# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1655678975 -32400
# Node ID d0938f220648a701d87e5580096254c82d133a62
# Parent  7c5a922931e5f0b405c9e859acf3ff5bb9871ee0
supf again

diff -r 7c5a922931e5 -r d0938f220648 src/zorn.agda
--- a/src/zorn.agda	Fri Jun 17 21:45:33 2022 +0900
+++ b/src/zorn.agda	Mon Jun 20 07:49:35 2022 +0900
@@ -233,21 +233,13 @@
    field
       x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x )
 
-record SupF (A : HOD) : Set n where
-   field
-      chain : Ordinal
-      -- sup : Ordinal 
-      -- asup : odef A sup
-      -- is-sup :  IsSup A (* chain) asup
-
-record ZChain ( A : HOD )  (x : Ordinal)  ( f : Ordinal → Ordinal ) (supf : Ordinal → SupF A) ( z : Ordinal ) : Set (Level.suc n) where
+record ZChain ( A : HOD )  (x : Ordinal)  ( f : Ordinal → Ordinal ) (supf : Ordinal → HOD) ( z : Ordinal ) : Set (Level.suc n) where
    chain : HOD
-   chain = * (SupF.chain (supf z ))
+   chain = supf z 
    field
       chain⊆A : chain ⊆' A
       chain∋x : odef chain x
       initial : {y : Ordinal } → odef chain y → * x ≤ * y
-      f-total : IsTotalOrderSet chain 
       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) 
@@ -256,9 +248,10 @@
 
 record ZChain1 ( A : HOD )  (x : Ordinal)  ( f : Ordinal → Ordinal )  ( z : Ordinal ) : Set (Level.suc n) where
    field
-      supf : Ordinal → SupF A
+      supf : Ordinal → HOD
       zc : ZChain A x f supf z
-      chain-mono : {x y : Ordinal} → x o≤ y → y o< osuc z →  * (SupF.chain (supf x )) ⊆' * (SupF.chain (supf y ))
+      chain-mono : {x y : Ordinal} → x o≤ y → y o≤ z →  supf x ⊆' supf y 
+      f-total : {x y : Ordinal} → x o≤ z → IsTotalOrderSet (supf x) 
 
 record Maximal ( A : HOD )  : Set (Level.suc n) where
    field
@@ -332,13 +325,13 @@
      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) →  (zc1 : ZChain1 A (& s) f (& A) ) → SUP A  (ZChain.chain (ZChain1.zc zc1)) 
-     zsup f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain.f-total zc  )   where
+     zsup f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) ( ZChain1.f-total zc1 {!!} )   where
            zc = ZChain1.zc zc1
      A∋zsup : (nmx : ¬ Maximal A ) (zc1 : ZChain1 A (& s) (cf nmx)  (& A) ) 
         →  A ∋ * ( & ( SUP.sup (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 )))
      A∋zsup nmx zc1 = subst (λ k → odef A (& k )) (sym *iso) ( SUP.A∋maximal  (zsup (cf nmx) (cf-is-≤-monotonic nmx) zc1 ) )
      sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc1 : ZChain1 A (& s) f  (& A) ) → SUP A (ZChain.chain (ZChain1.zc zc1))
-     sp0 f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain.f-total zc)   where
+     sp0 f mf zc1 = supP (ZChain.chain zc) (ZChain.chain⊆A zc) (ZChain1.f-total zc1 {!!} )   where
            zc = ZChain1.zc zc1
      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)
@@ -375,7 +368,7 @@
                    ... | 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 zc1))) ≡ & (SUP.sup (sp0 f mf zc1))
-           z14 with ZChain.f-total zc  (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12 
+           z14 with ZChain1.f-total zc1 {!!}  (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12 
            ... | tri< a ¬b ¬c = ⊥-elim z16 where
                z16 : ⊥
                z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
@@ -418,7 +411,7 @@
           open ZChain
           
           px = Oprev.oprev op
-          supf : Ordinal → SupF A
+          supf : Ordinal → HOD
           supf = ZChain1.supf (prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay)
           zc1 : ZChain1 A y f (Oprev.oprev op)
           zc1 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) ay 
@@ -436,30 +429,29 @@
                             * a < * b → odef (ZChain.chain zc0) b ) → ZChain1 A y f x
           no-extenion is-max = record { supf = supf0 ; zc = record { chain⊆A = subst (λ k → k ⊆' A ) seq (ZChain.chain⊆A zc0)
                      ; initial = subst (λ k →  {y₁ : Ordinal} → odef k y₁ → * y ≤ * y₁ ) seq (ZChain.initial zc0)
-                     ; f-total = subst (λ k → IsTotalOrderSet k ) seq (ZChain.f-total zc0)
                      ; f-next =  subst (λ k → {a : Ordinal} → odef k a → odef k (f a) ) seq (ZChain.f-next zc0) 
                      ; f-immediate =  subst (λ k →  {x₁ : Ordinal} {y₁ : Ordinal} → odef k x₁ → odef k y₁ →
                             ¬ (* x₁ < * y₁) ∧ (* y₁ < * (f x₁)) ) seq (ZChain.f-immediate zc0) ; chain∋x  = subst (λ k → odef k y ) seq (ZChain.chain∋x  zc0)
                              ; is-max = subst (λ k → {a b : Ordinal} → odef k a → b o< osuc x → (ab : odef A b) →
                                  HasPrev A k ab f ∨  IsSup A k ab → * a < * b → odef k b  ) seq is-max } 
                      ; chain-mono = mono } where
-                supf0 : Ordinal → SupF A
+                supf0 : Ordinal → HOD
                 supf0 z with trio< z x
                 ... | tri< a ¬b ¬c = supf z
-                ... | tri≈ ¬a b ¬c = record { chain = & (ZChain.chain zc0) } 
-                ... | tri> ¬a ¬b c = record { chain = & (ZChain.chain zc0) } 
-                seq : ZChain.chain zc0 ≡ * (SupF.chain (supf0 x)) 
+                ... | tri≈ ¬a b ¬c = ZChain.chain zc0
+                ... | tri> ¬a ¬b c = ZChain.chain zc0 
+                seq : ZChain.chain zc0 ≡ supf0 x 
                 seq with trio< x x
                 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl )
-                ... | tri≈ ¬a b ¬c = sym *iso
-                ... | tri> ¬a ¬b c = sym *iso
-                seq<x : {b : Ordinal } → b o< x →  * (SupF.chain (supf b))  ≡ * (SupF.chain (supf0 b))
+                ... | tri≈ ¬a b ¬c = refl 
+                ... | tri> ¬a ¬b c = refl 
+                seq<x : {b : Ordinal } → b o< x →  supf b  ≡ supf0 b
                 seq<x {b} b<x with trio< b x
                 ... | tri< a ¬b ¬c = refl
                 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a  b<x )
                 ... | tri> ¬a ¬b c  = ⊥-elim (¬a  b<x )
                 mono : {a b  : Ordinal}  → a o≤ b → b o< osuc x →
-                    * (SupF.chain (supf0 a )) ⊆' * (SupF.chain (supf0 b ))
+                    supf0 a  ⊆' supf0 b 
                 mono {a} {b} a≤b b<ox with osuc-≡< a≤b
                 ... | case1 refl = λ x → x
                 ... | case2 a<b with osuc-≡< b<ox 
@@ -470,10 +462,10 @@
                      ... | case2 b<x = ordtrans a<b b<x
                      a≤px : a o≤  px
                      a≤px = subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) a<x
-                     nc00 : * (SupF.chain (supf px))  ≡ * (SupF.chain (supf0 b))
+                     nc00 : supf px  ≡ supf0 b
                      nc00 with trio< b x
                      ... | tri< a ¬b ¬c = ⊥-elim ( ¬b b=x )
-                     ... | tri≈ ¬a b ¬c = sym *iso
+                     ... | tri≈ ¬a b ¬c = refl
                      ... | tri> ¬a ¬b c = ⊥-elim ( ¬b b=x )
                 ... | case2 b<x = subst₂ (λ j k → j ⊆' k ) (seq<x a<x ) (seq<x b<x )
                             ( ZChain1.chain-mono zc1 a≤b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) b<x ) )
@@ -503,8 +495,8 @@
                 ... | case1 b=x = subst (λ k → odef chain0 k ) (trans (sym (HasPrev.x=fy pr )) (trans &iso (sym b=x)) ) ( ZChain.f-next zc0 (HasPrev.ay pr))
           ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc0) ax )
           ... | case1 is-sup = -- x is a sup of zc0 
-                record { zc = record { chain⊆A = {!!}  ; f-total = {!!} ; f-next = {!!} ; chain∋sup = {!!}
-                     ; initial = {!!} ; f-immediate =  {!!} ; chain∋x  = {!!} ; is-max = {!!} ; fc∨sup = {!!} }  ; supf = supf0 ; chain-mono = mono } where 
+                record { zc = record { chain⊆A = {!!}  ; f-total = {!!} ; f-next = {!!} 
+                     ; initial = {!!} ; f-immediate =  {!!} ; chain∋x  = {!!} ; is-max = {!!} }  ; supf = supf0 ; chain-mono = mono } 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)
@@ -542,7 +534,7 @@
                         a<b : a < b
                         a<b = ptrans  (subst (λ k → a < k ) (sym *iso) a<sp ) ( subst₂ (λ j k → j < k ) refl *iso sp<b )
                 scmp : {a b : HOD} → odef schain (& a) → odef schain (& b) → Tri (a < b) (a ≡ b) (b < a )
-                scmp (case1 za) (case1 zb) = ZChain.f-total zc0 za zb
+                scmp (case1 za) (case1 zb) = ZChain1.f-total zc1 {!!} za zb
                 scmp {a} {b} (case1 za) (case2 fb) = cmp za fb 
                 scmp  (case2 fa) (case1 zb) with  cmp zb fa
                 ... | tri< a ¬b ¬c = tri> ¬c (λ eq → ¬b (sym eq))  a
@@ -603,23 +595,22 @@
                      z23 with IsSup.x<sup (z24 p) ( ZChain.chain∋x zc0 )
                      ... | case1 y=b  = subst (λ k → odef chain0 k )  y=b ( ZChain.chain∋x zc0 )
                      ... | case2 y<b  = ZChain.is-max zc0 (ZChain.chain∋x zc0 ) (zc0-b<x b b<x) ab (case2 (z24 p)) y<b
-                supf0 : Ordinal → SupF A
+                supf0 : Ordinal → HOD
                 supf0 z with trio< z x
                 ... | tri< a ¬b ¬c = supf z
-                ... | tri≈ ¬a b ¬c = record { chain = & schain }
-                ... | tri> ¬a ¬b c = record { chain = & schain }
-                seq : schain ≡ * (SupF.chain (supf0 x))
+                ... | tri≈ ¬a b ¬c = schain 
+                ... | tri> ¬a ¬b c = schain
+                seq : schain ≡ supf0 x 
                 seq with trio< x x
                 ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl )
-                ... | tri≈ ¬a b ¬c = sym *iso
-                ... | tri> ¬a ¬b c = sym *iso
-                seq<x : {b : Ordinal } → b o< x →  * (SupF.chain (supf b))  ≡ * (SupF.chain (supf0 b))
+                ... | tri≈ ¬a b ¬c = refl
+                ... | tri> ¬a ¬b c = refl
+                seq<x : {b : Ordinal } → b o< x →  supf b  ≡ supf0 b
                 seq<x {b} b<x with trio< b x
                 ... | tri< a ¬b ¬c = refl
                 ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a  b<x )
                 ... | tri> ¬a ¬b c  = ⊥-elim (¬a  b<x )
-                mono : {a b  : Ordinal}  → a o≤ b → b o< osuc x →
-                    * (SupF.chain (supf0 a )) ⊆' * (SupF.chain (supf0 b ))
+                mono : {a b  : Ordinal}  → a o≤ b → b o< osuc x → supf0 a ⊆' supf0 b 
                 mono {a} {b} a≤b b<ox = {!!}
 
           ... | case2 ¬x=sup =  no-extenion z18 where -- x is not f y' nor sup of former ZChain from y -- no extention
@@ -655,32 +646,32 @@
          u-initial {z} u = ZChain.initial ( uzc u )  (UZFChain.chain∋z u)
          u-chain∋x :  odef Uz y
          u-chain∋x = record { u = y ; u<x = {!!} ; chain∋z = ZChain.chain∋x (ZChain1.zc (prev y {!!} ay )) }
-         supf0 : Ordinal → SupF A
+         supf0 : Ordinal → HOD
          supf0 z with trio< z x
          ... | tri< a ¬b ¬c = ZChain1.supf (prev z a {y} ay) z
-         ... | tri≈ ¬a b ¬c = record { chain = & Uz }
-         ... | tri> ¬a ¬b c = record { chain = & Uz }
-         seq : Uz ≡ * (SupF.chain (supf0 x))
+         ... | tri≈ ¬a b ¬c = Uz 
+         ... | tri> ¬a ¬b c = Uz
+         seq : Uz ≡ supf0 x
          seq with trio< x x
          ... | tri< a ¬b ¬c = ⊥-elim ( ¬b refl )
-         ... | tri≈ ¬a b ¬c = sym *iso
-         ... | tri> ¬a ¬b c = sym *iso
-         seq<x : {b : Ordinal } → (b<x : b o< x ) →  * (SupF.chain (ZChain1.supf (prev b b<x {y} ay) b))  ≡ * (SupF.chain (supf0 b))
+         ... | tri≈ ¬a b ¬c = refl
+         ... | tri> ¬a ¬b c = refl
+         seq<x : {b : Ordinal } → (b<x : b o< x ) →  ZChain1.supf (prev b b<x {y} ay) b  ≡ supf0 b
          seq<x {b} b<x with trio< b x
-         ... | tri< a ¬b ¬c = cong (λ k → * (SupF.chain (ZChain1.supf (prev b k {y} ay) b)) ) o<-irr --  b<x ≡ a
+         ... | tri< a ¬b ¬c = cong (λ k → ZChain1.supf (prev b k {y} ay) b) o<-irr --  b<x ≡ a
          ... | tri≈ ¬a b₁ ¬c = ⊥-elim (¬a  b<x )
          ... | tri> ¬a ¬b c  = ⊥-elim (¬a  b<x )
          u-mono :  ( a b : Ordinal ) → b o< x → a o< osuc b → (za : ZChain1 A y f a) (zb : ZChain1 A y f b) → ZChain.chain (ZChain1.zc za)  ⊆' ZChain.chain (ZChain1.zc zb)
          u-mono a b b<x a≤b za zb {i} zai = ZChain1.chain-mono zb a≤b <-osuc (uz01 zai) where
-              uz01 : odef (* (SupF.chain (ZChain1.supf za a))) i → odef (* (SupF.chain (ZChain1.supf zb a))) i
+              uz01 : odef (ZChain1.supf za a) i → odef (ZChain1.supf zb a) i
               uz01 = {!!}
          u-total : IsTotalOrderSet Uz
          u-total {x} {y} ux uy  with trio< (UZFChain.u ux) (UZFChain.u uy)
-         ... | tri< a ¬b ¬c = ZChain.f-total (uzc uy) (u-mono (UZFChain.u ux) (UZFChain.u uy)
+         ... | tri< a ¬b ¬c = ZChain1.f-total (uzc1 uy) {!!} (u-mono (UZFChain.u ux) (UZFChain.u uy)
             (UZFChain.u<x uy) (ordtrans a <-osuc ) (uzc1 ux) (uzc1 uy) (UZFChain.chain∋z ux)) (UZFChain.chain∋z uy)
-         ... | tri≈ ¬a b ¬c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux)
+         ... | tri≈ ¬a b ¬c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux)
             (UZFChain.u<x ux) (subst (λ k → k o< osuc (UZFChain.u ux)) b <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy))
-         ... | tri> ¬a ¬b c = ZChain.f-total (uzc ux) (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux)
+         ... | tri> ¬a ¬b c = ZChain1.f-total (uzc1 ux) {!!} (UZFChain.chain∋z ux) (u-mono (UZFChain.u uy) (UZFChain.u ux)
             (UZFChain.u<x ux) (ordtrans c <-osuc) (uzc1 uy) (uzc1 ux) (UZFChain.chain∋z uy)) 
          
      SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ya : odef A y) → ZChain1 A y f (& A)