# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1654712219 -32400
# Node ID cc416fc0ef846bcda6fc64123730b6d6293013f4
# Parent  6e0789af0d63c61036356a3646b764ad1ba9abfb
chainf is now global on ZChain

diff -r 6e0789af0d63 -r cc416fc0ef84 src/zorn.agda
--- a/src/zorn.agda	Thu Jun 09 02:45:21 2022 +0900
+++ b/src/zorn.agda	Thu Jun 09 03:16:59 2022 +0900
@@ -233,9 +233,10 @@
       x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x )
 
 record ZChain ( A : HOD )  (x : Ordinal)  ( f : Ordinal → Ordinal ) 
-                  ( z : Ordinal ) : Set (Level.suc n) where
+                (chainf : Ordinal → HOD)  ( z : Ordinal ) : Set (Level.suc n) where
+   chain : HOD
+   chain = chainf z
    field
-      chain : HOD
       chain⊆A : chain ⊆' A
       chain∋x : odef chain x
       initial : {y : Ordinal } → odef chain y → * x ≤ * y
@@ -250,8 +251,8 @@
 record ZChain∧Chain  ( A : HOD )  (x : Ordinal)  ( f : Ordinal → Ordinal )
                   ( z : Ordinal ) : Set (Level.suc n) where
    field
-      zchain : ZChain A x f z
       chainf :  (b : Ordinal) → HOD
+      zchain : ZChain A x f chainf z
       chain-mono :  {a b : Ordinal} → a o< b → b o< osuc z  → chainf a  ⊆' chainf b
       chain=zchain : chainf z ≡ ZChain.chain zchain 
 
@@ -320,24 +321,24 @@
      cf-is-≤-monotonic : (nmx : ¬ Maximal A ) →  ≤-monotonic-f A ( cf nmx )
      cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫
 
-     zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) →  (zc : ZChain A (& s) f (& A) ) → SUP A  (ZChain.chain zc) 
-     zsup f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc  )   
-     A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A (& s) (cf nmx)  (& A) ) 
-        →  A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) ))
-     A∋zsup nmx zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc ) )
-     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (& s) f (& A) ) → SUP A (ZChain.chain zc)
-     sp0 f mf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) 
+     zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) → (chainf : Ordinal → HOD)  (zc : ZChain A (& s) f chainf (& A) ) → SUP A  (ZChain.chain zc) 
+     zsup f mf chainf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc  )   
+     A∋zsup :  (nmx : ¬ Maximal A ) (chainf : Ordinal → HOD) (zc : ZChain A (& s) (cf nmx) chainf (& A) ) 
+        →  A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) chainf zc ) ))
+     A∋zsup nmx chainf zc = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) chainf zc ) )
+     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (chainf : Ordinal → HOD) (zc : ZChain A (& s) f chainf (& A) ) → SUP A (ZChain.chain zc)
+     sp0 f mf chainf zc = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc) 
      zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
      zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y)
 
      ---
      --- the maximum chain  has fix point of any ≤-monotonic function
      ---
-     fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (& s) f (& A) )
-            → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc ))
-     fixpoint f mf zc = z14 where
+     fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (chainf : Ordinal → HOD) (zc : ZChain A (& s) f chainf (& A) )
+            → f (& (SUP.sup (sp0 f mf chainf zc ))) ≡ & (SUP.sup (sp0 f mf chainf zc ))
+     fixpoint f mf chainf zc = z14 where
            chain = ZChain.chain zc
-           sp1 = sp0 f mf zc
+           sp1 = sp0 f mf chainf zc
            z10 :  {a b : Ordinal } → (ca : odef chain a ) → b o< osuc (& A) → (ab : odef A b ) 
               →  HasPrev A chain ab f ∨  IsSup A chain {b} ab -- (supO  chain  (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b )
               → * a < * b  → odef chain b
@@ -360,7 +361,7 @@
                    ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p ))
                    ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p )
                    -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ?  (SUP.x<sup sp1 ? ) }
-           z14 :  f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc))
+           z14 :  f (& (SUP.sup (sp0 f mf chainf zc))) ≡ & (SUP.sup (sp0 f mf chainf zc))
            z14 with ZChain.f-total zc  (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12 
            ... | tri< a ¬b ¬c = ⊥-elim z16 where
                z16 : ⊥
@@ -381,12 +382,12 @@
      -- ZChain forces fix point on any ≤-monotonic function (fixpoint)
      -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain
      --
-     z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A (& s) (cf nmx)  (& A)) → ⊥
-     z04 nmx zc = <-irr0  {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal  sp1))))
+     z04 :  (nmx : ¬ Maximal A ) → (chainf : Ordinal → HOD) → (zc : ZChain A (& s) (cf nmx) chainf (& A)) → ⊥
+     z04 nmx chainf zc = <-irr0  {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal  sp1))))
                                                (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) )
-           (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ))) -- x ≡ f x ̄
+           (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) chainf zc ))) -- x ≡ f x ̄
            (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where          -- x < f x
-          sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc
+          sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) chainf zc
           c = & (SUP.sup sp1)
 
      --
@@ -400,18 +401,20 @@
           --
           -- we have previous ordinal to use induction
           --
-          prev : (z : Ordinal) → z o< x → {y : Ordinal} → odef A y → ZChain A y f z
+          prev : (z : Ordinal) → (z<x : z o< x ) → {y : Ordinal} → (ay : odef A y) → ZChain A y f (ZChain∧Chain.chainf (prevzc z z<x ay) ) z
           prev z z<x ay = ZChain∧Chain.zchain (prevzc z z<x ay)
           px = Oprev.oprev op
-          zc0 : ZChain A y f (Oprev.oprev op)
+          pchainf : Ordinal → HOD
+          pchainf = ZChain∧Chain.chainf (prevzc px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay)
+          zc0 : ZChain A y f pchainf (Oprev.oprev op)
           zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay
           zc0-b<x : (b : Ordinal ) → b o< x → b o< osuc px
           zc0-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt 
 
-          zc4 : ZChain A y f x 
+          zc4 : ZChain A y f {!!} x 
           zc4 with ODC.∋-p O A (* x)
           ... | no noax =  -- ¬ A ∋ p, just skip
-                 record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; initial = ZChain.initial zc0 
+                 record {  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 ; fc∨sup = {!!} }  where -- no extention
                 zc11 : {a b : Ordinal} → odef (ZChain.chain zc0) a → b o< osuc x → (ab : odef A b) →
@@ -429,13 +432,13 @@
                 zc17 {a} {b} za b<ox ab P a<b with osuc-≡< b<ox
                 ... | 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 y f x
+                zc9 :  ZChain A y f {!!} 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 -- 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 ; fc∨sup = {!!} } where 
+                record { chain⊆A = s⊆A  ; f-total = scmp ; f-next = scnext 
+                     ; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = {!!} ; 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)
@@ -549,15 +552,15 @@
                       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 with trio< x y
      ... | tri< a ¬b ¬c = record { zchain = 
-            record { chain = {!!} ; chain⊆A = {!!}  ; f-total = {!!}  ; f-next = {!!}
+            record { chain⊆A = {!!}  ; f-total = {!!}  ; f-next = {!!}
                      ; initial = {!!} ; f-immediate = {!!} ; chain∋x  = {!!} ; is-max = {!!} ; fc∨sup = {!!} }
             ; chainf = {!!} ; chain-mono = {!!} ; chain=zchain = {!!} }
      ... | tri≈ ¬a b ¬c = {!!}
      ... | tri> ¬a ¬b y<x = record { zchain = UnionZ 
             ; chainf = {!!} ; chain-mono = {!!} ; chain=zchain = {!!} } where
-       prev : (z : Ordinal) → z o< x → {y : Ordinal} → odef A y → ZChain A y f z
+       prev : (z : Ordinal) → (z<x : z o< x )→ {y : Ordinal} → (ay : odef A y) → ZChain A y f (ZChain∧Chain.chainf (prevzc z z<x ay))  z
        prev z z<x ay = ZChain∧Chain.zchain (prevzc z z<x ay)
-       UnionZ : ZChain A y f x
+       UnionZ : ZChain A y f {!!} 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 = {!!} ; fc∨sup = {!!} }  where --- limit ordinal case
          record UZFChain (z : Ordinal) : Set n where -- Union of ZFChain from y which has maximality o< x
@@ -567,7 +570,7 @@
                chain∋z : odef (ZChain.chain (prev u u<x {y} ay )) z
          Uz⊆A : {z : Ordinal} → UZFChain z → odef A z
          Uz⊆A {z} u = ZChain.chain⊆A ( prev (UZFChain.u u) (UZFChain.u<x u) {y} ay ) (UZFChain.chain∋z u)
-         uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f (UZFChain.u u)
+         uzc : {z : Ordinal} → (u : UZFChain z) → ZChain A y f {!!} (UZFChain.u u)
          uzc {z} u =  prev (UZFChain.u u) (UZFChain.u<x u) {y} ay
          Uz : HOD
          Uz = record { od = record { def = λ y → UZFChain y } ; odmax = & A
@@ -613,7 +616,7 @@
          zorn01  = proj1  zorn03  
          zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
          zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
-     ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where
+     ... | yes ¬Maximal = ⊥-elim ( z04 nmx chainf zorn04) where
          -- if we have no maximal, make ZChain, which contradict SUP condition
          nmx : ¬ Maximal A 
          nmx mx =  ∅< {HasMaximal} zc5 ( ≡o∅→=od∅  ¬Maximal ) where
@@ -621,8 +624,9 @@
               zc5 = ⟪  Maximal.A∋maximal mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫
          zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain∧Chain A (& s) f (& A)
          zorn03 f mf = TransFinite {λ z → {y : Ordinal } → (ya : odef A y ) → ZChain∧Chain  A y f z } (ind f mf) (& A)
-
-         zorn04 : ZChain A (& s) (cf nmx)  (& A)
+         chainf : Ordinal → HOD
+         chainf = ZChain∧Chain.chainf (zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ))
+         zorn04 : ZChain A (& s) (cf nmx) chainf (& A)
          zorn04 = ZChain∧Chain.zchain (zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ))
 
 -- usage (see filter.agda )