### changeset 223:2e1f19c949dc

sepration of ordinal from OD
author Shinji KONO Fri, 09 Aug 2019 17:57:58 +0900 59771eb07df0 afc864169325 OD.agda zf.agda 2 files changed, 159 insertions(+), 304 deletions(-) [+]
line wrap: on
line diff
```--- a/OD.agda	Fri Aug 09 16:54:30 2019 +0900
+++ b/OD.agda	Fri Aug 09 17:57:58 2019 +0900
@@ -1,8 +1,8 @@
open import Level
-module OD where
+open import Ordinals
+module OD {n : Level } (O : Ordinals {n} ) 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
@@ -14,248 +14,223 @@
open import logic
open import nat

+open inOrdinal O
+
-- Ordinal Definable Set

-record OD {n : Level}  : Set (suc n) where
+record OD : Set (suc n ) where
field
-    def : (x : Ordinal {n} ) → Set n
+    def : (x : Ordinal  ) → Set n

open OD

-open Ordinal
open _∧_
open _∨_
open Bool

-record _==_ {n : Level} ( a b :  OD {n} ) : Set n where
+record _==_  ( a b :  OD  ) : Set n where
field
-     eq→ : ∀ { x : Ordinal {n} } → def a x → def b x
-     eq← : ∀ { x : Ordinal {n} } → def b x → def a x
+     eq→ : ∀ { x : Ordinal  } → def a x → def b x
+     eq← : ∀ { x : Ordinal  } → 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 }
+eq-refl :  {  x :  OD  } → x == x
+eq-refl  {x} = record { eq→ = id ; eq← = id }

open  _==_

-eq-sym : {n : Level} {  x y :  OD {n} } → x == y → y == x
+eq-sym :  {  x y :  OD  } → 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 z :  OD  } → 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
+⇔→== :  {  x y :  OD  } → ( {z : Ordinal } → def x z ⇔  def y z) → x == y
+eq→ ( ⇔→==  {x} {y}  eq ) {z} m = proj1 eq m
+eq← ( ⇔→==  {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 }
+Ord : ( a : Ordinal  ) → OD
+Ord  a = record { def = λ y → y o< a }

-od∅ : {n : Level} → OD {n}
-od∅ {n} = Ord o∅
+od∅ : OD
+od∅  = 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
+  od→ord : OD  → Ordinal
+  ord→od : Ordinal  → OD
+  c<→o<  :  {x y : OD  }   → def y ( od→ord x ) → od→ord x o< od→ord y
+  oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
+  diso   :  {x : Ordinal } → 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
+  ==→o≡ : { x y : OD  } → (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
+  --   o<→c<  : {n : Level} {x y : Ordinal  } → 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
+  sup-o  :  ( Ordinal  → Ordinal ) →  Ordinal
+  sup-o< :  { ψ : Ordinal  →  Ordinal } → ∀ {x : Ordinal } →  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-x  : {n : Level } → ( Ordinal  → Ordinal ) →  Ordinal
+  -- sup-lb : {n : Level } → { ψ : Ordinal  →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
-- mimimul and x∋minimul is an Axiom of choice
-  minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n}
+  minimul : (x : OD  ) → ¬ (x == od∅ )→ OD
-- 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 ) )
+  x∋minimul : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
-- minimulity (may proved by ε-induction )
-  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) )
+  minimul-1 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( 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 )
+_∋_ : ( a x : OD  ) → Set n
+_∋_  a x  = def a ( od→ord x )

-_c<_ : { n : Level } → ( x a : OD {n} ) → Set n
+_c<_ : ( x a : OD  ) → Set n
x c< a = a ∋ x

-_c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
+_c≤_ : OD  →  OD  → Set (suc n)
a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )

-cseq : {n : Level} →  OD {n} →  OD {n}
+cseq : {n : Level} →  OD  →  OD
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 :  {Z : OD } {X : Ordinal  }{z : OD } {x : Ordinal  }→ 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 : ( OD  → OD ) →  OD
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 )}
+sup-c< : ( ψ : OD  →  OD ) → ∀ {x : OD } → def ( sup-od ψ ) (od→ord ( ψ x ))
+sup-c<  ψ {x} = def-subst  {_} {_} {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 y : Ordinal {n} } → def (Ord a) x → def (Ord x) y → def (Ord a) y
+otrans : {n : Level} {a x y : Ordinal  } → def (Ord a) x → def (Ord x) y → def (Ord a) y
otrans x<a y<x = ordtrans y<x x<a

-def→o< : {n : Level } {X : OD {suc n}} → {x : Ordinal {suc n}} → def X x → x o< od→ord X
-def→o< {n} {X} {x} lt = o<-subst {suc n} {_} {_} {x} {od→ord X} ( c<→o< ( def-subst {suc n} {X} {x}  lt (sym oiso) (sym diso) )) diso diso
-
-∅3 : {n : Level} →  { x : Ordinal {suc n}}  → ( ∀(y : Ordinal {suc n}) → ¬ (y o< x ) ) → x ≡ o∅ {suc n}
-∅3 {n} {x} = TransFinite {n} c2 c3 x where
-   c0 : Nat →  Ordinal {suc n}  → Set (suc n)
-   c0 lx x = (∀(y : Ordinal {suc n}) → ¬ (y o< x))  → x ≡ o∅ {suc n}
-   c2 : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → c0 (lv x₁) (record { lv = lv x₁ ; ord = ord x₁ }))→ 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
+def→o< :  {X : OD } → {x : Ordinal } → def X x → x o< od→ord X
+def→o<  {X} {x} lt = o<-subst  {_} {_} {x} {od→ord X} ( c<→o< ( def-subst  {X} {x}  lt (sym oiso) (sym diso) )) diso diso

-- avoiding lv != Zero error
-orefl : {n : Level} →  { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
+orefl : { x : OD  } → { y : Ordinal  } → 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 {
+==-iso : { x y : OD  } → ord→od (od→ord x) == ord→od (od→ord y)  →  x == y
+==-iso  {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 : OD  } {z : Ordinal } → 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)
+=-iso :  {x y : OD  } → (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)
+ord→== : { x y : OD  } → od→ord x ≡  od→ord y →  x == y
+ord→==  {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
+   lemma : ( ox oy : Ordinal  ) → 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
+o≡→== : { x y : Ordinal  } → x ≡  y →  ord→od x == ord→od y
+o≡→==  {x} {.x} refl = eq-refl

-c≤-refl : {n : Level} →  ( x : OD {n} ) → x c≤ x
+c≤-refl : {n : Level} →  ( x : OD  ) → x c≤ x
c≤-refl x = case1 refl

-o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
-o∅≡od∅ {n} = ==→o≡ lemma where
+o∅≡od∅ : ord→od (o∅ ) ≡ od∅
+o∅≡od∅  = ==→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
+     lemma0 {x} lt = o<-subst (c<→o<  {ord→od x} {ord→od o∅} (def-subst  {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 ())
+     lemma1 {x} lt = ⊥-elim (¬x<0 lt)
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∅ ) )
+ord-od∅ : od→ord (od∅ ) ≡ o∅
+ord-od∅  = sym ( subst (λ k → k ≡  od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )

-∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ } == od∅ {n}
+∅0 : record { def = λ x →  Lift n ⊥ } == od∅
eq→ ∅0 {w} (lift ())
-eq← ∅0 {w} (case1 ())
-eq← ∅0 {w} (case2 ())
+eq← ∅0 {w} lt = lift (¬x<0 lt)

-∅< : {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 ()
+∅< : { x y : OD  } → def x (od→ord y ) → ¬ (  x  == od∅  )
+∅<  {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
+∅<  {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 )
+∅6 : { x : OD  }  → ¬ ( x ∋ x )    --  no Russel paradox
+∅6  {x} x∋x = o<¬≡ refl ( c<→o<  {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 : {A B : OD } {x y : Ordinal } → 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())
+is-o∅ : ( x : Ordinal  ) → Dec ( x ≡ o∅  )
+is-o∅ x with trio< x o∅
+is-o∅ x | tri< a ¬b ¬c = no ¬b
+is-o∅ x | tri≈ ¬a b ¬c = yes b
+is-o∅ x | tri> ¬a ¬b c = no ¬b

-ppp : { n : Level } → { p : Set (suc n) } { a : OD {suc n} } → record { def = λ x → p } ∋ a → p
-ppp {n} {p} {a} d = d
+ppp :  { p : Set n } { a : OD  } → record { def = λ x → p } ∋ a → p
+ppp  {p} {a} d = d

--
-- Axiom of choice in intutionistic logic implies the exclude middle
--     https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
--
-p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p )         -- assuming axiom of choice
-p∨¬p {n} p with is-o∅ ( od→ord ( record { def = λ x → p } ))
-p∨¬p {n} p | yes eq = case2 (¬p eq) where
+p∨¬p : ( p : Set n ) → p ∨ ( ¬ p )         -- assuming axiom of choice
+p∨¬p  p with is-o∅ ( od→ord ( record { def = λ x → p } ))
+p∨¬p  p | yes eq = case2 (¬p eq) where
ps = record { def = λ x → p }
lemma : ps == od∅ → p → ⊥
-   lemma eq p0 = ¬x<0 {n} {od→ord ps} (eq→ eq p0 )
+   lemma eq p0 = ¬x<0  {od→ord ps} (eq→ eq p0 )
¬p : (od→ord ps ≡ o∅) → p → ⊥
¬p eq = lemma ( subst₂ (λ j k → j ==  k ) oiso o∅≡od∅ ( o≡→== eq ))
-p∨¬p {n} p | no ¬p = case1 (ppp {n} {p} {minimul ps (λ eq →  ¬p (eqo∅ eq))} lemma) where
+p∨¬p  p | no ¬p = case1 (ppp  {p} {minimul ps (λ eq →  ¬p (eqo∅ eq))} lemma) where
ps = record { def = λ x → p }
-   eqo∅ : ps ==  od∅ {suc n} → od→ord ps ≡  o∅ {suc n}
+   eqo∅ : ps ==  od∅  → od→ord ps ≡  o∅
eqo∅ eq = subst (λ k → od→ord ps ≡ k) ord-od∅ ( cong (λ k → od→ord k ) (==→o≡ eq))
lemma : ps ∋ minimul ps (λ eq →  ¬p (eqo∅ eq))
lemma = x∋minimul ps (λ eq →  ¬p (eqo∅ eq))

-∋-p : { n : Level } → ( p : Set (suc n) ) → Dec p   -- assuming axiom of choice
-∋-p {n} p with p∨¬p p
-∋-p {n} p | case1 x = yes x
-∋-p {n} p | case2 x = no x
+∋-p : ( p : Set n ) → Dec p   -- assuming axiom of choice
+∋-p  p with p∨¬p p
+∋-p  p | case1 x = yes x
+∋-p  p | case2 x = no x

-double-neg-eilm : {n  : Level } {A : Set (suc n)} → ¬ ¬ A → A      -- we don't have this in intutionistic logic
-double-neg-eilm {n} {A} notnot with ∋-p  A                         -- assuming axiom of choice
+double-neg-eilm : {A : Set n} → ¬ ¬ A → A      -- we don't have this in intutionistic logic
+double-neg-eilm  {A} notnot with ∋-p  A                         -- assuming axiom of choice
... | yes p = p
... | no ¬p = ⊥-elim ( notnot ¬p )

-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
+OrdP : ( x : Ordinal  ) ( y : OD  ) → Dec ( Ord x ∋ y )
+OrdP  x y with trio< x (od→ord y)
+OrdP  x y | tri< a ¬b ¬c = no ¬c
+OrdP  x y | tri≈ ¬a refl ¬c = no ( o<¬≡ refl )
+OrdP  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))
+-- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality n (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 )))))  }
+in-codomain : (X : OD  ) → ( ψ : OD  → OD  ) → OD
+in-codomain  X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( 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 : OD  ) → OD
ZFSubset A x =  record { def = λ y → def A y ∧  def x y }  --   roughly x = A → Set

-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 )) ) )
+Def :  (A :  OD ) → OD
+Def  A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )

-_⊆_ : {n : Level} ( A B : OD {suc n}  ) → ∀{ x : OD {suc n} } →  Set (suc n)
+_⊆_ :  ( A B : OD   ) → ∀{ x : OD  } →  Set n
_⊆_ A B {x} = A ∋ x →  B ∋ x

infixr  220 _⊆_

-subset-lemma : {n : Level} → {A x y : OD {suc n} } → (  x ∋ y → ZFSubset A x ∋  y ) ⇔  ( _⊆_ x A {y} )
-subset-lemma {n} {A} {x} {y} = record {
+subset-lemma : {A x y : OD  } → (  x ∋ y → ZFSubset A x ∋  y ) ⇔  ( _⊆_ x A {y} )
+subset-lemma  {A} {x} {y} = record {
proj1 = λ z lt → proj1 (z  lt)
; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt }
}
@@ -265,18 +240,18 @@
-- 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 }))))
+-- L : {n : Level} → (α : Ordinal ) → OD
+-- L   record { lv = Zero ; ord = (Φ .0) } = od∅
+-- L   record { lv = lx ; ord = (OSuc lv ox) } = Def ( L  ( record { lv = lx ; ord = ox } ) )
+-- L   record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
+--     cseq ( Ord (od→ord (L   (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
+-- L0 :  {n : Level} → (α : Ordinal ) → L (osuc α) ∋ L α
+-- L1 :  {n : Level} → (α β : Ordinal ) → α o< β → ∀ (x : OD )  → 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→ZF : ZF
+OD→ZF   = record {
+    ZFSet = OD
; _∋_ = _∋_
; _≈_ = _==_
; ∅  = od∅
@@ -288,35 +263,35 @@
; infinite = infinite
; isZF = isZF
} where
-    ZFSet = OD {suc n}
-    Select : (X : OD {suc n} ) → ((x : OD {suc n} ) → Set (suc n) ) → OD {suc n}
+    ZFSet = OD
+    Select : (X : OD  ) → ((x : OD  ) → Set n ) → OD
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 : OD  → (OD  → OD  ) → OD
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}
+    _,_ : OD  → OD  → OD
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 : OD  → OD
Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x)))  }
-    _∈_ : ( A B : ZFSet  ) → Set (suc n)
+    _∈_ : ( A B : ZFSet  ) → Set n
A ∈ B = B ∋ A
-    Power : OD {suc n} → OD {suc n}
+    Power : OD  → OD
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
+    data infinite-d  : ( x : Ordinal  ) → Set n where
iφ :  infinite-d o∅
-        isuc : {x : Ordinal {suc n} } →   infinite-d  x  →
+        isuc : {x : Ordinal  } →   infinite-d  x  →
infinite-d  (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))

-    infinite : OD {suc n}
+    infinite : OD
infinite = record { def = λ x → infinite-d x }

infixr  200 _∈_
-- infixr  230 _∩_ _∪_
-    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
+    isZF : IsZF (OD )  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
isZF = record {
isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
;   pair  = pair
@@ -326,7 +301,7 @@
;   power→ = power→
;   power← = power←
;   extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
-       ;   ε-induction = ε-induction
+       -- ;   ε-induction = {!!}
;   infinity∅ = infinity∅
;   infinity = infinity
;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
@@ -336,18 +311,17 @@
;   choice = choice
} 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)
+         pair : (A B : OD  ) → ((A , B) ∋ A) ∧  ((A , B) ∋ B)
+         proj1 (pair A B ) = omax-x  (od→ord A) (od→ord B)
+         proj2 (pair A B ) = omax-y  (od→ord A) (od→ord B)

-         empty : {n : Level } (x : OD {suc n} ) → ¬  (od∅ ∋ x)
-         empty x (case1 ())
-         empty x (case2 ())
+         empty : (x : OD  ) → ¬  (od∅ ∋ x)
+         empty x = ¬x<0

-         o<→c< :  {x y : Ordinal {suc n}} {z : OD {suc n}}→ x o< y → _⊆_  (Ord x) (Ord y) {z}
+         o<→c< :  {x y : Ordinal } {z : OD }→ 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 : Ordinal } → (∀ (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
@@ -362,9 +336,9 @@
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 : OD  → Set n} {x y : OD } → 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 : OD → Set 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  }
@@ -392,26 +366,26 @@
--- 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 : OD  } → ({x : OD  } → (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  } ;
+                 proj1 = def-subst  {_} {_} {b} {x} (inc (def-subst  {_} {_} {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}
+         ord-power← a t t→A  = def-subst  {_} {_} {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}}
+                 proj1 = def-subst  {_} {_} {(Ord a)} {z}
+                    ( t→A (def-subst  {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
+              lemma1 :  {a : Ordinal } { t : OD }
→ (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 ))
+              lemma1  {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<

@@ -442,7 +416,7 @@
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}
+              lt1 = sup-o<  {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)
@@ -453,7 +427,7 @@
∎
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})
+                  lemma4 (sup-o<  {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))
@@ -467,21 +441,21 @@
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)
+                 lemma3 = record { proj1 = def-subst  {_} {_} {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) ))
+             lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst  {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))

-         extensionality0 : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
-         eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
-         eq← (extensionality0 {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d
+         extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
+         eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso  {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
+         eq← (extensionality0 {A} {B} eq ) {x} d = def-iso  {B} {A} (sym diso) (proj2 (eq (ord→od x))) d

-         extensionality : {A B w : OD {suc n} } → ((z : OD {suc n}) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
+         extensionality : {A B w : OD  } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d

-         infinity∅ : infinite  ∋ od∅ {suc n}
-         infinity∅ = def-subst {suc n} {_} {_} {infinite} {od→ord (od∅ {suc n})} iφ refl lemma where
+         infinity∅ : infinite  ∋ od∅
+         infinity∅ = def-subst  {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
lemma : o∅ ≡ od→ord od∅
lemma =  let open ≡-Reasoning in begin
o∅
@@ -491,134 +465,15 @@
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
+         infinity x lt = def-subst  {_} {_} {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 : OD  ) → {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 : OD  ) → {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 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 :  (z : OD) → ord→od record { lv = ly ; ord = oy } ∋ z → od→ord z o< record { lv = lx ; ord = ox }
-                    lemma z lt with osuc-≡< y<x
-                    lemma z lt | case1 refl = o<-subst (c<→o< lt) refl diso
-                    lemma z 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)
-
-         ---
-         --- With assuption of OD is ordered,  p ∨ ( ¬ p ) <=> axiom of choice
-         ---
-         record choiced {n : Level} ( X : OD {suc n}) : Set (suc (suc n)) where
-            field
-                a-choice : OD {suc n}
-                is-in : X ∋ a-choice
-         choice-func' : (X : OD {suc n} ) → (p∨¬p : { n : Level } → ( p : Set (suc n) ) → p ∨ ( ¬ p )) → ¬ ( X == od∅ ) → choiced X
-         choice-func' X p∨¬p not = have_to_find
-           where
-            ψ : ( ox : Ordinal {suc n}) → Set (suc (suc n))
-            ψ ox = (( x : Ordinal {suc n}) → x o< ox  → ( ¬ def X x )) ∨ choiced X
-            lemma-ord : ( ox : Ordinal {suc n} ) → ψ ox
-            lemma-ord  ox = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc ox where
-               ∋-p' : (A x : OD {suc n} ) → Dec ( A ∋ x )
-               ∋-p' A x with p∨¬p ( A ∋ x )
-               ∋-p' A x | case1 t = yes t
-               ∋-p' A x | case2 t = no t
-               ∀-imply-or :  {n : Level}  {A : Ordinal {suc n} → Set (suc n) } {B : Set (suc (suc n)) }
-                   → ((x : Ordinal {suc n}) → A x ∨ B) →  ((x : Ordinal {suc n}) → A x) ∨ B
-               ∀-imply-or {n} {A} {B} ∀AB with p∨¬p  ((x : Ordinal {suc n}) → A x)
-               ∀-imply-or {n} {A} {B} ∀AB | case1 t = case1 t
-               ∀-imply-or {n} {A} {B} ∀AB | case2 x = case2 (lemma x) where
-                    lemma : ¬ ((x : Ordinal {suc n}) → A x) →  B
-                    lemma not with p∨¬p B
-                    lemma not | case1 b = b
-                    lemma not | case2 ¬b = ⊥-elim  (not (λ x → dont-orb (∀AB x) ¬b ))
-               caseΦ : (lx : Nat) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx) → ψ x) → ψ (ordinal lx (Φ lx) )
-               caseΦ lx prev with ∋-p' X ( ord→od (ordinal lx (Φ lx) ))
-               caseΦ lx prev | yes p = case2 ( record { a-choice = ord→od (ordinal lx (Φ lx)) ; is-in = p } )
-               caseΦ lx prev | no ¬p = lemma where
-                    lemma1 : (x : Ordinal) → (((Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X)
-                    lemma1 x with trio< x (ordinal lx (Φ lx))
-                    lemma1 x | tri< a ¬b ¬c with prev (osuc x) (lemma2 a) where
-                        lemma2 : x o< (ordinal lx (Φ lx)) →  osuc x o< ordinal lx (Φ lx)
-                        lemma2 (case1 lt) = case1 lt
-                    lemma1 x | tri< a ¬b ¬c | case2 found = case2 found
-                    lemma1 x | tri< a ¬b ¬c | case1 not_found = case1 ( λ lt df → not_found x <-osuc df )
-                    lemma1 x | tri≈ ¬a refl ¬c = case1 ( λ lt → ⊥-elim (o<¬≡ refl lt ))
-                    lemma1 x | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim (o<> lt c ))
-                    lemma : ((x : Ordinal) → (Suc (lv x) ≤ lx) ∨ (ord x d< Φ lx) → def X x → ⊥) ∨ choiced X
-                    lemma = ∀-imply-or lemma1
-               caseOSuc : (lx : Nat) (x : OrdinalD lx) → ψ ( ordinal lx x ) → ψ ( ordinal lx (OSuc lx x) )
-               caseOSuc lx x prev with ∋-p' X (ord→od record { lv = lx ; ord = x } )
-               caseOSuc lx x prev | yes p = case2 (record { a-choice = ord→od record { lv = lx ; ord = x } ; is-in = p })
-               caseOSuc lx x (case1 not_found) | no ¬p = case1 lemma where
-                    lemma : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx x) → def X y → ⊥
-                    lemma y lt with trio< y (ordinal lx x )
-                    lemma y lt | tri< a ¬b ¬c = not_found y a
-                    lemma y lt | tri≈ ¬a refl ¬c = subst (λ k → def X k → ⊥ ) diso ¬p
-                    lemma y lt | tri> ¬a ¬b c with osuc-≡< lt
-                    lemma y lt | tri> ¬a ¬b c | case1 refl = ⊥-elim ( ¬b refl )
-                    lemma y lt | tri> ¬a ¬b c | case2 lt1 = ⊥-elim (o<> c lt1 )
-               caseOSuc lx x (case2 found) | no ¬p = case2 found
-            have_to_find : choiced X
-            have_to_find with lemma-ord (od→ord X )
-            have_to_find | t = dont-or  t ¬¬X∋x where
-                ¬¬X∋x : ¬ ((x : Ordinal) → (Suc (lv x) ≤ lv (od→ord X)) ∨ (ord x d< ord (od→ord X)) → def X x → ⊥)
-                ¬¬X∋x nn = not record {
-                       eq→ = λ {x} lt → ⊥-elim  (nn x (def→o< lt) lt)
-                     ; eq← = λ {x} lt → ⊥-elim ( ¬x<0 lt )
-                   }
-```
```--- a/zf.agda	Fri Aug 09 16:54:30 2019 +0900
+++ b/zf.agda	Fri Aug 09 17:57:58 2019 +0900
@@ -52,9 +52,9 @@
-- minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet
-- regularity : ∀( x : ZFSet  ) → (not : ¬ (x ≈ ∅)) → (  minimul x not  ∈ x ∧  (  minimul x not  ∩ x  ≈ ∅ ) )
-- another form of regularity
-     ε-induction : { ψ : ZFSet → Set m}
-             → ( {x : ZFSet } → ({ y : ZFSet } →  x ∋ y → ψ y ) → ψ x )
-             → (x : ZFSet ) → ψ x
+     -- ε-induction : { ψ : ZFSet → Set m}
+     --         → ( {x : ZFSet } → ({ y : ZFSet } →  x ∋ y → ψ y ) → ψ x )
+     --         → (x : ZFSet ) → ψ x
-- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
infinity∅ :  ∅ ∈ infinite
infinity :  ∀( x : ZFSet  ) → x ∈ infinite →  ( x ∪ ｛ x ｝) ∈ infinite ```