Mercurial > hg > Members > kono > Proof > ZF-in-agda

changeset 218:eee983e4b402

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try func
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 06 Aug 2019 15:50:14 +0900
parents d5668179ee69
children 43021d2b8756
files OD.agda ordinal.agda
diffstat 2 files changed, 43 insertions(+), 17 deletions(-) [+]
line wrap: on
line diff
--- a/OD.agda	Mon Aug 05 17:02:37 2019 +0900
+++ b/OD.agda	Tue Aug 06 15:50:14 2019 +0900
@@ -274,7 +274,6 @@
 -- 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 
 
-
 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
 OD→ZF {n}  = record { 
     ZFSet = OD {suc n}
@@ -630,19 +629,18 @@
          --     X ---------------------------> Y
          --          ymap   <-  def Y y
          --
-         record Onto {n : Level } (X Y : OD {suc n})  : Set (suc (suc n)) where
+         record Onto {n : Level } (X Y : OD {n})  : Set (suc n) where
             field
-                xmap : (x : Ordinal {suc n}) → Ordinal {suc n} 
-                ymap : (y : Ordinal {suc n}) → Ordinal {suc n} 
-                xmap-on-Y  : (x :  Ordinal {suc n} ) → def X x  → def Y (xmap x)  
-                ymap-on-X  : (y :  Ordinal {suc n} ) → def Y y  → def X (ymap y)  
-                onto-iso : (y :  Ordinal {suc n} ) → def Y y → xmap ( ymap y ) ≡ y
+                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 {suc n}) : Set (suc (suc n)) where
+         record Cardinal {n : Level } (X  : OD {n}) : Set (suc n) where
             field
-                cardinal : Ordinal {suc n}
+                cardinal : Ordinal {n}
                 conto : Onto (Ord cardinal) X 
-                cmax : ( y : Ordinal {suc n} ) → cardinal o< y → ¬ Onto (Ord y) 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 {
@@ -654,24 +652,31 @@
              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
-                     ;  xmap-on-Y  = xmap-on-Y
                      ;  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))))
-                xmap : (x : Ordinal {suc n}) → Ordinal {suc n}
+                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}) → Ordinal {suc n}
+                ymap : (y : Ordinal {suc n}) → def X y → Ordinal {suc n}
                 ymap = {!!}
-                xmap-on-Y  : (x :  Ordinal {suc n} ) → def Y x  → def X (xmap x)
-                xmap-on-Y  = {!!}
-                ymap-on-X  : (y :  Ordinal {suc n} ) → def X y  → def Y (ymap y)
+                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} ) → def X y → xmap ( ymap y ) ≡ y
+                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))}
@@ -681,4 +686,19 @@
                    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 } 
+
+
+         -----
+         --  All cardinal is ℵ0,  since we are working on Countable Ordinal, 
+         --  Power ω is larger than ℵ0, so it has no cardinal.
+
+
+
--- a/ordinal.agda	Mon Aug 05 17:02:37 2019 +0900
+++ b/ordinal.agda	Tue Aug 06 15:50:14 2019 +0900
@@ -29,6 +29,12 @@
 _o<_ : {n : Level} ( x y : Ordinal ) → Set n
 _o<_ x y =  (lv x < lv y )  ∨ ( ord x d< ord y )
 
+o<-dom :  {n : Level} { x y : Ordinal {n}} → x o< y → Ordinal 
+o<-dom {n} {x} _ = x
+
+o<-cod :  {n : Level} { x y : Ordinal {n}} → x o< y → Ordinal 
+o<-cod {n} {_} {y} _ = y
+
 s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x
 s<refl {n} {lv} {Φ lv}  = Φ<
 s<refl {n} {lv} {OSuc lv x}  = s< s<refl