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

changeset 574:9084a26445a7

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ZC data won't work
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 05 Jun 2022 12:49:48 +0900
parents 9ec37757a5a5
children 3abf55c8e295 409215885fc0
files src/filter.agda src/zorn.agda
diffstat 2 files changed, 40 insertions(+), 17 deletions(-) [+]
line wrap: on
line diff
--- a/src/filter.agda	Tue May 03 00:59:52 2022 +0900
+++ b/src/filter.agda	Sun Jun 05 12:49:48 2022 +0900
@@ -230,22 +230,24 @@
      f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP {!!} {!!}) )
 
 _⊆'_ : ( A B : HOD ) → Set n
-_⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
+_⊆'_ A B = {x : Ordinal } → odef A x → odef B x
 
 import zorn 
-open zorn O _⊆'_
+open zorn O _⊆'_ hiding (  _⊆'_ )
+
+open import  Relation.Binary.Structures
 
 MaximumSubset : {L P : HOD} 
-      → o∅ o< & L →  o∅ o< & P → P ⊆ L
-      → IsPartialOrderSet P 
-      → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B  → SUP P B  )
-      → Maximal P 
-MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP  = Zorn-lemma 0<P PO SP
+      → o∅ o< & L →  o∅ o< & P → P ⊆' L
+      → (PO : IsStrictPartialOrder _≡_ _⊆'_ )
+      → ( (B : HOD) → B ⊆ P → IsTotalOrderSet PO B  → SUP PO P B  )
+      → Maximal PO P
+MaximumSubset {L} {P} 0<L 0<P P⊆L PO C = Zorn-lemma PO 0<P {!!}
 
 MaximumFilterExist : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q))
       → (F : Filter LP) → o∅ o< & L →  o∅ o< & (filter F)  →  (¬ (filter F ∋ od∅)) → MaximumFilter LP 
 MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = record { mf = {!!} ; proper = {!!} ; is-maximum = {!!} }  where
-     mf01 : Maximal  P 
-     mf01 = MaximumSubset  0<L {!!}  {!!} {!!} {!!} 
+     mf01 : Maximal  {!!}  {!!}
+     mf01 = MaximumSubset  0<L {!!}  {!!} {!!}  {!!}
 
 
--- a/src/zorn.agda	Tue May 03 00:59:52 2022 +0900
+++ b/src/zorn.agda	Sun Jun 05 12:49:48 2022 +0900
@@ -246,8 +246,28 @@
       f-next : {a : Ordinal } → odef chain a → odef chain (f a)
       f-immediate : { x y : Ordinal } → odef chain x → odef chain y → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) )
       is-max :  {a b : Ordinal } → (ca : odef chain a ) →  b o< osuc z  → (ab : odef A b) 
-          → HasPrev A chain ab f ∨  IsSup A chain ab        --  ((sup  chain  chain⊆A  f-total) ≡ b )
+          → HasPrev A chain ab f ∨  IsSup A chain ab  
           → * a < * b  → odef chain b
+      fc∨sup :  {a : Ordinal } → a o< osuc z →  ( ca : odef chain a ) → HasPrev A chain ( chain⊆A ca) f  ∨ IsSup A chain  ( chain⊆A ca)
+
+data ZC (A : HOD) (f : Ordinal → Ordinal ) : Ordinal → Set n where
+   zc-init : ZC A f o∅
+   zc-fc  : {s x : Ordinal} → ZC A f s → FClosure A f s x → ZC A f x
+   zc-sup : {s x : Ordinal} → ZC A f s → * s < * x  → ZC A f x
+
+ZC→ZChain : (A : HOD) {x z : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
+    → ZC A f z → (sup : (C : HOD ) → ( C ⊆' A) → IsTotalOrderSet C → Ordinal) → ZChain A ax f mf sup z
+ZC→ZChain A {x} {z} ax f mf zc sup = record {
+      chain = ? 
+    ; chain⊆A = ? 
+    ; chain∋x = ? 
+    ; initial = ? 
+    ; f-total = ? 
+    ; f-next = ? 
+    ; f-immediate = ? 
+    ; is-max = ? 
+    ; fc∨sup = ?
+   }
 
 record Maximal ( A : HOD )  : Set (Level.suc n) where
    field
@@ -408,7 +428,7 @@
           ... | no noax =  -- ¬ A ∋ p, just skip
                  record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 
                      ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
-                     ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc11 }  where -- no extention
+                     ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc11 ; fc∨sup = {!!} }  where -- no extention
                 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
                     HasPrev A (ZChain.chain zc0) ab f ∨  IsSup A (ZChain.chain zc0) ab →
                             * a < * b → odef (ZChain.chain zc0) b
@@ -425,12 +445,12 @@
                 ... | case2 lt = ZChain.is-max zc0 za (zc0-b<x b lt) ab P a<b
                 ... | case1 b=x = subst (λ k → odef chain k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr))
                 zc9 :  ZChain A ay f mf supO x
-                zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0
-                     ; initial = ZChain.initial zc0 ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc17 }  -- no extention
+                zc9 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  ZChain.f-next zc0 -- no extention
+                     ; initial = ZChain.initial zc0 ; f-immediate =  ZChain.f-immediate zc0 ; chain∋x  = ZChain.chain∋x zc0 ; is-max = zc17 ; fc∨sup = {!!}}  
           ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax )
           ... | case1 is-sup = -- x is a sup of zc0 
                 record { chain = schain ; chain⊆A = s⊆A  ; f-total = scmp ; f-next = scnext 
-                     ; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax } where 
+                     ; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = s-ismax ; fc∨sup = {!!}} where 
                 sup0 : SUP A (ZChain.chain zc0) 
                 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where
                         x21 :  {y : HOD} → ZChain.chain zc0 ∋ y → (y ≡ * x) ∨ (y < * x)
@@ -531,7 +551,8 @@
                      ... | case2 y<b  = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b
           ... | case2 ¬x=sup =  -- x is not f y' nor sup of former ZChain from y
                    record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0
-                     ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x  =  ZChain.chain∋x zc0 ; is-max = z18 }  where -- no extention
+                     ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x  =  ZChain.chain∋x zc0 ; is-max = z18 ; fc∨sup = {!!} }  where
+                      -- no extention
                 z18 :  {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
                             HasPrev A (ZChain.chain zc0) ab f ∨ IsSup A (ZChain.chain zc0)   ab →
                             * a < * b → odef (ZChain.chain zc0) b
@@ -541,10 +562,10 @@
                 ... | case1 pr = ⊥-elim ( ¬fy<x record {y = HasPrev.y pr ; ay = HasPrev.ay pr ; x=fy = trans (trans &iso (sym b=x) ) (HasPrev.x=fy pr ) } )
                 ... | case2 b=sup = ⊥-elim ( ¬x=sup record { 
                       x<sup = λ {y} zy → subst (λ k → (y ≡ k) ∨ (y << k)) (trans b=x (sym &iso)) (IsSup.x<sup b=sup zy)  } ) 
-     ... | no ¬ox = {!!} where
+     ... | no ¬ox = UnionZ where
        UnionZ : ZChain A ay f mf supO x
        UnionZ = record { chain = Uz ; chain⊆A = Uz⊆A  ; f-total = u-total  ; f-next = u-next
-                     ; initial = u-initial ; f-immediate = {!!} ; chain∋x  = {!!} ; is-max = {!!} }  where --- limit ordinal case
+                     ; initial = u-initial ; f-immediate = {!!} ; chain∋x  = {!!} ; is-max = {!!} ; fc∨sup = {!!} }  where --- limit ordinal case
          record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x
             field
                u : Ordinal