# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1563678710 -32400
# Node ID 9f3c0e0b2bc9c19b07379b6e2a72692db9509ffc
# Parent  7012158bf2d9c6cc6e745175bbad317957e4ac86
remove ordinal-definable

diff -r 7012158bf2d9 -r 9f3c0e0b2bc9 HOD.agda
--- a/HOD.agda	Sun Jul 21 12:09:50 2019 +0900
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,525 +0,0 @@
-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 OD {n : Level}  : Set (suc n) where
-  field
-    def : (x : Ordinal {n} ) → Set n
-
-open OD
-
-open Ordinal
-open _∧_
-
-record _==_ {n : Level} ( a b :  OD {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 :  OD {n} } → x == x
-eq-refl {n} {x} = record { eq→ = id ; eq← = id }
-
-open  _==_ 
-
-eq-sym : {n : Level} {  x y :  OD {n} } → x == y → y == x
-eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
-
-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) }
-
-⇔→== : {n : Level} {  x y :  OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔  def y z) → x == y 
-eq→ ( ⇔→== {n} {x} {y}  eq ) {z} m = proj1 eq m 
-eq← ( ⇔→== {n} {x} {y}  eq ) {z} m = proj2 eq m 
-
--- Ordinal in OD ( and ZFSet ) Transitive Set
-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} = Ord o∅ 
-
-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
-  oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
-  diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
-  -- we should prove this in agda, but simply put here
-  ==→o≡ : {n : Level} →  { x y : OD {suc n} } → (x == y) → x ≡ 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 
-  --   ord→od x ≡ Ord x results the same
-  -- 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 ) 
-  -- mimimul and x∋minimul is a weaker form of Axiom of choice
-  minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
-  -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox  ( minimum of x )
-  x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
-  minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord  y) )
-
-_∋_ : { n : Level } → ( a x : OD {n} ) → Set n
-_∋_ {n} a x  = def a ( od→ord x )
-
-_c<_ : { n : Level } → ( x a : OD {n} ) → Set n
-x c< a = a ∋ x 
-
-_c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
-a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
-
-cseq : {n : Level} →  OD {n} →  OD {n}
-cseq x = record { def = λ y → def x (osuc y) } where
-
-def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z  →  X ≡ x  →  def z x
-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-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)  )
-
-otrans : {n : Level} {a x : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y
-otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a
-
-∅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 | ()
-
-∅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 : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
-orefl refl = refl
-
-==-iso : {n : Level} →  { x y : OD {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 : OD {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 : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
-=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
-
-ord→== : {n : Level} →  { x y : OD {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 : OD {n} ) → x c≤ x
-c≤-refl x = case1 refl
-
-∋→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∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
-o∅≡od∅ {n} = ==→o≡ lemma where
-     lemma0 :  {x : Ordinal} → def (ord→od o∅) x → def od∅ x
-     lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
-     lemma1 :  {x : Ordinal} → def od∅ x → def (ord→od o∅) x
-     lemma1 (case1 ())
-     lemma1 (case2 ())
-     lemma : ord→od o∅ == od∅
-     lemma = record { eq→ = lemma0 ; eq← = lemma1 }
-
-ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n}
-ord-od∅ {n} = sym ( subst (λ k → k ≡  od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
-
-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) 
-
-∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ } == od∅ {n} 
-eq→ ∅0 {w} (lift ())
-eq← ∅0 {w} (case1 ())
-eq← ∅0 {w} (case2 ())
-
-∅< : {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 ()
-       
-∅6 : {n : Level} →  { x : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
-∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {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-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())
-
-OrdP : {n : Level} →  ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y )
-OrdP {n} x y with trio< x (od→ord y)
-OrdP {n} x y | tri< a ¬b ¬c = no ¬c
-OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
-OrdP {n} x y | tri> ¬a ¬b c = yes c
-
--- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
--- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
-
-in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n}
-in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧  ( x ≡ od→ord (ψ (ord→od y )))))  }
-
--- 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 }   where
-
-Def :  {n : Level} → (A :  OD {suc n}) → OD {suc n}
-Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )   -- Ord x does not help ord-power→
-
--- Constructible Set on α
--- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y <  od→ord x } 
--- L (Φ 0) = Φ
--- L (OSuc lv n) = { Def ( L n )  } 
--- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
-L : {n : Level} → (α : Ordinal {suc n}) → OD {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 )
-    cseq ( Ord (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }))))
-
--- L0 :  {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α
--- L1 :  {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n})  → L α ∋ x → L β ∋ x 
-
-
-OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
-OD→ZF {n}  = record { 
-    ZFSet = OD {suc n}
-    ; _∋_ = _∋_ 
-    ; _≈_ = _==_ 
-    ; ∅  = od∅
-    ; _,_ = _,_
-    ; Union = Union
-    ; Power = Power
-    ; Select = Select
-    ; Replace = Replace
-    ; infinite = infinite
-    ; isZF = isZF 
- } where
-    ZFSet = OD {suc n}
-    Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n}
-    Select X ψ = record { def = λ x →  ( def X x ∧ ψ ( ord→od x )) }
-    Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
-    Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
-    _,_ : OD {suc n} → OD {suc n} → OD {suc n}
-    x , y = Ord (omax (od→ord x) (od→ord y))
-    _∩_ : ( A B : ZFSet  ) → ZFSet
-    A ∩ B = record { def = λ x → def A x ∧ def B x } 
-    Union : OD {suc n} → OD {suc n}  
-    Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x)))  }
-    _∈_ : ( 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
-    Power : OD {suc n} → OD {suc n}
-    Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
-    ｛_｝ : ZFSet → ZFSet
-    ｛ x ｝ = ( x ,  x )
-
-    data infinite-d  : ( x : Ordinal {suc n} ) → Set (suc n) where
-        iφ :  infinite-d o∅
-        isuc : {x : Ordinal {suc n} } →   infinite-d  x  →
-                infinite-d  (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
-
-    infinite : OD {suc n}
-    infinite = record { def = λ x → infinite-d x }
-
-    infixr  200 _∈_
-    -- infixr  230 _∩_ _∪_
-    infixr  220 _⊆_
-    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
-    isZF = record {
-           isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
-       ;   pair  = pair
-       ;   union→ = union→
-       ;   union← = union←
-       ;   empty = empty
-       ;   power→ = power→  
-       ;   power← = power← 
-       ;   extensionality = extensionality
-       ;   minimul = minimul
-       ;   regularity = regularity
-       ;   infinity∅ = infinity∅
-       ;   infinity = infinity
-       ;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
-       ;   replacement← = replacement←
-       ;   replacement→ = replacement→
-     } where
-
-         pair : (A B : OD {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 : {n : Level } (x : OD {suc n} ) → ¬  (od∅ ∋ x)
-         empty x (case1 ())
-         empty x (case2 ())
-
-         ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x}
-         ord-⊆ t x lt = c<→o< lt
-         o<→c< :  {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_  (Ord x) (Ord y) {z}
-         o<→c< lt lt1 = ordtrans lt1 lt
-         
-         ⊆→o< :  {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_  (Ord x) (Ord y) {z} ) →  x o< osuc y
-         ⊆→o< {x} {y}  lt with trio< x y 
-         ⊆→o< {x} {y}  lt | tri< a ¬b ¬c = ordtrans a <-osuc
-         ⊆→o< {x} {y}  lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
-         ⊆→o< {x} {y}  lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
-         ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
-
-         union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
-         union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
-              ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
-         union← :  (X z : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
-         union← X z UX∋z =  TransFiniteExists _ lemma UX∋z where
-              lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
-              lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 
-
-         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 : {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) →  X ∋ x → Replace X ψ ∋ ψ x
-         replacement← {ψ} X x lt = record { proj1 =  sup-c< ψ {x} ; proj2 = lemma } where
-             lemma : def (in-codomain X ψ) (od→ord (ψ x))
-             lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
-         replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
-         replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
-            lemma2 :  ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
-                    → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
-            lemma2 not not2  = not ( λ y d →  not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
-                lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od  y))) → (ord→od (od→ord x) == ψ (ord→od y))  
-                lemma3 {y} eq = subst (λ k  → ord→od (od→ord x) == k ) oiso (o≡→== eq )
-            lemma :  ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
-            lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso  ( proj2 not2 ))
-
-         ---
-         --- Power Set
-         ---
-         ---    First consider ordinals in OD
-         ---
-         --- 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
-         --
-         --
-         ∩-≡ :  { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
-         ∩-≡ {a} {b} inc = record {
-            eq→ = λ {x} x<a → record { proj2 = x<a ;
-                 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso  } ;
-            eq← = λ {x} x<a∩b → proj2 x<a∩b }
-         -- 
-         -- 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
-         -- 
-         ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
-         ord-power← a t t→A  = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t}
-                 lemma refl (lemma1 lemma-eq )where
-              lemma-eq :  ZFSubset (Ord a) t == t
-              eq→ lemma-eq {z} w = proj2 w 
-              eq← lemma-eq {z} w = record { proj2 = w  ;
-                 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
-                    ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
-              lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}}
-                 → (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
-              lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
-              lemma :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
-              lemma = sup-o<   
-
-         -- double-neg-eilm : {n  : Level } {A : Set n} → ¬ ¬ A → A      -- we don't have this in intutionistic logic
-         -- 
-         -- Every set in OD is a subset of Ordinals
-         -- 
-         -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
-
-         -- we have oly double negation form because of the replacement axiom
-         --
-         power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
-         power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where
-              a = od→ord A
-              lemma2 : ¬ ( (y : OD) → ¬ (t ==  (A ∩ y)))
-              lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
-              lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
-              lemma3 y eq not = not (proj1 (eq→ eq t∋x))
-              lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
-              lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
-              lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) →  ¬ ¬  (def A (od→ord x))
-              lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
-
-         power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
-         power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 
-              a = od→ord A
-              lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
-              lemma0 {x} t∋x = c<→o< (t→A t∋x)
-              lemma3 : Def (Ord a) ∋ t
-              lemma3 = ord-power← a t lemma0
-              lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x))
-              lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}
-              lemma4 :  (A ∩ ord→od (od→ord t)) ≡ t
-              lemma4 = let open ≡-Reasoning in begin
-                    A ∩ ord→od (od→ord t)
-                 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
-                    A ∩ t
-                 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
-                    t
-                 ∎
-              lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x))
-              lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x)))
-                  lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t})
-              lemma2 :  def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
-              lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
-                  lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 
-                  lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A  )))
-
-         regularity :  (x : OD) (not : ¬ (x == od∅)) →
-            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
-         proj1 (regularity x not ) = x∋minimul x not
-         proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
-             lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
-             lemma1 {x₁} s = ⊥-elim  ( minimul-1 x not (ord→od x₁) lemma3 ) where
-                 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
-                 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso)
-                                 ; proj2 = proj2 (proj2 s) } 
-             lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁
-             lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
-
-         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  
-
-         infinity∅ : infinite  ∋ od∅ {suc n}
-         infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where
-              lemma : o∅ ≡ od→ord od∅
-              lemma =  let open ≡-Reasoning in begin
-                    o∅
-                 ≡⟨ sym diso ⟩
-                    od→ord ( ord→od o∅ )
-                 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
-                    od→ord od∅
-                 ∎
-         infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
-         infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
-               lemma :  od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
-                    ≡ od→ord (Union (x , (x , x)))
-               lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso 
-
-         -- Axiom of choice ( is equivalent to the existence of minimul in our case )
-         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] 
-         choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
-         choice-func X {x} not X∋x = minimul x not
-         choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A 
-         choice X {A} X∋A not = x∋minimul A not
-
-         -- another form of regularity 
-         --
-         ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
-             → ( {x : OD {suc n} } → ({ y : OD {suc n} } →  x ∋ y → ψ y ) → ψ x )
-             → (x : OD {suc n} ) → ψ x
-         ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x)))  <-osuc) where
-            ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
-                → (ly < lx) ∨ (oy d< ox  ) → ψ (ord→od (record { lv = ly ; ord = oy } ) )
-            ε-induction-ord Zero (Φ 0)  (case1 ())
-            ε-induction-ord Zero (Φ 0)  (case2 ())
-            ε-induction-ord lx  (OSuc lx ox) {ly} {oy} y<x = 
-                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where
-                    lemma :  (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox }
-                    lemma y lt with osuc-≡< y<x
-                    lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
-                    lemma y lt | case2 lt1 = ordtrans  (o<-subst (c<→o< lt) refl diso) lt1  
-            ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) =                    
-                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt )  where  
-                    --
-                    --     if lv of z if less than x Ok
-                    --     else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma
-                    --
-                    --                         lx    Suc lx      (1) lz(a) <lx by case1
-                    --                 ly(1)   ly(2)             (2) lz(b) <lx by case1
-                    --           lz(a) lz(b)   lz(c)                 lz(c) <lx by case2 ( ly==lz==lx)
-                    --
-                    lemma0 : { lx ly : Nat } → ly < Suc lx  → lx < ly → ⊥
-                    lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2
-                    lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
-                    lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin
-                            lv (od→ord (ord→od (record { lv = ly ; ord = oy })))
-                         ≡⟨ cong ( λ k → lv k ) diso ⟩
-                            lv (record { lv = ly ; ord = oy })
-                         ≡⟨⟩
-                            ly
-                         ∎
-                    lemma :  (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z
-                    lemma z lt with  c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt
-                    lemma z lt | case1 lz<ly with <-cmp lx ly
-                    lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
-                    lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c =    -- ly(1)
-                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) ))
-                    lemma z lt | case1 lz<ly | tri> ¬a ¬b c =       -- lz(a)
-                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c))))
-                    lemma z lt | case2 lz=ly with <-cmp lx ly
-                    lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
-                    lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly        -- lz(b)
-                    ... | eq = subst (λ k → ψ k ) oiso
-                         (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c )))
-                    lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly    -- lz(c)
-                    ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡  k) lemma1 eq)) where
-                          lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z
-                          lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt )
-                          lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly }  →
-                               lx ≡ ly → ly ≡ lv (od→ord z)  → ψ z 
-                          lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)
-
diff -r 7012158bf2d9 -r 9f3c0e0b2bc9 OD.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/OD.agda	Sun Jul 21 12:11:50 2019 +0900
@@ -0,0 +1,525 @@
+open import Level
+module OD 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 OD {n : Level}  : Set (suc n) where
+  field
+    def : (x : Ordinal {n} ) → Set n
+
+open OD
+
+open Ordinal
+open _∧_
+
+record _==_ {n : Level} ( a b :  OD {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 :  OD {n} } → x == x
+eq-refl {n} {x} = record { eq→ = id ; eq← = id }
+
+open  _==_ 
+
+eq-sym : {n : Level} {  x y :  OD {n} } → x == y → y == x
+eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }
+
+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) }
+
+⇔→== : {n : Level} {  x y :  OD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔  def y z) → x == y 
+eq→ ( ⇔→== {n} {x} {y}  eq ) {z} m = proj1 eq m 
+eq← ( ⇔→== {n} {x} {y}  eq ) {z} m = proj2 eq m 
+
+-- Ordinal in OD ( and ZFSet ) Transitive Set
+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} = Ord o∅ 
+
+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
+  oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
+  diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
+  -- we should prove this in agda, but simply put here
+  ==→o≡ : {n : Level} →  { x y : OD {suc n} } → (x == y) → x ≡ 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 
+  --   ord→od x ≡ Ord x results the same
+  -- 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 ) 
+  -- mimimul and x∋minimul is a weaker form of Axiom of choice
+  minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
+  -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox  ( minimum of x )
+  x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
+  minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord  y) )
+
+_∋_ : { n : Level } → ( a x : OD {n} ) → Set n
+_∋_ {n} a x  = def a ( od→ord x )
+
+_c<_ : { n : Level } → ( x a : OD {n} ) → Set n
+x c< a = a ∋ x 
+
+_c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
+a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
+
+cseq : {n : Level} →  OD {n} →  OD {n}
+cseq x = record { def = λ y → def x (osuc y) } where
+
+def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z  →  X ≡ x  →  def z x
+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-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)  )
+
+otrans : {n : Level} {a x : Ordinal {n} } → def (Ord a) x → { y : Ordinal {n} } → y o< x → def (Ord a) y
+otrans {n} {a} {x} x<a {y} y<x = ordtrans y<x x<a
+
+∅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 | ()
+
+∅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 : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
+orefl refl = refl
+
+==-iso : {n : Level} →  { x y : OD {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 : OD {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 : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
+=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)
+
+ord→== : {n : Level} →  { x y : OD {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 : OD {n} ) → x c≤ x
+c≤-refl x = case1 refl
+
+∋→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∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
+o∅≡od∅ {n} = ==→o≡ lemma where
+     lemma0 :  {x : Ordinal} → def (ord→od o∅) x → def od∅ x
+     lemma0 {x} lt = o<-subst (c<→o< {suc n} {ord→od x} {ord→od o∅} (def-subst {suc n} {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
+     lemma1 :  {x : Ordinal} → def od∅ x → def (ord→od o∅) x
+     lemma1 (case1 ())
+     lemma1 (case2 ())
+     lemma : ord→od o∅ == od∅
+     lemma = record { eq→ = lemma0 ; eq← = lemma1 }
+
+ord-od∅ : {n : Level} → od→ord (od∅ {suc n}) ≡ o∅ {suc n}
+ord-od∅ {n} = sym ( subst (λ k → k ≡  od→ord (od∅ {suc n}) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )
+
+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) 
+
+∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ } == od∅ {n} 
+eq→ ∅0 {w} (lift ())
+eq← ∅0 {w} (case1 ())
+eq← ∅0 {w} (case2 ())
+
+∅< : {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 ()
+       
+∅6 : {n : Level} →  { x : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
+∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {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-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())
+
+OrdP : {n : Level} →  ( x : Ordinal {suc n} ) ( y : OD {suc n} ) → Dec ( Ord x ∋ y )
+OrdP {n} x y with trio< x (od→ord y)
+OrdP {n} x y | tri< a ¬b ¬c = no ¬c
+OrdP {n} x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
+OrdP {n} x y | tri> ¬a ¬b c = yes c
+
+-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
+-- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
+
+in-codomain : {n : Level} → (X : OD {suc n} ) → ( ψ : OD {suc n} → OD {suc n} ) → OD {suc n}
+in-codomain {n} X ψ = record { def = λ x → ¬ ( (y : Ordinal {suc n}) → ¬ ( def X y ∧  ( x ≡ od→ord (ψ (ord→od y )))))  }
+
+-- 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 }   where
+
+Def :  {n : Level} → (A :  OD {suc n}) → OD {suc n}
+Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )   -- Ord x does not help ord-power→
+
+-- Constructible Set on α
+-- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y <  od→ord x } 
+-- L (Φ 0) = Φ
+-- L (OSuc lv n) = { Def ( L n )  } 
+-- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
+L : {n : Level} → (α : Ordinal {suc n}) → OD {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 )
+    cseq ( Ord (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }))))
+
+-- L0 :  {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α
+-- L1 :  {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n})  → L α ∋ x → L β ∋ x 
+
+
+OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
+OD→ZF {n}  = record { 
+    ZFSet = OD {suc n}
+    ; _∋_ = _∋_ 
+    ; _≈_ = _==_ 
+    ; ∅  = od∅
+    ; _,_ = _,_
+    ; Union = Union
+    ; Power = Power
+    ; Select = Select
+    ; Replace = Replace
+    ; infinite = infinite
+    ; isZF = isZF 
+ } where
+    ZFSet = OD {suc n}
+    Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n}
+    Select X ψ = record { def = λ x →  ( def X x ∧ ψ ( ord→od x )) }
+    Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
+    Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
+    _,_ : OD {suc n} → OD {suc n} → OD {suc n}
+    x , y = Ord (omax (od→ord x) (od→ord y))
+    _∩_ : ( A B : ZFSet  ) → ZFSet
+    A ∩ B = record { def = λ x → def A x ∧ def B x } 
+    Union : OD {suc n} → OD {suc n}  
+    Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x)))  }
+    _∈_ : ( 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
+    Power : OD {suc n} → OD {suc n}
+    Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
+    ｛_｝ : ZFSet → ZFSet
+    ｛ x ｝ = ( x ,  x )
+
+    data infinite-d  : ( x : Ordinal {suc n} ) → Set (suc n) where
+        iφ :  infinite-d o∅
+        isuc : {x : Ordinal {suc n} } →   infinite-d  x  →
+                infinite-d  (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))
+
+    infinite : OD {suc n}
+    infinite = record { def = λ x → infinite-d x }
+
+    infixr  200 _∈_
+    -- infixr  230 _∩_ _∪_
+    infixr  220 _⊆_
+    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
+    isZF = record {
+           isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
+       ;   pair  = pair
+       ;   union→ = union→
+       ;   union← = union←
+       ;   empty = empty
+       ;   power→ = power→  
+       ;   power← = power← 
+       ;   extensionality = extensionality
+       ;   minimul = minimul
+       ;   regularity = regularity
+       ;   infinity∅ = infinity∅
+       ;   infinity = infinity
+       ;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
+       ;   replacement← = replacement←
+       ;   replacement→ = replacement→
+     } where
+
+         pair : (A B : OD {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 : {n : Level } (x : OD {suc n} ) → ¬  (od∅ ∋ x)
+         empty x (case1 ())
+         empty x (case2 ())
+
+         ord-⊆ : ( t x : OD {suc n} ) → _⊆_ t (Ord (od→ord t )) {x}
+         ord-⊆ t x lt = c<→o< lt
+         o<→c< :  {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_  (Ord x) (Ord y) {z}
+         o<→c< lt lt1 = ordtrans lt1 lt
+         
+         ⊆→o< :  {x y : Ordinal {suc n}} → (∀ (z : OD) → _⊆_  (Ord x) (Ord y) {z} ) →  x o< osuc y
+         ⊆→o< {x} {y}  lt with trio< x y 
+         ⊆→o< {x} {y}  lt | tri< a ¬b ¬c = ordtrans a <-osuc
+         ⊆→o< {x} {y}  lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
+         ⊆→o< {x} {y}  lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
+         ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
+
+         union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
+         union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
+              ; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
+         union← :  (X z : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
+         union← X z UX∋z =  TransFiniteExists _ lemma UX∋z where
+              lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
+              lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx } 
+
+         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 : {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) →  X ∋ x → Replace X ψ ∋ ψ x
+         replacement← {ψ} X x lt = record { proj1 =  sup-c< ψ {x} ; proj2 = lemma } where
+             lemma : def (in-codomain X ψ) (od→ord (ψ x))
+             lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
+         replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
+         replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
+            lemma2 :  ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
+                    → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
+            lemma2 not not2  = not ( λ y d →  not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
+                lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od  y))) → (ord→od (od→ord x) == ψ (ord→od y))  
+                lemma3 {y} eq = subst (λ k  → ord→od (od→ord x) == k ) oiso (o≡→== eq )
+            lemma :  ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
+            lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso  ( proj2 not2 ))
+
+         ---
+         --- Power Set
+         ---
+         ---    First consider ordinals in OD
+         ---
+         --- 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
+         --
+         --
+         ∩-≡ :  { a b : OD {suc n} } → ({x : OD {suc n} } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
+         ∩-≡ {a} {b} inc = record {
+            eq→ = λ {x} x<a → record { proj2 = x<a ;
+                 proj1 = def-subst {suc n} {_} {_} {b} {x} (inc (def-subst {suc n} {_} {_} {a} {_} x<a refl (sym diso) )) refl diso  } ;
+            eq← = λ {x} x<a∩b → proj2 x<a∩b }
+         -- 
+         -- 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
+         -- 
+         ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
+         ord-power← a t t→A  = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t}
+                 lemma refl (lemma1 lemma-eq )where
+              lemma-eq :  ZFSubset (Ord a) t == t
+              eq→ lemma-eq {z} w = proj2 w 
+              eq← lemma-eq {z} w = record { proj2 = w  ;
+                 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
+                    ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
+              lemma1 : {n : Level } {a : Ordinal {suc n}} { t : OD {suc n}}
+                 → (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
+              lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
+              lemma :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
+              lemma = sup-o<   
+
+         -- double-neg-eilm : {n  : Level } {A : Set n} → ¬ ¬ A → A      -- we don't have this in intutionistic logic
+         -- 
+         -- Every set in OD is a subset of Ordinals
+         -- 
+         -- Power A = Replace (Def (Ord (od→ord A))) ( λ y → A ∩ y )
+
+         -- we have oly double negation form because of the replacement axiom
+         --
+         power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
+         power→ A t P∋t {x} t∋x = TransFiniteExists _ lemma5 lemma4 where
+              a = od→ord A
+              lemma2 : ¬ ( (y : OD) → ¬ (t ==  (A ∩ y)))
+              lemma2 = replacement→ (Def (Ord (od→ord A))) t P∋t
+              lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
+              lemma3 y eq not = not (proj1 (eq→ eq t∋x))
+              lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
+              lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
+              lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) →  ¬ ¬  (def A (od→ord x))
+              lemma5 {y} eq not = (lemma3 (ord→od y) eq) not
+
+         power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
+         power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where 
+              a = od→ord A
+              lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
+              lemma0 {x} t∋x = c<→o< (t→A t∋x)
+              lemma3 : Def (Ord a) ∋ t
+              lemma3 = ord-power← a t lemma0
+              lt1 : od→ord (A ∩ ord→od (od→ord t)) o< sup-o (λ x → od→ord (A ∩ ord→od x))
+              lt1 = sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t}
+              lemma4 :  (A ∩ ord→od (od→ord t)) ≡ t
+              lemma4 = let open ≡-Reasoning in begin
+                    A ∩ ord→od (od→ord t)
+                 ≡⟨ cong (λ k → A ∩ k) oiso ⟩
+                    A ∩ t
+                 ≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
+                    t
+                 ∎
+              lemma1 : od→ord t o< sup-o (λ x → od→ord (A ∩ ord→od x))
+              lemma1 = subst (λ k → od→ord k o< sup-o (λ x → od→ord (A ∩ ord→od x)))
+                  lemma4 (sup-o< {suc n} {λ x → od→ord (A ∩ ord→od x)} {od→ord t})
+              lemma2 :  def (in-codomain (Def (Ord (od→ord A))) (_∩_ A)) (od→ord t)
+              lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
+                  lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t)) 
+                  lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A  )))
+
+         regularity :  (x : OD) (not : ¬ (x == od∅)) →
+            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
+         proj1 (regularity x not ) = x∋minimul x not
+         proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
+             lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
+             lemma1 {x₁} s = ⊥-elim  ( minimul-1 x not (ord→od x₁) lemma3 ) where
+                 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
+                 lemma3 = record { proj1 = def-subst {suc n} {_} {_} {minimul x not} {_} (proj1 s) refl (sym diso)
+                                 ; proj2 = proj2 (proj2 s) } 
+             lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁
+             lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))
+
+         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  
+
+         infinity∅ : infinite  ∋ od∅ {suc n}
+         infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where
+              lemma : o∅ ≡ od→ord od∅
+              lemma =  let open ≡-Reasoning in begin
+                    o∅
+                 ≡⟨ sym diso ⟩
+                    od→ord ( ord→od o∅ )
+                 ≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
+                    od→ord od∅
+                 ∎
+         infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
+         infinity x lt = def-subst {suc n} {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
+               lemma :  od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
+                    ≡ od→ord (Union (x , (x , x)))
+               lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso 
+
+         -- Axiom of choice ( is equivalent to the existence of minimul in our case )
+         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] 
+         choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
+         choice-func X {x} not X∋x = minimul x not
+         choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A 
+         choice X {A} X∋A not = x∋minimul A not
+
+         -- another form of regularity 
+         --
+         ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
+             → ( {x : OD {suc n} } → ({ y : OD {suc n} } →  x ∋ y → ψ y ) → ψ x )
+             → (x : OD {suc n} ) → ψ x
+         ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x)))  <-osuc) where
+            ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
+                → (ly < lx) ∨ (oy d< ox  ) → ψ (ord→od (record { lv = ly ; ord = oy } ) )
+            ε-induction-ord Zero (Φ 0)  (case1 ())
+            ε-induction-ord Zero (Φ 0)  (case2 ())
+            ε-induction-ord lx  (OSuc lx ox) {ly} {oy} y<x = 
+                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where
+                    lemma :  (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox }
+                    lemma y lt with osuc-≡< y<x
+                    lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
+                    lemma y lt | case2 lt1 = ordtrans  (o<-subst (c<→o< lt) refl diso) lt1  
+            ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) =                    
+                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt )  where  
+                    --
+                    --     if lv of z if less than x Ok
+                    --     else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma
+                    --
+                    --                         lx    Suc lx      (1) lz(a) <lx by case1
+                    --                 ly(1)   ly(2)             (2) lz(b) <lx by case1
+                    --           lz(a) lz(b)   lz(c)                 lz(c) <lx by case2 ( ly==lz==lx)
+                    --
+                    lemma0 : { lx ly : Nat } → ly < Suc lx  → lx < ly → ⊥
+                    lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2
+                    lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
+                    lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin
+                            lv (od→ord (ord→od (record { lv = ly ; ord = oy })))
+                         ≡⟨ cong ( λ k → lv k ) diso ⟩
+                            lv (record { lv = ly ; ord = oy })
+                         ≡⟨⟩
+                            ly
+                         ∎
+                    lemma :  (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z
+                    lemma z lt with  c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt
+                    lemma z lt | case1 lz<ly with <-cmp lx ly
+                    lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
+                    lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c =    -- ly(1)
+                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) ))
+                    lemma z lt | case1 lz<ly | tri> ¬a ¬b c =       -- lz(a)
+                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c))))
+                    lemma z lt | case2 lz=ly with <-cmp lx ly
+                    lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
+                    lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly        -- lz(b)
+                    ... | eq = subst (λ k → ψ k ) oiso
+                         (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c )))
+                    lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly    -- lz(c)
+                    ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡  k) lemma1 eq)) where
+                          lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z
+                          lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt )
+                          lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly }  →
+                               lx ≡ ly → ly ≡ lv (od→ord z)  → ψ z 
+                          lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)
+
diff -r 7012158bf2d9 -r 9f3c0e0b2bc9 ordinal-definable.agda
--- a/ordinal-definable.agda	Sun Jul 21 12:09:50 2019 +0900
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,391 +0,0 @@
-{-# OPTIONS --allow-unsolved-metas #-}
-
-open import Level
-module ordinal-definable where
-
-open import zf
-open import ordinal
-open import HOD
-
-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
-
-open OD
-open import Data.Unit
-
-open Ordinal
-open _==_
-
-
-postulate      
-  od=ord : {n : Level } → { x : Ordinal {n}} → ord→od x ≡ Ord x
-  -- 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 ) 
-
-o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
-o<→c< {n} {x} {y} lt  = def-subst {n} {_} {_} {ord→od y} {x} lt (sym od=ord) refl
-
-ord=od : {n : Level } → { x : OD {n}} → x ≡ Ord (od→ord x)
-ord=od {n} {x} = subst ( λ k → k ≡ Ord (od→ord x) ) oiso od=ord 
-
-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 = 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
-
-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≡o : {n : Level} →  { x y : Ordinal {suc n} } → ord→od x == ord→od y →  x ≡ y 
-==→o≡o {n} {x} {y} eq with trio< {n} x y
-==→o≡o {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso )))
-==→o≡o {n} {x} {y} eq | tri≈ ¬a b ¬c = b
-==→o≡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≡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≡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≡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} } → 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<→¬== : {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
-
-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
-
-
-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
-
-
-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))
-
-csuc :  {n : Level} →  OD {suc n} → OD {suc n}
-csuc x = Ord ( osuc ( od→ord x ))
-
-Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
-Ord→ZF {n}  = record { 
-    ZFSet = OD {suc n}
-    ; _∋_ = _∋_ 
-    ; _≈_ = _==_ 
-    ; ∅  = od∅
-    ; _,_ = _,_
-    ; Union = Union
-    ; Power = Power
-    ; Select = Select
-    ; Replace = Replace
-    ; infinite = ord→od ( record { lv = Suc Zero ; ord = Φ 1 } )
-    ; isZF = isZF 
- } where
-    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 = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) }
-    _∩_ : ( A B : OD {suc n} ) → OD
-    A ∩ B = record { def = λ x → def A x  ∧ def B x }
-    Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
-    Replace X ψ = record { def = λ x → (x o< sup-o ( λ x → od→ord (ψ (ord→od x )))) ∧ def (in-codomain X ψ) x }
-    Union : OD {suc n} → OD {suc n}
-    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
-    ZFSet = OD {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 → (A ∋ x)  ∨ ( B ∋ x ) )
-    infixr  200 _∈_
-    -- infixr  230 _∩_ _∪_
-    infixr  220 _⊆_
-    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select 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
-       ;   union→ = union→
-       ;   union← = union←
-       ;   empty = empty
-       ;   power→ = power→
-       ;   power← = power← 
-       ;   extensionality = extensionality
-       ;   minimul = minimul
-       ;   regularity = regularity
-       ;   infinity∅ = infinity∅
-       ;   infinity = infinity
-       ;   selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
-       ;   replacement← = replacement←
-       ;   replacement→ = replacement→
-     } where
-
-         pair : (A B : OD {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 : OD {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 : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
-         power→ A t P∋t {x} t∋x = double-neg (proj1 lemma-s) where
-              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 = def-subst (o<→c< (o<-subst (sup-lb (o<-subst (c<→o< {!!}) 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< {!!}  ) refl diso  )
-              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}
-                  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) (==→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 = def-subst (lemma <-osuc ) od=ord refl 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} (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))
-         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 : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z ))) -- (X ∋ csuc z) ∧ (csuc z ∋ z )
-         union← X z X∋z not = not (csuc z) 
-             record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (trans (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 : {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) →  X ∋ x → Replace X ψ ∋ ψ x
-         replacement← {ψ} X x lt = record { proj1 =  sup-c< ψ {x} ; proj2 = lemma } where
-             lemma : def (in-codomain X ψ) (od→ord (ψ x))
-             lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ) )
-         replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
-         replacement→ {ψ} X x lt = contra-position lemma (lemma2 (def-subst (proj2 lt) {!!} refl )) where
-            lemma2 :  ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (Ord y))))
-                    → ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)))
-            lemma2 not not2  = not ( λ y d →  not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
-                lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (Ord y))) → (ord→od (od→ord x) == ψ (Ord y))  
-                lemma3 {y} eq = subst (λ k  → ord→od (od→ord x) == k ) oiso (o≡→== eq )
-            lemma :  ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (Ord y)) )
-            lemma not y not2 = not (Ord y) (subst (λ k → k == ψ (Ord y)) oiso  ( proj2 not2 ))
-
-         minimul-o : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
-         minimul-o x  not = od∅   
-         regularity :  (x : OD) (not : ¬ (x == od∅)) →
-            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
-         proj1 (regularity x not ) = def-subst (¬∅=→∅∈ not) {!!} refl
-         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 = ⊥-elim ( ¬x<0 (def-subst (proj1 (proj2 t )) {!!} refl ))
-
-         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 → 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 → 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 ) (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 } ) lemma where
-             lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
-             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) (==→o≡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→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 = 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)
-              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
-
-         -- Axiom of choice ( is equivalent to the existence of minimul in our case )
-         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] 
-         choice-func : (X : OD {suc n} ) → {x : OD } → ¬ ( x == od∅ ) → ( X ∋ x ) → OD
-         choice-func X {x} not X∋x = od∅ {suc n}
-         choice : (X : OD {suc n} ) → {A : OD } → ( X∋A : X ∋ A ) → (not : ¬ ( A == od∅ )) → A ∋ choice-func X not X∋A
-         choice X {A} X∋A not = ¬∅=→∅∈ not
-
-         -- another form of regularity 
-         --
-         ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
-             → ( {x : OD {suc n} } → ({ y : OD {suc n} } →  x ∋ y → ψ y ) → ψ x )
-             → (x : OD {suc n} ) → ψ x
-         ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x)))  <-osuc) where
-            ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
-                → (ly < lx) ∨ (oy d< ox  ) → ψ (ord→od (record { lv = ly ; ord = oy } ) )
-            ε-induction-ord Zero (Φ 0)  (case1 ())
-            ε-induction-ord Zero (Φ 0)  (case2 ())
-            ε-induction-ord lx  (OSuc lx ox) {ly} {oy} y<x =
-                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord lx ox (lemma y lt ))) where
-                    lemma :  (y : OD) → ord→od record { lv = ly ; ord = oy } ∋ y → od→ord y o< record { lv = lx ; ord = ox }
-                    lemma y lt with osuc-≡< y<x
-                    lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
-                    lemma y lt | case2 lt1 = ordtrans  (o<-subst (c<→o< lt) refl diso) lt1
-            ε-induction-ord (Suc lx) (Φ (Suc lx)) {ly} {oy} (case1 y<sox ) =
-                ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → lemma y lt )  where
-                    --
-                    --     if lv of z if less than x Ok
-                    --     else lv z = lv x. We can create OSuc of z which is larger than z and less than x in lemma
-                    --
-                    --                         lx    Suc lx      (1) lz(a) <lx by case1
-                    --                 ly(1)   ly(2)             (2) lz(b) <lx by case1
-                    --           lz(a) lz(b)   lz(c)                 lz(c) <lx by case2 ( ly==lz==lx)
-                    --
-                    lemma0 : { lx ly : Nat } → ly < Suc lx  → lx < ly → ⊥
-                    lemma0 {Suc lx} {Suc ly} (s≤s lt1) (s≤s lt2) = lemma0 lt1 lt2
-                    lemma1 : {n : Level } {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
-                    lemma1 {n} {ly} {oy} = let open ≡-Reasoning in begin
-                            lv (od→ord (ord→od (record { lv = ly ; ord = oy })))
-                         ≡⟨ cong ( λ k → lv k ) diso ⟩
-                            lv (record { lv = ly ; ord = oy })
-                         ≡⟨⟩
-                            ly
-                         ∎
-                    lemma :  (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → ψ z
-                    lemma z lt with  c<→o< {suc n} {z} {ord→od (record { lv = ly ; ord = oy })} lt
-                    lemma z lt | case1 lz<ly with <-cmp lx ly
-                    lemma z lt | case1 lz<ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen
-                    lemma z lt | case1 lz<ly | tri≈ ¬a refl ¬c =    -- ly(1)
-                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → lv (od→ord z) < k ) lemma1 lz<ly ) ))
-                    lemma z lt | case1 lz<ly | tri> ¬a ¬b c =       -- lz(a)
-                          subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (<-trans lz<ly (subst (λ k → k < lx ) (sym lemma1) c))))
-                    lemma z lt | case2 lz=ly with <-cmp lx ly
-                    lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can-t happen
-                    lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly        -- lz(b)
-                    ... | eq = subst (λ k → ψ k ) oiso
-                         (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c )))
-                    lemma z lt | case2 lz=ly | tri≈ ¬a lx=ly ¬c with d<→lv lz=ly    -- lz(c)
-                    ... | eq = lemma6 {ly} {Φ lx} {oy} lx=ly (sym (subst (λ k → lv (od→ord z) ≡  k) lemma1 eq)) where
-                          lemma5 : (ox : OrdinalD lx) → (lv (od→ord z) < lx) ∨ (ord (od→ord z) d< ox) → ψ z
-                          lemma5 ox lt = subst (λ k → ψ k ) oiso (ε-induction-ord lx ox lt )
-                          lemma6 : { ly : Nat } { ox : OrdinalD {suc n} lx } { oy : OrdinalD {suc n} ly }  →
-                               lx ≡ ly → ly ≡ lv (od→ord z)  → ψ z
-                          lemma6 {ly} {ox} {oy} refl refl = lemma5 (OSuc lx (ord (od→ord z)) ) (case2 s<refl)
-