# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1565568256 -32400
# Node ID fe8392f527eb3954f5522262128a0393a240ffda
# Parent  6bb5d57c9561b336b32c6b2b6afed29f50c63fae# Parent  cb6f025a991ea700ad16ea70ab516eaa29554d1f
axiomaized ordinals. filter and cardinal are incomplete

diff -r 6bb5d57c9561 -r fe8392f527eb .hgtags
--- a/.hgtags	Thu Aug 01 12:24:26 2019 +0900
+++ b/.hgtags	Mon Aug 12 09:04:16 2019 +0900
@@ -9,3 +9,7 @@
 b06f5d2f34b1a16ff39aae15680a1c0d640e6b93 current
 b06f5d2f34b1a16ff39aae15680a1c0d640e6b93 current
 ecb329ba38ac904913313f2dd03ae2329039ffa6 current
+ecb329ba38ac904913313f2dd03ae2329039ffa6 current
+2c7d45734e3be59a06d272a07fecdbf77ab8ce10 current
+2c7d45734e3be59a06d272a07fecdbf77ab8ce10 current
+1b1620e2053cfc340a4df0d63de65b9059b19b6f current
diff -r 6bb5d57c9561 -r fe8392f527eb OD.agda
--- a/OD.agda	Thu Aug 01 12:24:26 2019 +0900
+++ b/OD.agda	Mon Aug 12 09:04:16 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 
@@ -11,272 +11,247 @@
 open import Relation.Binary
 open import Relation.Binary.Core
 
+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
 
->→¬< : {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 : {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 }
    } 
 
-
 -- 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 }))))
+-- 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 +465,22 @@
                     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 )
-                   }
-
+_,_ = ZF._,_ OD→ZF
+Union = ZF.Union OD→ZF
+Power = ZF.Power OD→ZF
+Select = ZF.Select OD→ZF
+Replace = ZF.Replace OD→ZF
+isZF = ZF.isZF  OD→ZF
+TransFinite = IsOrdinals.TransFinite (Ordinals.isOrdinal O)
diff -r 6bb5d57c9561 -r fe8392f527eb Ordinals.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Ordinals.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -0,0 +1,200 @@
+open import Level
+module Ordinals where
+
+open import zf
+
+open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import Data.Empty
+open import  Relation.Binary.PropositionalEquality
+open import  logic
+open import  nat
+open import Data.Unit using ( ⊤ )
+open import Relation.Nullary
+open import Relation.Binary
+open import Relation.Binary.Core
+
+record IsOrdinals {n : Level} (ord : Set n)  (o∅ : ord ) (osuc : ord → ord )  (_o<_ : ord → ord → Set n) : Set (suc (suc n)) where
+   field
+     Otrans :  {x y z : ord }  → x o< y → y o< z → x o< z
+     OTri : Trichotomous {n} _≡_  _o<_ 
+     ¬x<0 :   { x  : ord  } → ¬ ( x o< o∅  )
+     <-osuc :  { x : ord  } → x o< osuc x
+     osuc-≡< :  { a x : ord  } → x o< osuc a  →  (x ≡ a ) ∨ (x o< a)  
+     TransFinite : { ψ : ord  → Set (suc n) }
+          → ( (x : ord)  → ( (y : ord  ) → y o< x → ψ y ) → ψ x )
+          →  ∀ (x : ord)  → ψ x
+
+
+record Ordinals {n : Level} : Set (suc (suc n)) where
+   field
+     ord : Set n
+     o∅ : ord
+     osuc : ord → ord
+     _o<_ : ord → ord → Set n
+     isOrdinal : IsOrdinals ord o∅ osuc _o<_
+
+module inOrdinal  {n : Level} (O : Ordinals {n} ) where
+
+        Ordinal : Set n
+        Ordinal  = Ordinals.ord O 
+
+        _o<_ :  Ordinal  → Ordinal  → Set n
+        _o<_ = Ordinals._o<_ O 
+
+        osuc :   Ordinal  → Ordinal 
+        osuc  = Ordinals.osuc O 
+
+        o∅ :   Ordinal  
+        o∅ = Ordinals.o∅ O
+
+        ¬x<0 = IsOrdinals.¬x<0 (Ordinals.isOrdinal O)
+        osuc-≡< = IsOrdinals.osuc-≡<  (Ordinals.isOrdinal O)
+        <-osuc = IsOrdinals.<-osuc  (Ordinals.isOrdinal O)
+        
+        o<-dom :   { x y : Ordinal } → x o< y → Ordinal 
+        o<-dom  {x} _ = x
+
+        o<-cod :   { x y : Ordinal } → x o< y → Ordinal 
+        o<-cod  {_} {y} _ = y
+
+        o<-subst : {Z X z x : Ordinal }  → Z o< X → Z ≡ z  →  X ≡ x  →  z o< x
+        o<-subst df refl refl = df
+
+        ordtrans :  {x y z : Ordinal  }   → x o< y → y o< z → x o< z
+        ordtrans = IsOrdinals.Otrans (Ordinals.isOrdinal O)
+
+        trio< : Trichotomous  _≡_  _o<_ 
+        trio< = IsOrdinals.OTri (Ordinals.isOrdinal O)
+
+        o<¬≡ :  { ox oy : Ordinal } → ox ≡ oy  → ox o< oy  → ⊥
+        o<¬≡ {ox} {oy} eq lt with trio< ox oy
+        o<¬≡ {ox} {oy} eq lt | tri< a ¬b ¬c = ¬b eq
+        o<¬≡ {ox} {oy} eq lt | tri≈ ¬a b ¬c = ¬a lt
+        o<¬≡ {ox} {oy} eq lt | tri> ¬a ¬b c = ¬b eq
+
+        o<> :   {x y : Ordinal   }  →  y o< x → x o< y → ⊥
+        o<> {ox} {oy} lt tl with trio< ox oy
+        o<> {ox} {oy} lt tl | tri< a ¬b ¬c = ¬c lt
+        o<> {ox} {oy} lt tl | tri≈ ¬a b ¬c = ¬a tl
+        o<> {ox} {oy} lt tl | tri> ¬a ¬b c = ¬a tl
+
+        osuc-< :  { x y : Ordinal  } → y o< osuc x  → x o< y → ⊥
+        osuc-< {x} {y} y<ox x<y with osuc-≡< y<ox
+        osuc-< {x} {y} y<ox x<y | case1 refl = o<¬≡ refl x<y
+        osuc-< {x} {y} y<ox x<y | case2 y<x = o<> x<y y<x
+
+        osucc :  {ox oy : Ordinal } → oy o< ox  → osuc oy o< osuc ox  
+        ----   y < osuc y < x < osuc x
+        ----   y < osuc y = x < osuc x
+        ----   y < osuc y > x < osuc x   -> y = x ∨ x < y → ⊥
+        osucc {ox} {oy} oy<ox with trio< (osuc oy) ox
+        osucc {ox} {oy} oy<ox | tri< a ¬b ¬c = ordtrans a <-osuc
+        osucc {ox} {oy} oy<ox | tri≈ ¬a refl ¬c = <-osuc
+        osucc {ox} {oy} oy<ox | tri> ¬a ¬b c with  osuc-≡< c
+        osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case1 eq = ⊥-elim (o<¬≡ (sym eq) oy<ox)
+        osucc {ox} {oy} oy<ox | tri> ¬a ¬b c | case2 lt = ⊥-elim (o<> lt oy<ox)
+
+        open _∧_
+
+        osuc2 :  ( x y : Ordinal  ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
+        proj2 (osuc2 x y) lt = osucc lt
+        proj1 (osuc2 x y) ox<ooy with osuc-≡< ox<ooy
+        proj1 (osuc2 x y) ox<ooy | case1 ox=oy = o<-subst <-osuc refl ox=oy
+        proj1 (osuc2 x y) ox<ooy | case2 ox<oy = ordtrans <-osuc ox<oy 
+
+        _o≤_ :  Ordinal → Ordinal → Set  n
+        a o≤ b  = (a ≡ b)  ∨ ( a o< b )
+
+
+        xo<ab :  {oa ob : Ordinal } → ( {ox : Ordinal } → ox o< oa  → ox o< ob ) → oa o< osuc ob
+        xo<ab   {oa} {ob} a→b with trio< oa ob
+        xo<ab   {oa} {ob} a→b | tri< a ¬b ¬c = ordtrans a <-osuc
+        xo<ab   {oa} {ob} a→b | tri≈ ¬a refl ¬c = <-osuc
+        xo<ab   {oa} {ob} a→b | tri> ¬a ¬b c = ⊥-elim ( o<¬≡ refl (a→b c )  )
+
+        maxα :   Ordinal  →  Ordinal  → Ordinal
+        maxα x y with trio< x y
+        maxα x y | tri< a ¬b ¬c = y
+        maxα x y | tri> ¬a ¬b c = x
+        maxα x y | tri≈ ¬a refl ¬c = x
+
+        minα :    Ordinal  →  Ordinal  → Ordinal
+        minα  x y with trio<  x  y
+        minα x y | tri< a ¬b ¬c = x
+        minα x y | tri> ¬a ¬b c = y
+        minα x y | tri≈ ¬a refl ¬c = x
+
+        min1 :   {x y z : Ordinal  } → z o< x → z o< y → z o< minα x y
+        min1  {x} {y} {z} z<x z<y with trio<  x y
+        min1  {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x
+        min1  {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x
+        min1  {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y
+
+        --
+        --  max ( osuc x , osuc y )
+        --
+
+        omax :  ( x y : Ordinal  ) → Ordinal 
+        omax  x y with trio< x y
+        omax  x y | tri< a ¬b ¬c = osuc y
+        omax  x y | tri> ¬a ¬b c = osuc x
+        omax  x y | tri≈ ¬a refl ¬c  = osuc x
+
+        omax< :  ( x y : Ordinal  ) → x o< y → osuc y ≡ omax x y
+        omax<  x y lt with trio< x y
+        omax<  x y lt | tri< a ¬b ¬c = refl
+        omax<  x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
+        omax<  x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
+
+        omax≡ :  ( x y : Ordinal  ) → x ≡ y → osuc y ≡ omax x y
+        omax≡  x y eq with trio< x y
+        omax≡  x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
+        omax≡  x y eq | tri≈ ¬a refl ¬c = refl
+        omax≡  x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
+
+        omax-x :  ( x y : Ordinal  ) → x o< omax x y
+        omax-x  x y with trio< x y
+        omax-x  x y | tri< a ¬b ¬c = ordtrans a <-osuc
+        omax-x  x y | tri> ¬a ¬b c = <-osuc
+        omax-x  x y | tri≈ ¬a refl ¬c = <-osuc
+
+        omax-y :  ( x y : Ordinal  ) → y o< omax x y
+        omax-y  x y with  trio< x y
+        omax-y  x y | tri< a ¬b ¬c = <-osuc
+        omax-y  x y | tri> ¬a ¬b c = ordtrans c <-osuc
+        omax-y  x y | tri≈ ¬a refl ¬c = <-osuc
+
+        omxx :  ( x : Ordinal  ) → omax x x ≡ osuc x
+        omxx  x with  trio< x x
+        omxx  x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
+        omxx  x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
+        omxx  x | tri≈ ¬a refl ¬c = refl
+
+        omxxx :  ( x : Ordinal  ) → omax x (omax x x ) ≡ osuc (osuc x)
+        omxxx  x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
+
+        open _∧_
+
+        OrdTrans :  Transitive  _o≤_
+        OrdTrans (case1 refl) (case1 refl) = case1 refl
+        OrdTrans (case1 refl) (case2 lt2) = case2 lt2
+        OrdTrans (case2 lt1) (case1 refl) = case2 lt1
+        OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y)
+
+        OrdPreorder :   Preorder n n n
+        OrdPreorder  = record { Carrier = Ordinal
+           ; _≈_  = _≡_ 
+           ; _∼_   = _o≤_
+           ; isPreorder   = record {
+                isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
+                ; reflexive     = case1 
+                ; trans         = OrdTrans 
+             }
+         }
+
+        TransFiniteExists : {m l : Level} → ( ψ : Ordinal  → Set m ) 
+          → {p : Set l} ( P : { y : Ordinal  } →  ψ y → ¬ p )
+          → (exists : ¬ (∀ y → ¬ ( ψ y ) ))
+          → ¬ p
+        TransFiniteExists  {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) 
+
diff -r 6bb5d57c9561 -r fe8392f527eb cardinal.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/cardinal.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -0,0 +1,140 @@
+open import Level
+open import Ordinals
+module cardinal {n : Level } (O : Ordinals {n}) where
+
+open import zf
+open import logic
+import OD 
+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
+
+open inOrdinal O
+open OD O
+open OD.OD
+
+open _∧_
+open _∨_
+open Bool
+
+-- we have to work on Ordinal to keep OD Level n
+-- since we use p∨¬p which works only on Level n
+
+func→od : (f : Ordinal → Ordinal ) → ( dom : OD ) → OD 
+func→od f dom = Replace dom ( λ x →  x , (ord→od (f (od→ord x) )))
+
+record _⊗_  (A B : Ordinal) : Set n where
+   field
+      π1 : Ordinal
+      π2 : Ordinal
+      A∋π1 : def (ord→od A)  π1
+      B∋π2 : def (ord→od B)  π2
+
+-- Clearly wrong. We need ordered pair
+Func :  ( A B : OD ) → OD
+Func A B = record { def = λ x → (od→ord A) ⊗ (od→ord B) }
+
+open  _⊗_
+
+func←od : { dom cod : OD } → (f : OD )  → Func dom cod ∋ f → (Ordinal → Ordinal )
+func←od {dom} {cod} f lt x = sup-o ( λ y → lemma  y ) where
+   lemma : Ordinal → Ordinal
+   lemma y with p∨¬p ( _⊗_.π1 lt ≡ x )
+   lemma y | case1 refl = _⊗_.π2 lt
+   lemma y | case2 not = o∅
+
+-- contra position of sup-o<
+--
+
+postulate
+  -- contra-position of mimimulity of supermum required in Cardinal
+  sup-x  : ( Ordinal  → Ordinal ) →  Ordinal 
+  sup-lb : { ψ : Ordinal  →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
+
+------------
+--
+-- Onto map
+--          def X x ->  xmap
+--     X ---------------------------> Y
+--          ymap   <-  def Y y
+--
+record Onto  (X Y : OD )  : Set n where
+   field
+       xmap : Ordinal 
+       ymap : Ordinal 
+       xfunc : def (Func X Y) xmap 
+       yfunc : def (Func Y X) ymap 
+       onto-iso   : {y :  Ordinal  } → (lty : def Y y ) → func←od (ord→od xmap) xfunc ( func←od (ord→od ymap) yfunc y )  ≡ y
+
+open Onto
+
+onto-restrict : {X Y Z : OD} → Onto X Y → ({x : OD} → _⊆_ Z Y {x}) → Onto X Z
+onto-restrict {X} {Y} {Z} onto  Z⊆Y = record {
+     xmap = xmap1
+   ; ymap = zmap
+   ; xfunc = xfunc1
+   ; yfunc = zfunc
+   ; onto-iso = onto-iso1
+  } where
+       xmap1 : Ordinal 
+       xmap1 = od→ord (Select (ord→od (xmap onto)) {!!} ) 
+       zmap : Ordinal 
+       zmap = {!!}
+       xfunc1 : def (Func X Z) xmap1
+       xfunc1 = {!!}
+       zfunc : def (Func Z X) zmap 
+       zfunc = {!!}
+       onto-iso1   : {z :  Ordinal  } → (ltz : def Z z ) → func←od (ord→od xmap1) xfunc1 ( func←od (ord→od zmap) zfunc z )  ≡ z
+       onto-iso1   = {!!}
+
+
+record Cardinal  (X  : OD ) : Set n where
+   field
+       cardinal : Ordinal 
+       conto : Onto X (Ord cardinal)  
+       cmax : ( y : Ordinal  ) → cardinal o< y → ¬ Onto X (Ord y)  
+
+cardinal :  (X  : OD ) → Cardinal X
+cardinal  X = record {
+       cardinal = sup-o ( λ x → proj1 ( cardinal-p x) )
+     ; conto = onto
+     ; cmax = cmax
+   } where
+    cardinal-p : (x  : Ordinal ) →  ( Ordinal  ∧ Dec (Onto X (Ord x) ) )
+    cardinal-p x with p∨¬p ( Onto X (Ord x)  ) 
+    cardinal-p x | case1 True  = record { proj1 = x  ; proj2 = yes True }
+    cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False }
+    S = sup-o (λ x → proj1 (cardinal-p x))
+    lemma1 :  (x : Ordinal) → ((y : Ordinal) → y o< x → Lift (suc n) (y o< (osuc S) → Onto X (Ord y))) →
+                    Lift (suc n) (x o< (osuc S) → Onto X (Ord x) )
+    lemma1 x prev with trio< x (osuc S)
+    lemma1 x prev | tri< a ¬b ¬c with osuc-≡< a
+    lemma1 x prev | tri< a ¬b ¬c | case1 x=S = lift ( λ lt → {!!} )
+    lemma1 x prev | tri< a ¬b ¬c | case2 x<S = lift ( λ lt → lemma2 ) where
+         lemma2 : Onto X (Ord x) 
+         lemma2 with prev {!!} {!!}
+         ... | lift t = t {!!}
+    lemma1 x prev | tri≈ ¬a b ¬c = lift ( λ lt → ⊥-elim ( o<¬≡ b lt ))
+    lemma1 x prev | tri> ¬a ¬b c = lift ( λ lt → ⊥-elim ( o<> c lt ))
+    onto : Onto X (Ord S) 
+    onto with TransFinite {λ x → Lift (suc n) ( x o< osuc S → Onto X (Ord x) ) } lemma1 S 
+    ... | lift t = t <-osuc  
+    cmax : (y : Ordinal) → S o< y → ¬ Onto X (Ord y) 
+    cmax y lt ontoy = o<> lt (o<-subst  {_} {_} {y} {S}
+       (sup-o<  {λ x → proj1 ( cardinal-p x)}{y}  ) lemma refl ) where
+          lemma : proj1 (cardinal-p y) ≡ y
+          lemma with  p∨¬p ( Onto X (Ord y) )
+          lemma | case1 x = refl
+          lemma | case2 not = ⊥-elim ( not ontoy )
+
+
+-----
+--  All cardinal is ℵ0,  since we are working on Countable Ordinal, 
+--  Power ω is larger than ℵ0, so it has no cardinal.
+
+
+
diff -r 6bb5d57c9561 -r fe8392f527eb logic.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/logic.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -0,0 +1,50 @@
+module logic where
+
+open import Level
+open import Relation.Nullary
+open import Relation.Binary
+open import Data.Empty
+
+
+data Bool : Set where
+   true : Bool
+   false : Bool
+
+record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
+   field
+      proj1 : A
+      proj2 : B
+
+data  _∨_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
+   case1 : A → A ∨ B
+   case2 : B → A ∨ B
+
+_⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m )  → Set (n ⊔ m)
+_⇔_ A B =  ( A → B ) ∧ ( B → A )
+
+contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A
+contra-position {n} {m} {A} {B}  f ¬b a = ¬b ( f a )
+
+double-neg : {n  : Level } {A : Set n} → A → ¬ ¬ A
+double-neg A notnot = notnot A
+
+double-neg2 : {n  : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
+double-neg2 notnot A = notnot ( double-neg A )
+
+de-morgan : {n  : Level } {A B : Set n} →  A ∧ B  → ¬ ( (¬ A ) ∨ (¬ B ) )
+de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
+de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
+
+dont-or :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ A → B
+dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
+dont-or {A} {B} (case2 b) ¬A = b
+
+dont-orb :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ B → A
+dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
+dont-orb {A} {B} (case1 a) ¬B = a
+
+
+infixr  130 _∧_
+infixr  140 _∨_
+infixr  150 _⇔_
+
diff -r 6bb5d57c9561 -r fe8392f527eb nat.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/nat.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -0,0 +1,46 @@
+module nat where
+
+open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import Data.Empty
+open import Relation.Nullary
+open import  Relation.Binary.PropositionalEquality
+open import  logic
+
+
+nat-<> : { x y : Nat } → x < y → y < x → ⊥
+nat-<>  (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
+
+nat-<≡ : { x : Nat } → x < x → ⊥
+nat-<≡  (s≤s lt) = nat-<≡ lt
+
+nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥
+nat-≡< refl lt = nat-<≡ lt
+
+¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
+¬a≤a  (s≤s lt) = ¬a≤a  lt
+
+a<sa : {la : Nat} → la < Suc la 
+a<sa {Zero} = s≤s z≤n
+a<sa {Suc la} = s≤s a<sa 
+
+=→¬< : {x : Nat  } → ¬ ( x < x )
+=→¬< {Zero} ()
+=→¬< {Suc x} (s≤s lt) = =→¬< lt
+
+>→¬< : {x y : Nat  } → (x < y ) → ¬ ( y < x )
+>→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x
+
+<-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) )
+<-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl
+<-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n)
+<-∨ {Suc x} {Zero} (s≤s ())
+<-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt
+<-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq)
+<-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
+
+max : (x y : Nat) → Nat
+max Zero Zero = Zero
+max Zero (Suc x) = (Suc x)
+max (Suc x) Zero = (Suc x)
+max (Suc x) (Suc y) = Suc ( max x y )
+
diff -r 6bb5d57c9561 -r fe8392f527eb ordinal.agda
--- a/ordinal.agda	Thu Aug 01 12:24:26 2019 +0900
+++ b/ordinal.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -7,6 +7,8 @@
 open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
 open import Data.Empty
 open import  Relation.Binary.PropositionalEquality
+open import  logic
+open import  nat
 
 data OrdinalD {n : Level} :  (lv : Nat) → Set n where
    Φ : (lv : Nat) → OrdinalD  lv
@@ -58,12 +60,6 @@
       lv x  ≡ lv y → ord x ≅ ord y →  x ≡ y
 ordinal-cong refl refl = refl
 
-ordinal-lv : {n : Level} {x y : Ordinal {n}}  → x ≡ y → lv x  ≡ lv y 
-ordinal-lv refl = refl
-
-ordinal-d : {n : Level} {x y : Ordinal {n}}  → x ≡ y → ord x  ≅ ord y 
-ordinal-d refl = refl
-
 ≡→¬d< : {n : Level} →  {lv : Nat} → {x  : OrdinalD {n}  lv }  → x d< x → ⊥
 ≡→¬d<  {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
 
@@ -73,9 +69,6 @@
 trio>≡ : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
 trio>≡ refl = ≡→¬d<
 
-triO : {n : Level} →  {lx ly : Nat} → OrdinalD {n} lx  →  OrdinalD {n} ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
-triO  {n} {lx} {ly} x y = <-cmp lx ly
-
 triOrdd : {n : Level} →  {lx : Nat}   → Trichotomous  _≡_ ( _d<_  {n} {lx} {lx} )
 triOrdd  {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d<
 triOrdd  {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
@@ -92,50 +85,6 @@
 <-osuc {n} {record { lv = lx ; ord = Φ .lx }} =  case2 Φ<
 <-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl )
 
-osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x )
-osuc-lveq {n} = refl
-
-osucc : {n : Level} {ox oy : Ordinal {n}} → oy o< ox  → osuc oy o< osuc ox
-osucc {n} {ox} {oy} (case1 x) = case1 x
-osucc {n} {ox} {oy} (case2 x) with d<→lv x
-... | refl = case2 (s< x)
-
-nat-<> : { x y : Nat } → x < y → y < x → ⊥
-nat-<>  (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
-
-nat-<≡ : { x : Nat } → x < x → ⊥
-nat-<≡  (s≤s lt) = nat-<≡ lt
-
-nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥
-nat-≡< refl lt = nat-<≡ lt
-
-¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
-¬a≤a  (s≤s lt) = ¬a≤a  lt
-
-a<sa : {la : Nat} → la < Suc la 
-a<sa {Zero} = s≤s z≤n
-a<sa {Suc la} = s≤s a<sa 
-
-=→¬< : {x : Nat  } → ¬ ( x < x )
-=→¬< {Zero} ()
-=→¬< {Suc x} (s≤s lt) = =→¬< lt
-
-<-∨ : { x y : Nat } → x < Suc y → ( (x ≡ y ) ∨ (x < y) )
-<-∨ {Zero} {Zero} (s≤s z≤n) = case1 refl
-<-∨ {Zero} {Suc y} (s≤s lt) = case2 (s≤s z≤n)
-<-∨ {Suc x} {Zero} (s≤s ())
-<-∨ {Suc x} {Suc y} (s≤s lt) with <-∨ {x} {y} lt
-<-∨ {Suc x} {Suc y} (s≤s lt) | case1 eq = case1 (cong (λ k → Suc k ) eq)
-<-∨ {Suc x} {Suc y} (s≤s lt) | case2 lt1 = case2 (s≤s lt1)
-
-case12-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord x d< ord y → ⊥
-case12-⊥ {x} {y} lt1 lt2 with d<→lv lt2
-... | refl = nat-≡< refl lt1
-
-case21-⊥ : {n : Level} {x y : Ordinal {suc n}} → lv x < lv y → ord y d< ord x → ⊥
-case21-⊥ {x} {y} lt1 lt2 with d<→lv lt2
-... | refl = nat-≡< refl lt1
-
 o<¬≡ : {n : Level } { ox oy : Ordinal {suc n}} → ox ≡ oy  → ox o< oy  → ⊥
 o<¬≡ {_} {ox} {ox} refl (case1 lt) =  =→¬< lt
 o<¬≡ {_} {ox} {ox} refl (case2 (s< lt)) = trio<≡ refl lt
@@ -176,26 +125,6 @@
 osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂
 osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x
 
-max : (x y : Nat) → Nat
-max Zero Zero = Zero
-max Zero (Suc x) = (Suc x)
-max (Suc x) Zero = (Suc x)
-max (Suc x) (Suc y) = Suc ( max x y )
-
-maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx  →  OrdinalD  lx  →  OrdinalD  lx
-maxαd x y with triOrdd x y
-maxαd x y | tri< a ¬b ¬c = y
-maxαd x y | tri≈ ¬a b ¬c = x
-maxαd x y | tri> ¬a ¬b c = x
-
-minαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx  →  OrdinalD  lx  →  OrdinalD  lx
-minαd x y with triOrdd x y
-minαd x y | tri< a ¬b ¬c = x
-minαd x y | tri≈ ¬a b ¬c = y
-minαd x y | tri> ¬a ¬b c = x
-
-_o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
-a o≤ b  = (a ≡ b)  ∨ ( a o< b )
 
 ordtrans : {n : Level} {x y z : Ordinal {n} }   → x o< y → y o< z → x o< z
 ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ )
@@ -234,105 +163,15 @@
    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
-maxα x y | tri> ¬a ¬b c = x
-maxα x y | tri≈ ¬a refl ¬c = x
-
-minα : {n : Level} →  Ordinal {suc n} →  Ordinal  → Ordinal
-minα {n} x y with trio< {n} x  y
-minα x y | tri< a ¬b ¬c = x
-minα x y | tri> ¬a ¬b c = y
-minα x y | tri≈ ¬a refl ¬c = x
-
-min1 : {n : Level} →  {x y z : Ordinal {suc n} } → z o< x → z o< y → z o< minα x y
-min1 {n} {x} {y} {z} z<x z<y with trio< {n} x y
-min1 {n} {x} {y} {z} z<x z<y | tri< a ¬b ¬c = z<x
-min1 {n} {x} {y} {z} z<x z<y | tri≈ ¬a refl ¬c = z<x
-min1 {n} {x} {y} {z} z<x z<y | tri> ¬a ¬b c = z<y
-
---
---  max ( osuc x , osuc y )
---
-
-omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n}
-omax {n} x y with trio< x y
-omax {n} x y | tri< a ¬b ¬c = osuc y
-omax {n} x y | tri> ¬a ¬b c = osuc x
-omax {n} x y | tri≈ ¬a refl ¬c  = osuc x
-
-omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y
-omax< {n} x y lt with trio< x y
-omax< {n} x y lt | tri< a ¬b ¬c = refl
-omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
-omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
-
-omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y
-omax≡ {n} x y eq with trio< x y
-omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
-omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl
-omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
-
-omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y
-omax-x {n} x y with trio< x y
-omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc
-omax-x {n} x y | tri> ¬a ¬b c = <-osuc
-omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc
-
-omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y
-omax-y {n} x y with  trio< x y
-omax-y {n} x y | tri< a ¬b ¬c = <-osuc
-omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc
-omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc
-
-omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x
-omxx {n} x with  trio< x x
-omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
-omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
-omxx {n} x | tri≈ ¬a refl ¬c = refl
-
-omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x)
-omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
 
 open _∧_
 
-osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
-proj1 (osuc2 {n} x y) (case1 lt) = case1 lt
-proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt
-proj2 (osuc2 {n} x y) (case1 lt) = case1 lt
-proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt
-... | refl = case2 (s< lt)
-
-OrdTrans : {n : Level} → Transitive {suc n} _o≤_
-OrdTrans (case1 refl) (case1 refl) = case1 refl
-OrdTrans (case1 refl) (case2 lt2) = case2 lt2
-OrdTrans (case2 lt1) (case1 refl) = case2 lt1
-OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y)
-
-OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n)
-OrdPreorder {n} = record { Carrier = Ordinal
-   ; _≈_  = _≡_ 
-   ; _∼_   = _o≤_
-   ; isPreorder   = record {
-        isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
-        ; reflexive     = case1 
-        ; trans         = OrdTrans 
-     }
- }
-
 TransFinite : {n m : Level} → { ψ : Ordinal {suc n} → Set m }
   → ( ∀ (lx : Nat ) → ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx)  → ψ x ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
-  → ( ∀ (lx : Nat ) → (x : OrdinalD lx )  → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
+  → ( ∀ (lx : Nat ) → (x : OrdinalD lx ) → ( (y : Ordinal {suc n} ) → y o< ordinal lx (OSuc lx x)  → ψ y )   → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
   →  ∀ (x : Ordinal)  → ψ x
 TransFinite {n} {m} {ψ} caseΦ caseOSuc x = proj1 (TransFinite1 (lv x) (ord x) ) where
-  TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx (Φ lx)  → ψ x ) )
+  TransFinite1 : (lx : Nat) (ox : OrdinalD lx ) → ψ (ordinal lx ox) ∧ ( ( (x : Ordinal {suc n} ) → x o< ordinal lx ox  → ψ x ) )
   TransFinite1 Zero (Φ 0) = record { proj1 = caseΦ Zero lemma ; proj2 = lemma1 } where
       lemma : (x : Ordinal) → x o< ordinal Zero (Φ Zero) → ψ x
       lemma x (case1 ())
@@ -344,12 +183,26 @@
       lemma0 : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy  o< ordinal lx (Φ lx) → ψ (ordinal ly oy)
       lemma0 ly oy lt = proj2 ( TransFinite1 lx (Φ lx) ) (ordinal ly oy) lt
       lemma : (ly : Nat) (oy : OrdinalD ly ) → ordinal ly oy  o< ordinal (Suc lx) (Φ (Suc lx)) → ψ (ordinal ly oy)
-      lemma lx1 ox1 (case1 lt) with <-∨ lt
-      lemma lx (Φ lx) (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) )
-      lemma lx (OSuc lx ox1) (case1 lt) | case1 refl = caseOSuc lx ox1 ( lemma lx ox1 (case1 a<sa)) 
-      lemma lx (Φ lx) (case1 lt) | case2 (s≤s lt1) = lemma0  lx (Φ lx) (case1 (s≤s lt1))
-      lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 ( lemma lx1 ox1 (case1 (<-trans lt1 a<sa ))) 
-  TransFinite1 lx (OSuc lx ox)  = record { proj1 = caseOSuc lx ox (proj1 (TransFinite1 lx ox )) ; proj2 = proj2 (TransFinite1 lx ox )}
+      lemma lx1 ox1            (case1 lt) with <-∨ lt
+      lemma lx (Φ lx)          (case1 lt) | case1 refl = proj1 ( TransFinite1 lx (Φ lx) )
+      lemma lx (Φ lx)          (case1 lt) | case2 lt1 = lemma0  lx (Φ lx) (case1 lt1)
+      lemma lx (OSuc lx ox1)   (case1 lt) | case1 refl = caseOSuc lx ox1 lemma2 where
+          lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx) ∨ (ord y d< OSuc lx ox1) → ψ y
+          lemma2 y lt1 with osuc-≡< lt1
+          lemma2 y lt1 | case1 refl = lemma lx ox1 (case1 a<sa)
+          lemma2 y lt1 | case2 t = proj2 (TransFinite1 lx ox1) y t 
+      lemma lx1 (OSuc lx1 ox1) (case1 lt) | case2 lt1 = caseOSuc lx1 ox1 lemma2 where
+          lemma2 : (y : Ordinal) → (Suc (lv y) ≤ lx1) ∨ (ord y d< OSuc lx1 ox1) → ψ y
+          lemma2 y lt2 with osuc-≡< lt2
+          lemma2 y lt2 | case1 refl = lemma lx1 ox1 (ordtrans lt2 (case1 lt))
+          lemma2 y lt2 | case2 (case1 lt3) = proj2 (TransFinite1 lx (Φ lx)) y (case1 (<-trans lt3 lt1 ))
+          lemma2 y lt2 | case2 (case2 lt3) with d<→lv lt3
+          ... | refl = proj2 (TransFinite1 lx (Φ lx)) y (case1 lt1)
+  TransFinite1 lx (OSuc lx ox)  = record { proj1 = caseOSuc lx ox lemma ; proj2 = lemma } where
+      lemma : (y : Ordinal) → y o< ordinal lx (OSuc lx ox) → ψ y
+      lemma y lt with osuc-≡< lt
+      lemma y lt | case1 refl = proj1 ( TransFinite1 lx ox ) 
+      lemma y lt | case2 lt1 = proj2 ( TransFinite1 lx ox ) y lt1
 
 -- we cannot prove this in intutionistic logic 
 --  (¬ (∀ y → ¬ ( ψ y ))) → (ψ y → p )  → p
@@ -367,3 +220,171 @@
   → ¬ p
 TransFiniteExists {n} {m} {l} ψ {p} P = contra-position ( λ p y ψy → P {y} ψy p ) 
 
+open import Ordinals 
+
+C-Ordinal : {n : Level} →  Ordinals {suc n} 
+C-Ordinal {n} = record {
+     ord = Ordinal {suc n}
+   ; o∅ = o∅
+   ; osuc = osuc
+   ; _o<_ = _o<_
+   ; isOrdinal = record {
+       Otrans = ordtrans
+     ; OTri = trio<
+     ; ¬x<0 = ¬x<0 
+     ; <-osuc = <-osuc
+     ; osuc-≡< = osuc-≡<
+     ; TransFinite = TransFinite1
+   }
+  } where
+     ord1 : Set (suc n)
+     ord1 = Ordinal {suc n}
+     TransFinite1 : { ψ : ord1  → Set (suc (suc n)) }
+          → ( (x : ord1)  → ( (y : ord1  ) → y o< x → ψ y ) → ψ x )
+          →  ∀ (x : ord1)  → ψ x
+     TransFinite1 {ψ} lt x = TransFinite {n} {suc (suc n)} {ψ} caseΦ caseOSuc x where
+        caseΦ : (lx : Nat) → ((x₁ : Ordinal) → x₁ o< ordinal lx (Φ lx) → ψ x₁) →
+            ψ (record { lv = lx ; ord = Φ lx })
+        caseΦ lx prev = lt (ordinal lx (Φ lx) ) prev
+        caseOSuc :  (lx : Nat) (x₁ : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x₁) → ψ y) →
+            ψ (record { lv = lx ; ord = OSuc lx x₁ })
+        caseOSuc lx ox prev = lt (ordinal lx (OSuc lx ox)) prev 
+
+module C-Ordinal-with-choice {n : Level} where
+
+  import OD 
+  -- open inOrdinal C-Ordinal 
+  open OD (C-Ordinal {n})
+  open OD.OD
+  
+  --
+  -- another form of regularity 
+  --
+  ε-induction : {m : Level} { ψ : OD  → Set m}
+   → ( {x : OD } → ({ y : OD } →  x ∋ y → ψ y ) → ψ x )
+   → (x : OD ) → ψ x
+  ε-induction {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 : {ly : Nat} {oy : OrdinalD {suc n} ly} → lv (od→ord (ord→od (record { lv = ly ; ord = oy }))) ≡ ly
+          lemma1  {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<  {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 =  subst (λ k → ψ k ) oiso ( ε-induction-ord lx (dz oz=lx) {lv (od→ord z)} {ord (od→ord z)} (case2 (dz<dz oz=lx) )) where
+              oz=lx : lv (od→ord z) ≡ lx 
+              oz=lx = let open ≡-Reasoning in begin
+                  lv (od→ord z)
+                 ≡⟨ eq ⟩
+                  lv (od→ord (ord→od (ordinal ly oy)))
+                 ≡⟨ cong ( λ k → lv k ) diso ⟩
+                  lv (ordinal ly oy)
+                 ≡⟨ sym lx=ly  ⟩
+                  lx
+                 ∎
+              dz : lv (od→ord z) ≡ lx → OrdinalD lx
+              dz refl = OSuc lx (ord (od→ord z))
+              dz<dz  : (z=x : lv (od→ord z) ≡ lx ) → ord (od→ord z) d< dz z=x
+              dz<dz refl = s<refl 
+  
+  ---
+  --- With assuption of OD is ordered,  p ∨ ( ¬ p ) <=> axiom of choice
+  ---
+  record choiced  ( X : OD) : Set (suc (suc n)) where
+   field
+      a-choice : OD
+      is-in : X ∋ a-choice
+  
+  choice-func' :  (X : OD ) → (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 caseOSuc1 ox where
+             ∋-p' : (A x : OD ) → 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
+             caseOSuc1 : (lx : Nat) (x : OrdinalD lx) → ((y : Ordinal) → y o< ordinal lx (OSuc lx x) → ψ y) →
+                 ψ (record { lv = lx ; ord = OSuc lx x })
+             caseOSuc1 lx x lt =  caseOSuc lx x (lt ( ordinal lx x ) (case2 s<refl))
+          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 )
+                 }
+  
diff -r 6bb5d57c9561 -r fe8392f527eb zf.agda
--- a/zf.agda	Thu Aug 01 12:24:26 2019 +0900
+++ b/zf.agda	Mon Aug 12 09:04:16 2019 +0900
@@ -2,56 +2,12 @@
 
 open import Level
 
-data Bool : Set where
-   true : Bool
-   false : Bool
-
-record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
-   field 
-      proj1 : A
-      proj2 : B
-
-data  _∨_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
-   case1 : A → A ∨ B
-   case2 : B → A ∨ B
-
-_⇔_ : {n m : Level } → ( A : Set n ) ( B : Set m )  → Set (n ⊔ m)
-_⇔_ A B =  ( A → B ) ∧ ( B → A )
-
+open import logic
 
 open import Relation.Nullary
 open import Relation.Binary
 open import Data.Empty
 
-
-contra-position : {n m : Level } {A : Set n} {B : Set m} → (A → B) → ¬ B → ¬ A 
-contra-position {n} {m} {A} {B}  f ¬b a = ¬b ( f a ) 
-
-double-neg : {n  : Level } {A : Set n} → A → ¬ ¬ A
-double-neg A notnot = notnot A
-
-double-neg2 : {n  : Level } {A : Set n} → ¬ ¬ ¬ A → ¬ A
-double-neg2 notnot A = notnot ( double-neg A )
-
-de-morgan : {n  : Level } {A B : Set n} →  A ∧ B  → ¬ ( (¬ A ) ∨ (¬ B ) )
-de-morgan {n} {A} {B} and (case1 ¬A) = ⊥-elim ( ¬A ( _∧_.proj1 and ))
-de-morgan {n} {A} {B} and (case2 ¬B) = ⊥-elim ( ¬B ( _∧_.proj2 and ))
-
-dont-or :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ A → B
-dont-or {A} {B} (case1 a) ¬A = ⊥-elim ( ¬A a )
-dont-or {A} {B} (case2 b) ¬A = b
-
-dont-orb :　{n m : Level} {A  : Set n} { B : Set m } →  A ∨ B → ¬ B → A
-dont-orb {A} {B} (case2 b) ¬B = ⊥-elim ( ¬B b )
-dont-orb {A} {B} (case1 a) ¬B = a
-
--- mid-ex-neg : {n  : Level } {A : Set n} → ( ¬ ¬ A ) ∨ (¬ A)
--- mid-ex-neg {n} {A} = {!!}
-
-infixr  130 _∧_
-infixr  140 _∨_
-infixr  150 _⇔_
-
 record IsZF {n m : Level }
      (ZFSet : Set n)
      (_∋_ : ( A x : ZFSet  ) → Set m)
@@ -96,9 +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