### diff zf.agda @ 3:e7990ff544bf

reocrd ZF
author Shinji KONO Sat, 11 May 2019 10:47:23 +0900 c12d964a04c0
line wrap: on
line diff
```--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/zf.agda	Sat May 11 10:47:23 2019 +0900
@@ -0,0 +1,78 @@
+module zf where
+
+open import Level
+
+
+record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
+   field
+      proj1 : A
+      proj2 : B
+
+open _∧_
+
+
+data  _∨_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
+   case1 : A → A ∨ B
+   case2 : B → A ∨ B
+
+open import Relation.Binary.PropositionalEquality
+
+_⇔_ : {n : Level } → ( A B : Set n )  → Set  n
+_⇔_ A B =  ( A → B ) ∧ ( B  → A )
+
+infixr  130 _∧_
+infixr  140 _∨_
+infixr  150 _⇔_
+
+open import Data.Empty
+open import Relation.Nullary
+
+record ZF (n m : Level ) : Set (suc (n ⊔ m)) where
+  coinductive
+  field
+     ZFSet : Set n
+     _∋_ : ( A x : ZFSet  ) → Set m
+     _≈_ : ( A B : ZFSet  ) → Set m
+  -- ZF Set constructor
+     ∅  : ZFSet
+     _×_ : ( A B : ZFSet  ) → ZFSet
+     Union : ( A : ZFSet  ) → ZFSet
+     Power : ( A : ZFSet  ) → ZFSet
+     Restrict : ( ZFSet → Set m ) → ZFSet
+  infixl  200 _∋_
+  infixr  210 _×_
+  infixr  220 _≈_
+  field
+     -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
+     pair : ( A B : ZFSet  ) →  A × B  ∋ A  ∧ A × B  ∋ B
+     -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t  ∈ x))
+     union→ : ( X x y : ZFSet  ) → X ∋ x  → x ∋ y → Union X  ∋ y
+     union← : ( X x y : ZFSet  ) → Union X  ∋ y → X ∋ x  → x ∋ y
+  _∈_ : ( A B : ZFSet  ) → Set m
+  A ∈ B = B ∋ A
+  _⊆_ : ( A B : ZFSet  ) → { x : ZFSet } → { A∋x : Set m } → Set m
+  _⊆_ A B {x} {A∋x} = B ∋ x
+  _∩_ : ( A B : ZFSet  ) → ZFSet
+  A ∩ B = Restrict ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) )
+  _∪_ : ( A B : ZFSet  ) → ZFSet
+  A ∪ B = Restrict ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) )
+  infixr  200 _∈_
+  infixr  230 _∩_ _∪_
+  infixr  220 _⊆_
+  field
+     empty : ( x : ZFSet  ) → ¬ ( ∅ ∋ x )
+     -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
+     power→ : ( A X t : ZFSet  ) → A ∋ t → ∀ {x} {y} →  _⊆_ t X {x} {y}
+     power← : ( A X t : ZFSet  ) → ∀ {x} {y} →  _⊆_ t X {x} {y} → A ∋ t
+     -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
+     extentionality :  ( A B z  : ZFSet  ) → A ∋ z ⇔ B ∋ z → A ≈ B
+     -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
+     smaller : ZFSet → ZFSet
+     regularity : ( x : ZFSet  ) → ¬ (x ≈ ∅) → smaller x  ∩ x  ≈ ∅
+     -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
+     infinite : ZFSet
+     infinity∅ :  ∅ ∈ infinite
+     infinity :  ( x : ZFSet  ) → x ∈ infinite →  ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite
+     -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
+     replacement : ( ψ :  ZFSet → Set m ) → ( y : ZFSet  ) →  y  ∈  Restrict ψ  → ψ y
+```