# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1557626897 -32400
# Node ID cb014a103467f896522b42194b0d3b78624da80d
# Parent  813f1b3b000bb71a3b5d4b26014d3af5146f60a9
fix

diff -r 813f1b3b000b -r cb014a103467 zf.agda
--- a/zf.agda	Sat May 11 19:14:16 2019 +0900
+++ b/zf.agda	Sun May 12 11:08:17 2019 +0900
@@ -62,21 +62,21 @@
   field
      empty :  ∀( x : ZFSet  ) → ¬ ( ∅ ∋ x )
      -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
-     power→ : ( A t : ZFSet  ) → Power A ∋ t → ∀ {x} {y} →  _⊆_ t A {x} {y}
-     power← : ( A t : ZFSet  ) → ∀ {x} {y} →  _⊆_ t A {x} {y} → Power A ∋ t 
+     power→ : ∀( A t : ZFSet  ) → Power A ∋ t → ∀ {x} {y} →  _⊆_ t A {x} {y}
+     power← : ∀( A t : ZFSet  ) → ∀ {x} {y} →  _⊆_ t A {x} {y} → Power 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  ≈ ∅ 
+     regularity : ∀( x : ZFSet  ) → ¬ (x ≈ ∅) → (  smaller x ∈ x ∧  ( smaller x  ∩ x  ≈ ∅ ) )
      -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
      infinity∅ :  ∅ ∈ infinite
-     infinity :  ( x : ZFSet  ) → x ∈ infinite →  ( x ∪ Restrict ( λ y → x ≈ y )) ∈ 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
+     -- this form looks like specification
+     replacement : ( ψ :  ZFSet → Set m ) → ( x : ZFSet  ) →  x  ∈  Restrict ψ  → ψ x
 
 record ZF {n m : Level } : Set (suc (n ⊔ m)) where
-  coinductive
   infixr  210 _×_
   infixl  200 _∋_ 
   infixr  220 _≈_
@@ -126,9 +126,10 @@
 open import  Relation.Binary.PropositionalEquality
 
 
--- data Transtive {n : Level } ( lv : Nat) : Set n where
---   Φ : lv ≡ Zero → Transtive lv
---   pair : (s : Transtive (prev lv) ) → (t : Transtive (prev lv) ) → Transtive lv
+data Transtive {n : Level }  : ( lv : Nat) → Set n where
+   Φ : {lv : Nat} → lv ≡ Zero → Transtive lv
+   T-suc : {lv : Nat} → lv ≡ Zero → Transtive {n} lv → Transtive lv
+   ℵ :  {lv : Nat} → Transtive {n} lv → Transtive (Suc lv)