# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1656796251 -32400
# Node ID 9142e834c4c671870824aac2270c7ebb555902e8
# Parent  db9477c80dce10a82d639fe96f2a51fb058716ba
fix

diff -r db9477c80dce -r 9142e834c4c6 src/zorn.agda
--- a/src/zorn.agda	Sat Jul 02 07:52:05 2022 +0900
+++ b/src/zorn.agda	Sun Jul 03 06:10:51 2022 +0900
@@ -239,12 +239,12 @@
       A∋maximal : A ∋ sup
       x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup )   -- B is Total, use positive
 
-record UChain (chain : Ordinal → HOD) (x : Ordinal) (z : Ordinal) : Set n where 
+record UChain (x : Ordinal) (chain : (z : Ordinal ) → z o< x → HOD)  (z : Ordinal) : Set n where 
    -- Union of supf z which o< x
    field
       u : Ordinal
       u<x : u o< x
-      chain∋z : odef (chain u) z
+      chain∋z : odef (chain u u<x) z
 
 ∈∧P→o< :  {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A
 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
@@ -259,37 +259,19 @@
     ch-skip : {x : Ordinal } { chain : HOD } ( op : Oprev Ordinal osuc x ) ( ax : odef A x ) 
         ( c : Chain A f ay (Oprev.oprev op) chain) ( nh : ¬  HasPrev A chain ax f ) ( nsup : ¬ IsSup A chain ax ) → Chain A f ay x chain
     ch-union : {x : Ordinal } { chain : HOD } ( nop : ¬ Oprev Ordinal osuc x ) ( ax : odef A x ) 
-         → ( chainf : Ordinal → HOD ) → ( lt : ( z : Ordinal ) → z o< x → Chain A f ay z ( chainf z ))
+         → ( chainf : ( z : Ordinal ) → z o< x → HOD ) → ( lt : ( z : Ordinal ) → (z<x : z o< x ) → Chain A f ay z ( chainf z z<x ))
          → Chain A f ay x 
-             record { od = record { def = λ z → odef A z ∧ (UChain chainf x z ∨ FClosure A f y z ) } 
-                ; odmax = & A ; <odmax = λ {y} sy → {!!}   }
-
-Chain-uniq : (A : HOD ) ( f : Ordinal → Ordinal ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal)
-     → ( Ordinal → HOD ) → Set (Level.suc n)
-Chain-uniq A f {y} ay x chain  with Oprev-p x
-... | yes op = st1 where
-      px = Oprev.oprev op
-      st1 : Set (Level.suc n)
-      st1 with ODC.∋-p O A (* x)
-      ... | no noax = chain x ≡ chain px
-      ... | yes ax with ODC.p∨¬p O ( HasPrev A (chain px) ax f )   
-      ... | case1 pr = chain x ≡ chain px
-      ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (chain px) ax )
-      ... | case1 is-sup = chain x ≡ schain where
-            schain : HOD
-            schain = record { od = record { def = λ x → odef (chain px) x ∨ (FClosure A f y x) } ; odmax = & A ; <odmax = λ {y} sy → {!!}  }
-      ... | case2 ¬x=sup = chain x ≡ chain px
-... | no ¬ox = chain x ≡ record { od = record { def = λ z → odef A z ∧ ( UChain chain x z ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = λ {y} sy → {!!}   }
+             record { od = record { def = λ z → odef A z ∧ (UChain x chainf z ∨ FClosure A f y z ) } 
+                ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
 
 record ZChain1 ( A : HOD )    ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( z : Ordinal ) : Set (Level.suc n) where
    field
-      chain : Ordinal → HOD
-      chain-mono : {x : Ordinal} → x o≤ z → chain x ⊆' chain z 
-      chain-uniq : Chain-uniq A f ay z chain
+      chain : HOD
+      chain-uniq : Chain A f ay z chain 
 
-record ZChain ( A : HOD )    ( f : Ordinal → Ordinal ) {init : Ordinal} (ay : odef A init)  (zc0 : ZChain1 A f ay (& A) ) ( z : Ordinal ) : Set (Level.suc n) where
+record ZChain ( A : HOD )    ( f : Ordinal → Ordinal ) {init : Ordinal} (ay : odef A init)  (zc0 : (x : Ordinal) → ZChain1 A f ay x ) ( z : Ordinal ) : Set (Level.suc n) where
    chain : HOD
-   chain = ZChain1.chain zc0 z 
+   chain = ZChain1.chain (zc0 z)
    field
       chain⊆A : chain ⊆' A
       chain∋init : odef chain init
@@ -363,7 +345,7 @@
      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  ) ⟫
 
-     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : ZChain1 A f as0 (& A) ) (zc : ZChain A f as0 zc0 (& A) ) 
+     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : (x : Ordinal) → ZChain1 A f as0 x ) (zc : ZChain A f as0 zc0 (& A) ) 
         (total : IsTotalOrderSet (ZChain.chain zc) )  → SUP A (ZChain.chain zc)
      sp0 f mf zc0 zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total 
      zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
@@ -372,7 +354,7 @@
      ---
      --- the maximum chain  has fix point of any ≤-monotonic function
      ---
-     fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : ZChain1 A f as0 (& A)) (zc : ZChain A f as0 zc0 (& A) )
+     fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 : (x : Ordinal) → ZChain1 A f as0 x) (zc : ZChain A f as0 zc0 (& A) )
             → (total : IsTotalOrderSet (ZChain.chain zc) )
             → f (& (SUP.sup (sp0 f mf zc0 zc total ))) ≡ & (SUP.sup (sp0 f mf zc0 zc  total))
      fixpoint f mf zc0 zc total = z14 where
@@ -421,7 +403,7 @@
      -- 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 ) → (zc0 : ZChain1 A (cf nmx) as0 (& A)) (zc : ZChain A (cf nmx) as0 zc0 (& A)) → IsTotalOrderSet (ZChain.chain zc) → ⊥
+     z04 :  (nmx : ¬ Maximal A ) → (zc0 : (x : Ordinal) → ZChain1 A (cf nmx) as0 x) (zc : ZChain A (cf nmx) as0 zc0 (& A)) → IsTotalOrderSet (ZChain.chain zc) → ⊥
      z04 nmx zc0 zc total = <-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 ) zc0 zc total ))) -- x ≡ f x ̄
@@ -444,42 +426,33 @@
           px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc 
           sc : ZChain1 A f ay px
           sc = prev px px<x
-          no-ext : ZChain1 A f ay x
-          no-ext = record { chain = s01 ; chain-mono = ? ; chain-uniq = s02 } where
-                s01 : Ordinal → HOD
-                s01 z with trio< z x
-                ... | tri< a ¬b ¬c = chain (prev z a ) z
-                ... | tri≈ ¬a b ¬c = chain (prev px px<x ) px
-                ... | tri> ¬a ¬b c = chain (prev px px<x ) px
-                s02 : Chain-uniq A f ay x s01
-                s02 with trio< x x
-                ... | tri< a ¬b ¬c = ?
-                ... | tri≈ ¬a refl ¬c = ?
-                ... | tri> ¬a ¬b c = ?
           sc4 : ZChain1 A f ay x
           sc4 with ODC.∋-p O A (* x)
-          ... | no noax = {!!}
-          ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain1.chain sc x) ax f )   
+          ... | no noax = record { chain = ? ; chain-uniq = ? } 
+          ... | yes ax with ODC.p∨¬p O ( HasPrev A (ZChain1.chain sc ) ax f )   
           ... | case1 pr = {!!}
-          ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain1.chain sc x) ax )
+          ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain1.chain sc ) ax )
           ... | case1 is-sup = {!!} where
                 -- A∋sc -- x is a sup of zc 
-                sup0 : SUP A (ZChain1.chain sc x )
+                sup0 : SUP A (ZChain1.chain sc  )
                 sup0 = record { sup = * x ; A∋maximal = ax ; x<sup = x21 } where
-                        x21 :  {y : HOD} → (ZChain1.chain sc x) ∋ y → (y ≡ * x) ∨ (y < * x)
+                        x21 :  {y : HOD} → (ZChain1.chain sc ) ∋ y → (y ≡ * x) ∨ (y < * x)
                         x21 {y} zy with IsSup.x<sup is-sup zy 
                         ... | case1 y=x = case1 (subst₂ (λ j k → j ≡ * k ) *iso &iso ( cong (*) y=x)  )
                         ... | case2 y<x = case2 (subst₂ (λ j k → j < * k ) *iso &iso y<x  )
                 sp : HOD
                 sp = SUP.sup sup0 
                 schain : HOD
-                schain = record { od = record { def = λ x → odef (ZChain1.chain sc x) x ∨ (FClosure A f (& sp) x) } ; odmax = & A ; <odmax = λ {y} sy → {!!}  }
+                schain = record { od = record { def = λ x → odef A x ∧ ( odef (ZChain1.chain sc ) x ∨ (FClosure A f (& sp) x)) } 
+                    ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
           ... | case2 ¬x=sup = {!!}
      ... | no ¬ox = ? where
+          supf : (z : Ordinal) → z o< x → HOD
+          supf = ?
           sc5 : HOD
-          sc5 = record { od = record { def = λ z → odef A z ∧ (UChain ? x z ∨ FClosure A f y z)} ; odmax = & A ; <odmax = λ {y} sy → {!!}  }
+          sc5 = record { od = record { def = λ z → odef A z ∧ (UChain x supf z ∨ FClosure A f y z)} ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
 
-     ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → (zc0 : ZChain1 A f ay (& A)) 
+     ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → (zc0 : (x : Ordinal) → ZChain1 A f ay x) 
          → ((z : Ordinal) → z o< x → ZChain A f ay zc0 z) → ZChain A f ay zc0 x
      ind f mf {y} ay x zc0 prev with Oprev-p x
      ... | yes op = zc4 where
@@ -488,7 +461,7 @@
           --
           px = Oprev.oprev op
           supf : Ordinal → HOD
-          supf = ZChain1.chain zc0
+          supf x = ZChain1.chain (zc0 x)
           zc : ZChain A f ay zc0 (Oprev.oprev op)
           zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) 
           zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px
@@ -656,11 +629,11 @@
      ... | no ¬ox = record { chain⊆A = {!!} ; f-next = {!!} ; f-total = ?
                      ; initial = {!!} ; chain∋init  = {!!} ; is-max = {!!} }   where --- limit ordinal case
          supf : Ordinal → HOD
-         supf = ZChain1.chain zc0
-         uzc : {z : Ordinal} → (u : UChain supf x z) → ZChain A f ay zc0 (UChain.u u)
+         supf x = ZChain1.chain (zc0 x)
+         uzc : {z : Ordinal} → (u : UChain x ? z) → ZChain A f ay zc0 (UChain.u u)
          uzc {z} u =  prev (UChain.u u) (UChain.u<x u) 
          Uz : HOD
-         Uz = record { od = record { def = λ z → odef A z ∧ ( UChain supf z x ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = ?  }
+         Uz = record { od = record { def = λ z → odef A z ∧ ( UChain z ? x ∨ FClosure A f y z ) } ; odmax = & A ; <odmax = ?  }
          u-next : {z : Ordinal} → odef Uz z → odef Uz (f z)
          u-next {z} = ?
          -- (case1 u) = case1 record { u = UChain.u u ; u<x = UChain.u<x u ; chain∋z = ZChain.f-next ( uzc u ) (UChain.chain∋z u)  }
@@ -673,7 +646,7 @@
          u-chain∋init = ? -- case2 ( init ay )
          supf0 : Ordinal → HOD
          supf0 z with trio< z x
-         ... | tri< a ¬b ¬c = ZChain1.chain zc0 z
+         ... | tri< a ¬b ¬c = ZChain1.chain (zc0 z)
          ... | tri≈ ¬a b ¬c = Uz 
          ... | tri> ¬a ¬b c = Uz
          u-mono :  {z : Ordinal} {w : Ordinal} → z o≤ w → w o≤ x → supf0 z ⊆' supf0 w
@@ -685,9 +658,9 @@
          ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl )
          ... | tri≈ ¬a b ¬c = refl
          ... | tri> ¬a ¬b c = refl
-         seq<x : {b : Ordinal } → (b<x : b o< x ) →  ZChain1.chain zc0 b  ≡ supf0 b
+         seq<x : {b : Ordinal } → (b<x : b o< x ) →  ZChain1.chain (zc0 b)  ≡ supf0 b
          seq<x {b} b<x with trio< b x
-         ... | tri< a ¬b ¬c = cong (λ k → ZChain1.chain zc0 b) o<-irr --  b<x ≡ a
+         ... | tri< a ¬b ¬c = cong (λ k → ZChain1.chain (zc0 b)) o<-irr --  b<x ≡ a
          ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a  b<x )
          ... | tri> ¬a ¬b c  = ⊥-elim (¬a  b<x )
          ord≤< : {x y z : Ordinal} → x o< z → z o≤ y → x o< y
@@ -695,8 +668,8 @@
          ... | case1 z=y  = subst (λ k → x o< k ) z=y x<z
          ... | case2 z<y  = ordtrans x<z z<y
          
-     SZ0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain1 A f ay (& A)
-     SZ0 f mf ay = TransFinite {λ z → ZChain1 A f ay z} (sind f mf ay ) (& A)
+     SZ0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain1 A f ay x
+     SZ0 f mf ay x = TransFinite {λ z → ZChain1 A f ay z} (sind f mf ay ) x
 
      SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f ay (SZ0 f mf ay)  (& A)
      SZ f mf {y} ay = TransFinite {λ z → ZChain A f ay (SZ0 f mf ay)  z  } (λ x → ind f mf ay x (SZ0 f mf ay)  ) (& A)
@@ -717,8 +690,8 @@
          nmx mx =  ∅< {HasMaximal} zc5 ( ≡o∅→=od∅  ¬Maximal ) where
               zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) →  odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) 
               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) ) ⟫
-         zc0 : ZChain1 A  (cf nmx) as0 (& A)
-         zc0 = TransFinite {λ z → ZChain1 A (cf nmx) as0 z} (sind (cf nmx) (cf-is-≤-monotonic nmx)  as0) (& A)
+         zc0 : (x : Ordinal) → ZChain1 A  (cf nmx) as0 x
+         zc0 x = TransFinite {λ z → ZChain1 A (cf nmx) as0 z} (sind (cf nmx) (cf-is-≤-monotonic nmx)  as0) x
          zorn04 : ZChain A (cf nmx) as0 zc0 (& A)
          zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) 
          total : IsTotalOrderSet (ZChain.chain zorn04)