transtive {n : Level } ( lv : Nat) : Set n where 129 data Transtive {n : Level } : ( lv : Nat) → Set n where
130 -- Φ : lv ≡ Zero → Transtive lv 130 Φ : {lv : Nat} → lv ≡ Zero → Transtive lv
131 -- pair : (s : Transtive (prev lv) ) → (t : Transtive (prev lv) ) → Transtive lv 131 T-suc : {lv : Nat} → lv ≡ Zero → Transtive {n} lv → Transtive lv
132 ℵ : {lv : Nat} → Transtive {n} lv → Transtive (Suc lv)
132 133
133 134
134 135
135 136