--- a/cardinal.agda	Fri Aug 09 17:57:58 2019 +0900
+++ b/cardinal.agda	Sat Aug 10 12:31:25 2019 +0900
@@ -1,21 +1,22 @@
 open import Level
-module cardinal where
+open import Ordinals
+module cardinal {n : Level } (O : Ordinals {n}) where
 
 open import zf
-open import ordinal
 open import logic
-open import OD
+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 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 Ordinal
 open _∧_
 open _∨_
 open Bool
@@ -27,30 +28,30 @@
 --     X ---------------------------> Y
 --          ymap   <-  def Y y
 --
-record Onto {n : Level } (X Y : OD {n})  : Set (suc n) where
+record Onto  (X Y : OD )  : Set n where
    field
-       xmap : (x : Ordinal {n}) → def X x → Ordinal {n} 
-       ymap : (y : Ordinal {n}) → def Y y → Ordinal {n} 
-       ymap-on-X  : {y :  Ordinal {n} } → (lty : def Y y ) → def X (ymap y lty)  
-       onto-iso   : {y :  Ordinal {n} } → (lty : def Y y ) → xmap  ( ymap y lty ) (ymap-on-X lty ) ≡ y
+       xmap : (x : Ordinal ) → def X x → Ordinal  
+       ymap : (y : Ordinal ) → def Y y → Ordinal  
+       ymap-on-X  : {y :  Ordinal  } → (lty : def Y y ) → def X (ymap y lty)  
+       onto-iso   : {y :  Ordinal  } → (lty : def Y y ) → xmap  ( ymap y lty ) (ymap-on-X lty ) ≡ y
 
-record Cardinal {n : Level } (X  : OD {n}) : Set (suc n) where
+record Cardinal  (X  : OD ) : Set n where
    field
-       cardinal : Ordinal {n}
+       cardinal : Ordinal 
        conto : Onto (Ord cardinal) X 
-       cmax : ( y : Ordinal {n} ) → cardinal o< y → ¬ Onto (Ord y) X 
+       cmax : ( y : Ordinal  ) → cardinal o< y → ¬ Onto (Ord y) X 
 
-cardinal : {n : Level } (X  : OD {suc n}) → Cardinal X
-cardinal {n} X = record {
+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 {suc n}) →  ( Ordinal {suc n} ∧ Dec (Onto (Ord x) X) )
+    cardinal-p : (x  : Ordinal ) →  ( Ordinal  ∧ Dec (Onto (Ord x) X) )
     cardinal-p x with p∨¬p ( Onto (Ord x) X ) 
     cardinal-p x | case1 True = record { proj1 = x ; proj2 = yes True }
     cardinal-p x | case2 False = record { proj1 = o∅ ; proj2 = no False }
-    onto-set : OD {suc n}
+    onto-set : OD 
     onto-set = record { def = λ x →  {!!} } -- Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X }
     onto : Onto (Ord (sup-o (λ x → proj1 (cardinal-p x)))) X
     onto = record {
@@ -63,37 +64,37 @@
        -- Ord cardinal itself has no onto map, but if we have x o< cardinal, there is one
        --    od→ord X o< cardinal, so if we have def Y y or def X y, there is an Onto (Ord y) X
        Y = (Ord (sup-o (λ x → proj1 (cardinal-p x))))
-       lemma1 : (y : Ordinal {suc n}) → def Y y  →  Onto (Ord y) X
-       lemma1 y y<Y with sup-o< {suc n} {λ x → proj1 ( cardinal-p x)} {y} 
+       lemma1 : (y : Ordinal ) → def Y y  →  Onto (Ord y) X
+       lemma1 y y<Y with sup-o<  {λ x → proj1 ( cardinal-p x)} {y} 
        ... | t = {!!}
        lemma2 :  def Y (od→ord X)
        lemma2 = {!!}
-       xmap : (x : Ordinal {suc n}) → def Y x → Ordinal {suc n}
+       xmap : (x : Ordinal ) → def Y x → Ordinal 
        xmap = {!!}
-       ymap : (y : Ordinal {suc n}) → def X y → Ordinal {suc n}
+       ymap : (y : Ordinal ) → def X y → Ordinal 
        ymap = {!!}
-       ymap-on-X  : {y :  Ordinal {suc n} } → (lty : def X y ) → def Y (ymap y lty)  
+       ymap-on-X  : {y :  Ordinal  } → (lty : def X y ) → def Y (ymap y lty)  
        ymap-on-X  = {!!}
-       onto-iso   : {y :  Ordinal {suc n} } → (lty : def X y ) → xmap  (ymap y lty) (ymap-on-X lty ) ≡ y
+       onto-iso   : {y :  Ordinal  } → (lty : def X y ) → xmap  (ymap y lty) (ymap-on-X lty ) ≡ y
        onto-iso = {!!}
     cmax : (y : Ordinal) → sup-o (λ x → proj1 (cardinal-p x)) o< y → ¬ Onto (Ord y) X
-    cmax y lt ontoy = o<> lt (o<-subst {suc n} {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))}
-       (sup-o< {suc n} {λ x → proj1 ( cardinal-p x)}{y}  ) lemma refl ) where
+    cmax y lt ontoy = o<> lt (o<-subst  {_} {_} {y} {sup-o (λ x → proj1 (cardinal-p x))}
+       (sup-o<  {λ x → proj1 ( cardinal-p x)}{y}  ) lemma refl ) where
           lemma : proj1 (cardinal-p y) ≡ y
           lemma with  p∨¬p ( Onto (Ord y) X )
           lemma | case1 x = refl
           lemma | case2 not = ⊥-elim ( not ontoy )
 
-func : {n : Level}  → (f : Ordinal {suc n} → Ordinal {suc n}) → OD {suc n}
-func {n} f = record { def = λ y → (x : Ordinal {suc n}) → y ≡ f x }
+func : (f : Ordinal  → Ordinal ) → OD 
+func  f = record { def = λ y → (x : Ordinal ) → y ≡ f x }
 
-Func : {n : Level}  → OD {suc n}
-Func {n} = record { def = λ x →  (f : Ordinal {suc n} → Ordinal {suc n}) → x ≡ od→ord (func f) }
+Func : OD 
+Func  = record { def = λ x →  (f : Ordinal  → Ordinal ) → x ≡ od→ord (func f) }
 
-odmap : {n : Level}  → { x : OD {suc n} } → Func ∋ x → Ordinal {suc n} → OD {suc n}
-odmap {n} {f} lt x = record { def = λ y → def f y } 
+odmap : { x : OD  } → Func ∋ x → Ordinal  → OD 
+odmap  {f} lt x = record { def = λ y → def f y } 
 
-lemma1 :  {n : Level}  → { x : OD {suc n} } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal {suc n} → Ordinal {suc n}) →  ¬ ( x ≡ od→ord (func f)  ))
+lemma1 :  { x : OD  } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal  → Ordinal ) →  ¬ ( x ≡ od→ord (func f)  ))
 lemma1 = {!!}