# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1563599132 -32400
# Node ID 11490a3170d4005292b01af651733614892c5393
# Parent  aa89d1b8ce9639c51e0012c823eb36f00000ff3d
new ordinal-definable

diff -r aa89d1b8ce96 -r 11490a3170d4 HOD.agda
--- a/HOD.agda	Sat Jul 20 08:21:54 2019 +0900
+++ b/HOD.agda	Sat Jul 20 14:05:32 2019 +0900
@@ -502,23 +502,20 @@
                          ≡⟨⟩
                             ly
                          ∎
-                    lemma2 : { lx : Nat } → lx < Suc lx  
-                    lemma2 {Zero} = s≤s z≤n
-                    lemma2 {Suc lx} = s≤s (lemma2 {lx})
                     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 =     -- (1)
+                    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 =       -- z(a)
+                    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       -- z(b)
+                    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    -- z(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 )
diff -r aa89d1b8ce96 -r 11490a3170d4 ordinal-definable.agda
--- a/ordinal-definable.agda	Sat Jul 20 08:21:54 2019 +0900
+++ b/ordinal-definable.agda	Sat Jul 20 14:05:32 2019 +0900
@@ -5,6 +5,7 @@
 
 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
@@ -16,102 +17,19 @@
 
 -- Ordinal Definable Set
 
-record OD {n : Level}  : Set (suc n) where
-  field
-    def : (x : Ordinal {n} ) → Set n
-
 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 
-     eq← : ∀ { x : Ordinal {n} } → def b x → def a x 
+open _==_
 
-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) }
-
-ord→od : {n : Level} → Ordinal {n} → OD {n} 
-ord→od a = Ord a
-
-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}
-  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
-  -- 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 
   -- 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 ) 
-
-_∋_ : { 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 )
-
-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→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} {_} {_} {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
-   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 | ()
+  o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
 
 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 ) 
@@ -121,43 +39,6 @@
    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
 
-∅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<→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 )
@@ -168,14 +49,14 @@
 ... | 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 )))
+==→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≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where
+≡-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))
@@ -186,43 +67,22 @@
 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 )
+==→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≡ eq) z>x
+==-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} } → 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 (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)
-
 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) 
-
 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 )
@@ -235,16 +95,6 @@
 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 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
-
-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
@@ -252,10 +102,6 @@
 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 () )
-is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())
 
 open _∧_
 
@@ -278,24 +124,6 @@
 csuc :  {n : Level} →  OD {suc n} → OD {suc n}
 csuc x = Ord ( osuc ( od→ord x ))
 
-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 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 }  
-
-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 )) ) )  
-
--- Constructible Set on α
-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 )
-    record { def = λ y → osuc y o< (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }) )) }
-
 Ord→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
 Ord→ZF {n}  = record { 
     ZFSet = OD {suc n}
@@ -373,9 +201,9 @@
               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 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
          -- 
@@ -384,8 +212,7 @@
          -- 
          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}
-                  ( 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
+                  {!!} refl lemma1 where
               lemma-eq :  ZFSubset A t == t
               eq→ lemma-eq {z} w = proj2 w 
               eq← lemma-eq {z} w = record { proj2 = w  ;
@@ -396,7 +223,7 @@
               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
+         union-lemma-u {X} {z} U>z = {!!} where -- 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} (o<→c< lt) refl refl
          union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
@@ -409,7 +236,7 @@
          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 (sym diso) ; proj2 = union-lemma-u X∋z } 
+             record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} (o<→c< X∋z) oiso (sym {!!}) ; 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
@@ -424,7 +251,7 @@
              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
+         replacement→ {ψ} X x lt = contra-position lemma (lemma2 {!!}) 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
@@ -433,17 +260,17 @@
             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 : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
-         minimul x  not = od∅   
+         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 ) = ¬∅=→∅∈ not 
+         proj1 (regularity x 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 = ⊥-elim ( ¬x<0 (proj1 (proj2 t )) )
+            reg {y} t = ⊥-elim ( ¬x<0 {!!} )
 
          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  
@@ -467,7 +294,7 @@
          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≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) 
+         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 )
@@ -492,3 +319,67 @@
               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)
+
diff -r aa89d1b8ce96 -r 11490a3170d4 ordinal.agda
--- a/ordinal.agda	Sat Jul 20 08:21:54 2019 +0900
+++ b/ordinal.agda	Sat Jul 20 14:05:32 2019 +0900
@@ -224,6 +224,12 @@
    lemma1 (case1 x) = ¬a x
    lemma1 (case2 x) = ≡→¬d< x
 
+xo<ab : {n : Level} {oa ob : Ordinal {suc n}} → ( {ox : Ordinal {suc n}} → ox o< oa  → ox o< ob ) → oa o< osuc ob
+xo<ab {n}  {oa} {ob} a→b with trio< oa ob
+xo<ab {n}  {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
+xo<ab {n}  {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
+xo<ab {n}  {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c )  )
+
 maxα : {n : Level} →  Ordinal {suc n} →  Ordinal  → Ordinal
 maxα x y with trio< x y
 maxα x y | tri< a ¬b ¬c = y