--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/cardinal.agda	Wed Aug 07 09:50:51 2019 +0900
@@ -0,0 +1,105 @@
+open import Level
+module cardinal where
+
+open import zf
+open import ordinal
+open import logic
+open 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 OD.OD
+
+open Ordinal
+open _∧_
+open _∨_
+open Bool
+
+------------
+--
+-- Onto map
+--          def X x ->  xmap
+--     X ---------------------------> Y
+--          ymap   <-  def Y y
+--
+record Onto {n : Level } (X Y : OD {n})  : Set (suc 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
+
+record Cardinal {n : Level } (X  : OD {n}) : Set (suc n) where
+   field
+       cardinal : Ordinal {n}
+       conto : Onto (Ord cardinal) X 
+       cmax : ( y : Ordinal {n} ) → cardinal o< y → ¬ Onto (Ord y) X 
+
+cardinal : {n : Level } (X  : OD {suc n}) → Cardinal X
+cardinal {n} 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 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 = 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 {
+               xmap = xmap
+            ;  ymap = ymap
+            ;  ymap-on-X  = ymap-on-X
+            ;  onto-iso = onto-iso
+      } where
+       --
+       -- 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} 
+       ... | t = {!!}
+       lemma2 :  def Y (od→ord X)
+       lemma2 = {!!}
+       xmap : (x : Ordinal {suc n}) → def Y x → Ordinal {suc n}
+       xmap = {!!}
+       ymap : (y : Ordinal {suc n}) → def X y → Ordinal {suc n}
+       ymap = {!!}
+       ymap-on-X  : {y :  Ordinal {suc n} } → (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 = {!!}
+    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
+          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 : {n : Level}  → OD {suc n}
+Func {n} = record { def = λ x →  (f : Ordinal {suc n} → Ordinal {suc n}) → 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 } 
+
+lemma1 :  {n : Level}  → { x : OD {suc n} } → Func ∋ x → {!!} -- ¬ ( (f : Ordinal {suc n} → Ordinal {suc n}) →  ¬ ( x ≡ od→ord (func f)  ))
+lemma1 = {!!}
+
+
+-----
+--  All cardinal is ℵ0,  since we are working on Countable Ordinal, 
+--  Power ω is larger than ℵ0, so it has no cardinal.
+
+
+