# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1561409422 -32400
# Node ID c42352a7ee077ee8bc81a4d88541e4af15664b78
# Parent  1daa1d24348c1ed89ad296383fe68271d8c031eb
HOD

diff -r 1daa1d24348c -r c42352a7ee07 HOD.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/HOD.agda	Tue Jun 25 05:50:22 2019 +0900
@@ -0,0 +1,425 @@
+open import Level
+module HOD where
+
+open import zf
+open import ordinal
+
+open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import  Relation.Binary.PropositionalEquality
+open import Data.Nat.Properties 
+open import Data.Empty
+open import Relation.Nullary
+open import Relation.Binary
+open import Relation.Binary.Core
+
+-- Ordinal Definable Set
+
+record HOD {n : Level}  : Set (suc n) where
+  field
+    def : (x : Ordinal {n} ) → Set n
+    otrans : {x y : Ordinal {n} } → def x → y o< x → def y
+
+open HOD
+open import Data.Unit
+
+open Ordinal
+
+record _==_ {n : Level} ( a b :  HOD {n} ) : Set n where
+  field
+     eq→ : ∀ { x : Ordinal {n} } → def a x → def b x 
+     eq← : ∀ { x : Ordinal {n} } → def b x → def a x 
+
+id : {n : Level} {A : Set n} → A → A
+id x = x
+
+eq-refl : {n : Level} {  x :  HOD {n} } → x == x
+eq-refl {n} {x} = record { eq→ = id ; eq← = id }
+
+open  _==_ 
+
+eq-sym : {n : Level} {  x y :  HOD {n} } → x == y → y == x
+eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
+
+eq-trans : {n : Level} {  x y z :  HOD {n} } → x == y → y == z → x == z
+eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y  t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
+
+-- Ordinal in HOD ( and ZFSet )
+Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n}
+Ord {n} a = record { def = λ y → y o< a ; otrans = lemma }  where
+   lemma : {x y : Ordinal} → x o< a → y o< x → y o< a
+   lemma {x} {y} x<a y<x = ordtrans {n} {y} {x} {a} y<x x<a
+
+-- od∅ : {n : Level} → HOD {n} 
+-- od∅ {n} = record { def = λ _ → Lift n ⊥ }
+od∅ : {n : Level} → HOD {n} 
+od∅ {n} = Ord o∅ 
+
+data SinO {n : Level} : (ox : Ordinal {n}) (x : HOD {n}) → Set (suc n) where
+  o-in-o : {ox : Ordinal {n} } → SinO ox (Ord ox)
+  s-in-o : {ox : Ordinal {n} } → {y x : HOD {n} } → SinO ox y → x == y → SinO ox x
+
+postulate      
+  -- HOD can be iso to a subset of Ordinal ( by means of Godel Set )
+  od→ord : {n : Level} → HOD {n} → Ordinal {n}
+  c<→o<  : {n : Level} {x y : HOD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
+  -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set
+  -- o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
+  sino   : {n : Level} {x : HOD {n}}     → SinO ( od→ord x ) x
+  -- supermum as Replacement Axiom
+  sup-o  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
+  sup-o< : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → ∀ {x : Ordinal {n}} →  ψ x  o<  sup-o ψ 
+  -- contra-position of mimimulity of supermum required in Power Set Axiom
+  sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
+  sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
+  -- sup-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( sup-o ψ ) 
+
+
+_∋_ : { n : Level } → ( a x : HOD {n} ) → Set n
+_∋_ {n} a x  = def a ( od→ord x )
+
+_c<_ : { n : Level } → ( x a : HOD {n} ) → Set n
+x c< a = a ∋ x 
+
+postulate      
+  o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y  → Ord y ∋ Ord x 
+
+ord→od : {n : Level} → Ordinal {n} → HOD {n} 
+ord→od = ?
+
+oiso   : {n : Level} {x : HOD {n}} → ? ≡ x
+oiso   = ?
+
+diso   : {n : Level} {x : Ordinal {n}} → od→ord ? ≡ x
+diso   = ?
+
+_c≤_ : {n : Level} →  HOD {n} →  HOD {n} → Set (suc n)
+a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
+
+def-subst : {n : Level } {Z : HOD {n}} {X : Ordinal {n} }{z : HOD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z  →  X ≡ x  →  def z x
+def-subst df refl refl = df
+
+sup-od : {n : Level } → ( HOD {n} → HOD {n}) →  HOD {n}
+sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
+
+sup-c< : {n : Level } → ( ψ : HOD {n} →  HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
+sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
+        lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
+    lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
+    lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso)  )
+
+∅3 : {n : Level} →  { x : Ordinal {n}}  → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
+∅3 {n} {x} = TransFinite {n} c2 c3 x where
+   c0 : Nat →  Ordinal {n}  → Set n
+   c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x))  → x ≡ o∅ {n}
+   c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } )
+   c2 Zero not = refl
+   c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } )
+   ... | t with t (case1 ≤-refl )
+   c2 (Suc lx) not | t | ()
+   c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx  (record { lv = lx ; ord = x₁ })  → c0 lx (record { lv = lx ; ord = OSuc lx x₁ })
+   c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } )
+   ... | t with t (case2 Φ< )
+   c3 lx (Φ .lx) d not | t | ()
+   c3 lx (OSuc .lx x₁) d not with not (  record { lv = lx ; ord = OSuc lx x₁ } )
+   ... | t with t (case2 (s< s<refl ) )
+   c3 lx (OSuc .lx x₁) d not | t | ()
+
+transitive : {n : Level } { z y x : HOD {suc n} } → z ∋ y → y ∋ x → z  ∋ x
+transitive  {n} {z} {y} {x} z∋y x∋y  with  ordtrans ( c<→o< {suc n} {x} {y} x∋y ) (  c<→o< {suc n} {y} {z} z∋y ) 
+... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y ) 
+
+record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
+  field
+     mino : Ordinal {n}
+     min<x :  mino o< x
+
+∅5 : {n : Level} →  { x : Ordinal {n} }  → ¬ ( x  ≡ o∅ {n} ) → o∅ {n} o< x
+∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) 
+∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ<
+∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n)
+
+ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y }
+ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso
+
+-- avoiding lv != Zero error
+orefl : {n : Level} →  { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
+orefl refl = refl
+
+==-iso : {n : Level} →  { x y : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y)  →  x == y
+==-iso {n} {x} {y} eq = record {
+      eq→ = λ d →  lemma ( eq→  eq (def-subst d (sym oiso) refl )) ;
+      eq← = λ d →  lemma ( eq←  eq (def-subst d (sym oiso) refl ))  }
+        where
+           lemma : {x : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
+           lemma {x} {z} d = def-subst d oiso refl
+
+=-iso : {n : Level } {x y : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
+=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
+
+ord→== : {n : Level} →  { x y : HOD {n} } → od→ord x ≡  od→ord y →  x == y
+ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
+   lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy →  (ord→od ox) == (ord→od oy)
+   lemma ox ox  refl = eq-refl
+
+o≡→== : {n : Level} →  { x y : Ordinal {n} } → x ≡  y →  ord→od x == ord→od y
+o≡→== {n} {x} {.x} refl = eq-refl
+
+>→¬< : {x y : Nat  } → (x < y ) → ¬ ( y < x )
+>→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
+
+c≤-refl : {n : Level} →  ( x : HOD {n} ) → x c≤ x
+c≤-refl x = case1 refl
+
+∋→o< : {n : Level} →  { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a
+∋→o< {n} {a} {x} lt = t where
+         t : (od→ord x) o< (od→ord a)
+         t = c<→o< {suc n} {x} {a} lt 
+
+o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
+o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
+o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
+    lemma :  o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
+    lemma lt with def-subst {!!} oiso refl
+    lemma lt | t = {!!}
+o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
+o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
+
+ord-od∅ :  {n : Level} → o∅ {suc n} ≡ od→ord (Ord (o∅ {suc n}))
+ord-od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (Ord (o∅ {suc n})))
+ord-od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
+    lemma :  o∅ {suc n } o< (od→ord (Ord (o∅ {suc n}))) → ⊥
+    lemma lt with  o<→c< lt
+    lemma lt | t = o<¬≡ refl t
+ord-od∅ {n} | tri≈ ¬a b ¬c = b
+ord-od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
+
+o<→¬c> : {n : Level} →  { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (y c< x )
+o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
+
+o≡→¬c< : {n : Level} →  { x y : HOD {n} } →  (od→ord x ) ≡ ( od→ord y) →   ¬ x c< y
+o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡  (orefl oeq ) (c<→o< lt) 
+
+∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ ; otrans = λ () } == od∅ {n} 
+eq→ ∅0 {w} (lift ())
+eq← ∅0 {w} (case1 ())
+eq← ∅0 {w} (case2 ())
+
+∅< : {n : Level} →  { x y : HOD {n} } → def x (od→ord y ) → ¬ (  x  == od∅ {n} )
+∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
+∅< {n} {x} {y} d eq | lift ()
+       
+-- ∅6 : {n : Level} →  { x : HOD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
+-- ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
+
+def-iso : {n : Level} {A B : HOD {n}} {x y : Ordinal {n}} → x ≡ y  → (def A y → def B y)  → def A x → def B x
+def-iso refl t = t
+
+is-o∅ : {n : Level} →  ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
+is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
+is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
+is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
+
+open _∧_
+
+-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
+-- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
+
+csuc :  {n : Level} →  HOD {suc n} → HOD {suc n}
+csuc x = ord→od ( osuc ( od→ord x ))
+
+-- Power Set of X ( or constructible by λ y → def X (od→ord y )
+
+ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
+ZFSubset A x =  record { def = λ y → def A y ∧  def x y ; otrans = {!!} }  
+
+Def :  {n : Level} → (A :  HOD {suc n}) → HOD {suc n}
+Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )  
+
+-- Constructible Set on α
+L : {n : Level} → (α : Ordinal {suc n}) → HOD {suc n}
+L {n}  record { lv = Zero ; ord = (Φ .0) } = od∅
+L {n}  record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) 
+L {n}  record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
+    record { def = λ y → osuc y o< (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} }
+
+omega : { n : Level } → Ordinal {n}
+omega = record { lv = Suc Zero ; ord = Φ 1 }
+
+HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
+HOD→ZF {n}  = record { 
+    ZFSet = HOD {suc n}
+    ; _∋_ = _∋_ 
+    ; _≈_ = _==_ 
+    ; ∅  = od∅
+    ; _,_ = _,_
+    ; Union = Union
+    ; Power = Power
+    ; Select = Select
+    ; Replace = Replace
+    ; infinite = Ord omega
+    ; isZF = isZF 
+ } where
+    Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n}
+    Replace X ψ = sup-od ψ
+    Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → X ∋ x → Set (suc n) ) → HOD {suc n}
+    Select X ψ = record { def = λ x →  ( (d : def X x ) →  ψ (ord→od x) (subst (λ k → def X k ) (sym diso) d)) ; otrans = lemma }  where
+       lemma : {x y : Ordinal} → ((d : def X x) → ψ (ord→od x) (subst (def X) (sym diso) d)) →
+            y o< x → (d : def X y) → ψ (ord→od y) (subst (def X) (sym diso) d)
+       lemma {x} {y} f y<x d = {!!}
+    _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n}
+    x , y = Ord (omax (od→ord x) (od→ord y))
+    Union : HOD {suc n} → HOD {suc n}
+    Union U = record { def = λ y → osuc y o< (od→ord U) ; otrans = {!!} }
+    -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x →  X ∋ x )
+    Power : HOD {suc n} → HOD {suc n}
+    Power A = Def A
+    ZFSet = HOD {suc n}
+    _∈_ : ( A B : ZFSet  ) → Set (suc n)
+    A ∈ B = B ∋ A
+    _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set (suc n)
+    _⊆_ A B {x} = A ∋ x →  B ∋ x
+    _∩_ : ( A B : ZFSet  ) → ZFSet
+    A ∩ B = Select (A , B) (  λ x d → ( A ∋ x ) ∧ (B ∋ x) )
+    -- _∪_ : ( A B : ZFSet  ) → ZFSet
+    -- A ∪ B = Select (A , B) (  λ x → (A ∋ x)  ∨ ( B ∋ x ) )
+    ｛_｝ : ZFSet → ZFSet
+    ｛ x ｝ = ( x ,  x )
+
+    infixr  200 _∈_
+    -- infixr  230 _∩_ _∪_
+    infixr  220 _⊆_
+    isZF : IsZF (HOD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace (Ord omega)
+    isZF = record {
+           isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
+       ;   pair  = pair
+       ;   union-u = λ _ z _ → csuc z
+       ;   union→ = union→
+       ;   union← = union←
+       ;   empty = empty
+       ;   power→ = power→
+       ;   power← = power← 
+       ;   extensionality = extensionality
+       ;   minimul = minimul
+       ;   regularity = regularity
+       ;   infinity∅ = infinity∅
+       ;   infinity = λ _ → infinity
+       ;   selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
+       ;   replacement = replacement
+     } where
+         open _∧_ 
+         open Minimumo
+         pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧  ((A , B) ∋ B)
+         proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
+         proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)
+         empty : (x : HOD {suc n} ) → ¬  (od∅ ∋ x)
+         empty x (case1 ())
+         empty x (case2 ())
+         ---
+         --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
+         --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )       Power X is a sup of all subset of A
+         --
+         --  if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t 
+         --    then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
+         --    In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity
+         --
+         power→ : (A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
+         power→ A t P∋t {x} t∋x = proj1 lemma-s where
+              minsup :  HOD
+              minsup =  ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) 
+              lemma-t : csuc minsup ∋ t
+              lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) 
+              lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))  ∋ x
+              lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso  )
+              lemma-s | case1 eq = def-subst {!!} oiso refl
+              lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) t∋x
+         -- 
+         -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
+         -- Power A is a sup of ZFSubset A t, so Power A ∋ t
+         -- 
+         power← : (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
+         power← A t t→A  = def-subst {suc n} {_} {_} {Power A} {od→ord t}
+                  {!!} refl lemma1 where
+              lemma-eq :  ZFSubset A t == t
+              eq→ lemma-eq {z} w = proj2 w 
+              eq← lemma-eq {z} w = record { proj2 = w  ;
+                 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
+              lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t
+              lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) {!!}
+              lemma :  od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x)))
+              lemma = sup-o<   
+         union-lemma-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z
+         union-lemma-u {X} {z} U>z = lemma <-osuc where
+             lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
+             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} {!!} refl refl
+         union→ :  (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
+         union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
+         union→ X y u xx | tri< a ¬b ¬c with  osuc-< a (c<→o< (proj2 xx))
+         union→ X y u xx | tri< a ¬b ¬c | ()
+         union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where
+             lemma : {oX ou ooy : Ordinal {suc n}} →  ou ≡ ooy  → ou o< oX   → ooy  o< oX
+             lemma refl lt = lt
+         union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) 
+         union← :  (X z : HOD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z )
+         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
+         ψiso :  {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y   → ψ y
+         ψiso {ψ} t refl = t
+         selection : {X : HOD } {ψ : (x : HOD ) →  x ∈ X  → Set (suc n)} {y : HOD} → ((d : X ∋ y ) → ψ y d ) ⇔ (Select X ψ ∋ y)
+         selection {ψ} {X} {y} =  {!!}
+         replacement : {ψ : HOD → HOD} (X x : HOD) → Replace X ψ ∋ ψ x
+         replacement {ψ} X x = sup-c< ψ {x}
+         ∅-iso :  {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 
+         ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq  
+         minimul : (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n} 
+         minimul x  not = {!!}
+         regularity :  (x : HOD) (not : ¬ (x == od∅)) →
+            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ d → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
+         proj1 (regularity x not ) = {!!}
+         proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where
+            reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ d → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
+            reg {y} t = {!!}
+         extensionality : {A B : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
+         eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d  
+         eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d  
+         xx-union : {x  : HOD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
+         xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x))
+         xxx-union : {x  : HOD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
+         xxx-union {x} = cong ( λ k → Ord k ) lemma where
+             lemma1 : {x  : HOD {suc n}} → od→ord x o< od→ord (x , x)
+             lemma1 {x} = c<→o< ( proj1 (pair x x ) )
+             lemma2 : {x  : HOD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
+             lemma2 = trans ( cong ( λ k →  od→ord k ) xx-union ) {!!}
+             lemma : {x  : HOD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
+             lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 )
+         uxxx-union : {x  : HOD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) }
+         uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where
+             lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
+             lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) {!!}
+         uxxx-2 : {x  : HOD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) }
+         eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt
+         eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt
+         uxxx-ord : {x  : HOD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x)
+         uxxx-ord {x} = trans (cong (λ k →  od→ord k ) uxxx-union) {!!} 
+         infinite : HOD {suc n}
+         infinite = Ord omega 
+         infinity∅ : Ord omega  ∋ od∅ {suc n}
+         infinity∅ = {!!}
+         infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
+         infinity x lt = {!!} where
+              lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega 
+              lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n)
+              lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n)
+              lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ()))
+              lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
+              lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
+              lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
+         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set
+         -- ∀ z [ ∀ x ( x ∈ z  → ¬ ( x ≈ ∅ ) )  ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y )  → x ∩ y ≈ ∅  ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ]
+         record Choice (z : HOD {suc n}) : Set (suc (suc n)) where
+             field
+                 u : {x : HOD {suc n}} ( x∈z  : x ∈ z ) → HOD {suc n}
+                 t : {x : HOD {suc n}} ( x∈z  : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n}
+                 choice : { x : HOD {suc n} } → ( x∈z  : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝
+         -- choice : {x :  HOD {suc n}} ( x ∈ z  → ¬ ( x ≈ ∅ ) ) →
+         -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) →  choice ¬x∅ A∈X ∈ A 
+         -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}
+
diff -r 1daa1d24348c -r c42352a7ee07 ordinal-definable.agda
--- a/ordinal-definable.agda	Thu Jun 20 13:18:18 2019 +0900
+++ b/ordinal-definable.agda	Tue Jun 25 05:50:22 2019 +0900
@@ -1,3 +1,5 @@
+{-# OPTIONS --allow-unsolved-metas #-}
+
 open import Level
 module ordinal-definable where
 
@@ -17,13 +19,21 @@
 record OD {n : Level}  : Set (suc n) where
   field
     def : (x : Ordinal {n} ) → Set n
-    otrans : {x y : Ordinal {n} } → def x → y o< x → def y
 
 open OD
 open import Data.Unit
 
 open Ordinal
 
+-- Ordinal in OD ( and ZFSet )
+Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n}
+Ord {n} a = record { def = λ y → y o< a }  
+
+-- od∅ : {n : Level} → OD {n} 
+-- od∅ {n} = record { def = λ _ → Lift n ⊥ }
+od∅ : {n : Level} → OD {n} 
+od∅ {n} = Ord o∅ 
+
 record _==_ {n : Level} ( a b :  OD {n} ) : Set n where
   field
      eq→ : ∀ { x : Ordinal {n} } → def a x → def b x 
@@ -43,30 +53,22 @@
 eq-trans : {n : Level} {  x y z :  OD {n} } → x == y → y == z → x == z
 eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y  t) ; eq← = λ t → eq← x=y ( eq← y=z t) }
 
--- Ordinal in OD ( and ZFSet )
-Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n}
-Ord {n} a = record { def = λ y → y o< a ; otrans = lemma }  where
-   lemma : {x y : Ordinal} → x o< a → y o< x → y o< a
-   lemma {x} {y} x<a y<x = ordtrans {n} {y} {x} {a} y<x x<a
+ord→od : {n : Level} → Ordinal {n} → OD {n} 
+ord→od a = Ord a
 
--- od∅ : {n : Level} → OD {n} 
--- od∅ {n} = record { def = λ _ → Lift n ⊥ }
-od∅ : {n : Level} → OD {n} 
-od∅ {n} = Ord o∅ 
+o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
+o<→c< {n} {x} {y} lt = lt 
 
 postulate      
   -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
   od→ord : {n : Level} → OD {n} → Ordinal {n}
-  ord→od : {n : Level} → Ordinal {n} → OD {n} 
   c<→o<  : {n : Level} {x y : OD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
-  -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set
-  -- o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
   oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
   diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
   -- supermum as Replacement Axiom
   sup-o  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
   sup-o< : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → ∀ {x : Ordinal {n}} →  ψ x  o<  sup-o ψ 
-  -- contra-position of mimimulity of supermum required in Power Set Axiom
+  -- a property of supermum required in Power Set Axiom
   sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
   sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
   -- sup-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( sup-o ψ ) 
@@ -77,9 +79,6 @@
 _c<_ : { n : Level } → ( x a : OD {n} ) → Set n
 x c< a = a ∋ x 
 
-postulate      
-  o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y  → Ord y ∋ Ord x 
-
 _c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
 a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
 
@@ -87,13 +86,15 @@
 def-subst df refl refl = df
 
 sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
-sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
+sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )
 
 sup-c< : {n : Level } → ( ψ : OD {n} →  OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
-sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
-        lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
-    lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
-    lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso)  )
+sup-c< {n} ψ {x} = def-subst {n} {_} {_} {sup-od ψ} {od→ord ( ψ x )}
+        ( o<→c< sup-o< ) refl (cong ( λ k → od→ord (ψ k) ) oiso)
+
+∅1 : {n : Level} →  ( x : OD {n} )  → ¬ ( x c< od∅ {n} )
+∅1 {n} x (case1 ())
+∅1 {n} x (case2 ())
 
 ∅3 : {n : Level} →  { x : Ordinal {n}}  → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
 ∅3 {n} {x} = TransFinite {n} c2 c3 x where
@@ -114,7 +115,11 @@
 
 transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z  ∋ x
 transitive  {n} {z} {y} {x} z∋y x∋y  with  ordtrans ( c<→o< {suc n} {x} {y} x∋y ) (  c<→o< {suc n} {y} {z} z∋y ) 
-... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y ) 
+... | t = lemma0 (lemma t) where
+   lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x)
+   lemma xo<z = o<→c< xo<z
+   lemma0 :  def ( ord→od ( od→ord z )) ( od→ord x) →  def z (od→ord x)
+   lemma0 dz  = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso)  refl
 
 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
   field
@@ -158,50 +163,100 @@
 c≤-refl : {n : Level} →  ( x : OD {n} ) → x c≤ x
 c≤-refl x = case1 refl
 
+o<→o> : {n : Level} →  { x y : OD {n} } →  (x == y) → (od→ord x ) o< ( od→ord y) → ⊥
+o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl )
+... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx )
+... | ()
+o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl )
+... | oyx with o<¬≡ refl (c<→o< {n} {x} oyx )
+... | ()
+
+==→o≡ : {n : Level} →  { x y : Ordinal {suc n} } → ord→od x == ord→od y →  x ≡ y 
+==→o≡ {n} {x} {y} eq with trio< {n} x y
+==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso )))
+==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
+==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso )))
+
+≡-def : {n : Level} →  { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } )
+≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where
+    lemma :  ord→od x == record { def = λ z → z o< x }
+    eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where 
+        t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
+        t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso))
+    eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl
+
+od≡-def : {n : Level} →  { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } 
+od≡-def {n} {x} = subst (λ k  → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} ))
+
+==→o≡1 : {n : Level} →  { x y : OD {suc n} } → x == y →  od→ord x ≡ od→ord y 
+==→o≡1 eq = ==→o≡ (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq )
+
+==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y
+==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡ eq) z>x
+
+==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z
+==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x  
+
 ∋→o< : {n : Level} →  { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a
 ∋→o< {n} {a} {x} lt = t where
          t : (od→ord x) o< (od→ord a)
          t = c<→o< {suc n} {x} {a} lt 
 
+o<∋→ : {n : Level} →  { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x 
+o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t  where
+         t : def (ord→od (od→ord a)) (od→ord x)
+         t = o<→c< {suc n} {od→ord x} {od→ord a} lt 
+
 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
 o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
 o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
     lemma :  o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
-    lemma lt with def-subst {!!} oiso refl
-    lemma lt | t = {!!}
+    lemma lt with def-subst (o<→c< lt) oiso refl
+    lemma lt | case1 ()
+    lemma lt | case2 ()
 o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
 o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
 
-ord-od∅ :  {n : Level} → o∅ {suc n} ≡ od→ord (Ord (o∅ {suc n}))
-ord-od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (Ord (o∅ {suc n})))
-ord-od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
-    lemma :  o∅ {suc n } o< (od→ord (Ord (o∅ {suc n}))) → ⊥
-    lemma lt with  o<→c< lt
-    lemma lt | t = o<¬≡ refl t
-ord-od∅ {n} | tri≈ ¬a b ¬c = b
-ord-od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
+o<→¬== : {n : Level} →  { x y : OD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (x == y )
+o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
 
 o<→¬c> : {n : Level} →  { x y : OD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (y c< x )
 o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
 
 o≡→¬c< : {n : Level} →  { x y : OD {n} } →  (od→ord x ) ≡ ( od→ord y) →   ¬ x c< y
-o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡  (orefl oeq ) (c<→o< lt) 
+o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡   (orefl oeq ) (c<→o< lt) 
 
-∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ ; otrans = λ () } == od∅ {n} 
-eq→ ∅0 {w} (lift ())
-eq← ∅0 {w} (case1 ())
-eq← ∅0 {w} (case2 ())
+tri-c< : {n : Level} →  Trichotomous _==_ (_c<_ {suc n})
+tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) 
+tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a )
+tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b))
+tri-c< {n} x y | tri> ¬a ¬b c = tri>  ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso refl)
+
+c<> : {n : Level } { x y : OD {suc n}} → x c<  y  → y c< x  →  ⊥
+c<> {n} {x} {y} x<y y<x with tri-c< x y
+c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
+c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y )
+c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
 
 ∅< : {n : Level} →  { x y : OD {n} } → def x (od→ord y ) → ¬ (  x  == od∅ {n} )
-∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
-∅< {n} {x} {y} d eq | lift ()
+∅< {n} {x} {y} d eq with eq→ eq d
+∅< {n} {x} {y} d eq | case1 ()
+∅< {n} {x} {y} d eq | case2 ()
        
--- ∅6 : {n : Level} →  { x : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
--- ∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
+∅6 : {n : Level} →  { x : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
+∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x
 
 def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y  → (def A y → def B y)  → def A x → def B x
 def-iso refl t = t
 
+is-∋ : {n : Level} →  ( x y : OD {suc n} ) → Dec ( x ∋ y )
+is-∋ {n} x y with tri-c< x y
+is-∋ {n} x y | tri< a ¬b ¬c = no ¬c
+is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c
+is-∋ {n} x y | tri> ¬a ¬b c = yes c
+
 is-o∅ : {n : Level} →  ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
 is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
 is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
@@ -209,6 +264,19 @@
 
 open _∧_
 
+--
+-- This menas OD is Ordinal here
+--
+¬∅=→∅∈ :  {n : Level} →  { x : OD {suc n} } → ¬ (  x  == od∅ {suc n} ) → x ∋ od∅ {suc n} 
+¬∅=→∅∈  {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
+     lemma : (ox : Ordinal {suc n}) →  ¬ (ord→od  ox  == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
+     lemma ox ne with is-o∅ ox
+     lemma ox ne | yes refl with ne ( ord→== lemma1 ) where
+         lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅
+         lemma1 = cong ( λ k → od→ord k ) o∅≡od∅
+     lemma o∅ ne | yes refl | ()
+     lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) )  
+
 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
 -- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
 
@@ -218,7 +286,7 @@
 -- Power Set of X ( or constructible by λ y → def X (od→ord y )
 
 ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n}
-ZFSubset A x =  record { def = λ y → def A y ∧  def x y ; otrans = {!!} }  
+ZFSubset A x =  record { def = λ y → def A y ∧  def x y }  
 
 Def :  {n : Level} → (A :  OD {suc n}) → OD {suc n}
 Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )  
@@ -228,13 +296,10 @@
 L {n}  record { lv = Zero ; ord = (Φ .0) } = od∅
 L {n}  record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) 
 L {n}  record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
-    record { def = λ y → osuc y o< (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} }
+    record { def = λ y → osuc y o< (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }) )) }
 
-omega : { n : Level } → Ordinal {n}
-omega = record { lv = Suc Zero ; ord = Φ 1 }
-
-OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
-OD→ZF {n}  = record { 
+Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
+Ord→ZF {n}  = record { 
     ZFSet = OD {suc n}
     ; _∋_ = _∋_ 
     ; _≈_ = _==_ 
@@ -242,22 +307,19 @@
     ; _,_ = _,_
     ; Union = Union
     ; Power = Power
-    ; Select = Select
+    ; Select = {!!}
     ; Replace = Replace
-    ; infinite = Ord omega
+    ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } )
     ; isZF = isZF 
  } where
     Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
     Replace X ψ = sup-od ψ
-    Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → X ∋ x → Set (suc n) ) → OD {suc n}
-    Select X ψ = record { def = λ x →  ( (d : def X x ) →  ψ (ord→od x) (subst (λ k → def X k ) (sym diso) d)) ; otrans = lemma }  where
-       lemma : {x y : Ordinal} → ((d : def X x) → ψ (ord→od x) (subst (def X) (sym diso) d)) →
-            y o< x → (d : def X y) → ψ (ord→od y) (subst (def X) (sym diso) d)
-       lemma {x} {y} f y<x d = {!!}
+    Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n}
+    Select X ψ = record { def = λ x →  ( def X  x ∧  ψ ( ord→od x )) } 
     _,_ : OD {suc n} → OD {suc n} → OD {suc n}
-    x , y = Ord (omax (od→ord x) (od→ord y))
+    x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) }
     Union : OD {suc n} → OD {suc n}
-    Union U = record { def = λ y → osuc y o< (od→ord U) ; otrans = {!!} }
+    Union U = record { def = λ y → osuc y o< (od→ord U) }
     -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x →  X ∋ x )
     Power : OD {suc n} → OD {suc n}
     Power A = Def A
@@ -266,17 +328,14 @@
     A ∈ B = B ∋ A
     _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set (suc n)
     _⊆_ A B {x} = A ∋ x →  B ∋ x
-    _∩_ : ( A B : ZFSet  ) → ZFSet
-    A ∩ B = Select (A , B) (  λ x d → ( A ∋ x ) ∧ (B ∋ x) )
+    -- _∩_ : ( A B : ZFSet  ) → ZFSet
+    -- A ∩ B = Select (A , B) (  λ x → ( A ∋ x ) ∧ (B ∋ x) )
     -- _∪_ : ( A B : ZFSet  ) → ZFSet
     -- A ∪ B = Select (A , B) (  λ x → (A ∋ x)  ∨ ( B ∋ x ) )
-    ｛_｝ : ZFSet → ZFSet
-    ｛ x ｝ = ( x ,  x )
-
     infixr  200 _∈_
     -- infixr  230 _∩_ _∪_
     infixr  220 _⊆_
-    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace (Ord omega)
+    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power {!!} Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} ))
     isZF = record {
            isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
        ;   pair  = pair
@@ -291,7 +350,7 @@
        ;   regularity = regularity
        ;   infinity∅ = infinity∅
        ;   infinity = λ _ → infinity
-       ;   selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
+       ;   selection = λ {ψ} {X} {y} → {!!}
        ;   replacement = replacement
      } where
          open _∧_ 
@@ -315,30 +374,31 @@
               minsup :  OD
               minsup =  ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) 
               lemma-t : csuc minsup ∋ t
-              lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) 
+              lemma-t = o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) 
               lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))  ∋ x
               lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso  )
-              lemma-s | case1 eq = def-subst {!!} oiso refl
-              lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) t∋x
+              lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl
+              lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) t∋x
          -- 
          -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
          -- Power A is a sup of ZFSubset A t, so Power A ∋ t
          -- 
          power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
          power← A t t→A  = def-subst {suc n} {_} {_} {Power A} {od→ord t}
-                  {!!} refl lemma1 where
+                  ( o<→c< {suc n} {od→ord (ZFSubset A (ord→od (od→ord t)) )} {sup-o (λ x → od→ord (ZFSubset A (ord→od x)))}
+                      lemma ) refl lemma1 where
               lemma-eq :  ZFSubset A t == t
               eq→ lemma-eq {z} w = proj2 w 
               eq← lemma-eq {z} w = record { proj2 = w  ;
                  proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
               lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t
-              lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) {!!}
+              lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq))
               lemma :  od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x)))
               lemma = sup-o<   
          union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc z ∋ z
          union-lemma-u {X} {z} U>z = lemma <-osuc where
              lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
-             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} {!!} refl refl
+             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} (o<→c< lt) refl refl
          union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
          union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
          union→ X y u xx | tri< a ¬b ¬c with  osuc-< a (c<→o< (proj2 xx))
@@ -348,51 +408,68 @@
              lemma refl lt = lt
          union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) 
          union← :  (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z )
-         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
+         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
          ψiso :  {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y   → ψ y
          ψiso {ψ} t refl = t
-         selection : {X : OD } {ψ : (x : OD ) →  x ∈ X  → Set (suc n)} {y : OD} → ((d : X ∋ y ) → ψ y d ) ⇔ (Select X ψ ∋ y)
-         selection {ψ} {X} {y} =  {!!}
+         selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
+         selection {ψ} {X} {y} = record {
+              proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso)  }
+            ; proj2 = λ select → record { proj1 = proj1 select  ; proj2 =  ψiso {ψ} (proj2 select) oiso  }
+           }
          replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x
          replacement {ψ} X x = sup-c< ψ {x}
          ∅-iso :  {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 
          ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq  
          minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
-         minimul x  not = {!!}
+         minimul x  not = od∅   
          regularity :  (x : OD) (not : ¬ (x == od∅)) →
-            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ d → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
-         proj1 (regularity x not ) = {!!}
-         proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where
-            reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ d → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
-            reg {y} t = {!!}
+            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
+         proj1 (regularity x not ) = ¬∅=→∅∈ not 
+         proj2 (regularity x not ) = record { eq→ = reg ; eq← = lemma } where
+            lemma : {ox : Ordinal} → def od∅ ox → def (Select (minimul x not) (λ y → (minimul x not ∋ y) ∧ (x ∋ y))) ox
+            lemma (case1 ())
+            lemma (case2 ())
+            reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
+            reg {y} t with proj1 t
+            ... | x∈∅ = x∈∅
          extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
          eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d  
          eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d  
          xx-union : {x  : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
-         xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x))
+         xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x))
          xxx-union : {x  : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
-         xxx-union {x} = cong ( λ k → Ord k ) lemma where
+         xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where
              lemma1 : {x  : OD {suc n}} → od→ord x o< od→ord (x , x)
              lemma1 {x} = c<→o< ( proj1 (pair x x ) )
              lemma2 : {x  : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
-             lemma2 = trans ( cong ( λ k →  od→ord k ) xx-union ) {!!}
+             lemma2 = trans ( cong ( λ k →  od→ord k ) xx-union ) (sym ≡-def)
              lemma : {x  : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
              lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 )
          uxxx-union : {x  : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) }
-         uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where
+         uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where
              lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
-             lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) {!!}
+             lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def )
          uxxx-2 : {x  : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) }
          eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt
          eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt
          uxxx-ord : {x  : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x)
-         uxxx-ord {x} = trans (cong (λ k →  od→ord k ) uxxx-union) {!!} 
+         uxxx-ord {x} = trans (cong (λ k →  od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) 
+         omega = record { lv = Suc Zero ; ord = Φ 1 }
          infinite : OD {suc n}
-         infinite = Ord omega 
-         infinity∅ : Ord omega  ∋ od∅ {suc n}
-         infinity∅ = {!!}
+         infinite = ord→od ( omega )
+         infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
+         infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅}
+              (o<→c< ( case1 (s≤s z≤n )))  refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k →  od→ord k) o∅≡od∅ ))
+         infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega
+         infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where
+              t  : od→ord x o< od→ord (ord→od (omega))
+              t  = ∋→o< {n} {infinite} {x} lt
+         infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x ))
+         infinite∋uxxx x lt = o<∋→ t where
+              t  :  od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega))
+              t  = subst (λ k →  od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) ( sym  (uxxx-ord {x} ) ) lt ) 
          infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
-         infinity x lt = {!!} where
+         infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt ))   where
               lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega 
               lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n)
               lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n)
@@ -400,14 +477,4 @@
               lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
               lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
               lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
-         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set
-         -- ∀ z [ ∀ x ( x ∈ z  → ¬ ( x ≈ ∅ ) )  ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y )  → x ∩ y ≈ ∅  ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ]
-         record Choice (z : OD {suc n}) : Set (suc (suc n)) where
-             field
-                 u : {x : OD {suc n}} ( x∈z  : x ∈ z ) → OD {suc n}
-                 t : {x : OD {suc n}} ( x∈z  : x ∈ z ) → (x : OD {suc n} ) → OD {suc n}
-                 choice : { x : OD {suc n} } → ( x∈z  : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝
-         -- choice : {x :  OD {suc n}} ( x ∈ z  → ¬ ( x ≈ ∅ ) ) →
-         -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) →  choice ¬x∅ A∈X ∈ A 
-         -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}