# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1651024903 -32400
# Node ID 3234a5f6bfcf4192f5bb76a0bb12e2ab59023063
# Parent  f8eb56442f2ca4cb95735f86b40a5ea103ecc8f8
...

diff -r f8eb56442f2c -r 3234a5f6bfcf src/zorn.agda
--- a/src/zorn.agda	Wed Apr 27 09:08:25 2022 +0900
+++ b/src/zorn.agda	Wed Apr 27 11:01:43 2022 +0900
@@ -191,17 +191,16 @@
 SupCond : ( A B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → Set (Level.suc n)
 SupCond A B _ _ = SUP A B  
 
-record ZChain ( A : HOD ) {x : Ordinal} (ax : A ∋ * x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
-         (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) (z : Ordinal)  : Set (Level.suc n) where
+record ZChain ( A : HOD )  {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
+                 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) : Set (Level.suc n) where
    field
       chain : HOD
       chain⊆A : chain ⊆ A
       chain∋x : odef chain x
-      ¬chain∋x>z : { a : Ordinal } → z o< osuc a → ¬ odef chain a
       f-total : IsTotalOrderSet chain 
-      f-next : {a : Ordinal } → odef chain a → a o< z  → odef chain (f a)
+      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 ) → (ba : odef A b) → a o< z
+      is-max :  {a b : Ordinal } → (ca : odef chain a ) → (ba : odef A b) 
           → Prev< A chain ba f
                ∨  (sup (& chain) (subst (λ k → k  ⊆ A) (sym *iso) chain⊆A)  (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b )
           → * a < * b  → odef chain b
@@ -224,8 +223,6 @@
      s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) 
      sa : A ∋ * ( & s  )
      sa =  subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))  )
-     sa0 : odef A (& s)
-     sa0 =  subst (λ k → odef A (& k) ) {!!} ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))  )
      HasMaximal : HOD
      HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) →  odef A m → ¬ (* x < * m)) }  ; odmax = & A ; <odmax = z07 } 
      no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) →  odef A m →  odef A x ∧ (¬ (* x < * m) )) →  ⊥
@@ -259,12 +256,12 @@
      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 sa f mf supO (& A)) → SUP A  (ZChain.chain zc) 
+     zsup :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f) →  (zc : ZChain A sa f mf supO ) → 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 sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) 
+     A∋zsup :  (nmx : ¬ Maximal A ) (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO ) 
         →  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 sa f mf supO (& A)) → SUP A (* (& (ZChain.chain zc)))
+     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO ) → SUP A (* (& (ZChain.chain zc)))
      sp0 f mf zc = supP (* (& (ZChain.chain zc))) (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc))
                (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc) )
      zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
@@ -273,12 +270,12 @@
      ---
      --- sup is fix point in maximum chain
      ---
-     z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A sa f mf supO (& A))
-            → f (& (SUP.sup (sp0 f mf zc  ))) ≡ & (SUP.sup (sp0 f mf zc  ))
+     z03 :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) f mf supO )
+            → f (& (SUP.sup (sp0 f mf zc ))) ≡ & (SUP.sup (sp0 f mf zc ))
      z03 f mf zc = z14 where
            chain = ZChain.chain zc
            sp1 = sp0 f mf zc
-           z10 :  {a b : Ordinal } → (ca : odef chain a ) → (ab : odef A b ) → a o< (& A)
+           z10 :  {a b : Ordinal } → (ca : odef chain a ) → (ab : odef A b ) 
               →  Prev< A chain ab f
                    ∨  (supO (& chain) (subst (λ k → k  ⊆ A) (sym *iso) (ZChain.chain⊆A zc))  (subst (λ k → IsTotalOrderSet k) (sym *iso) (ZChain.f-total zc)) ≡ b )
               → * a < * b  → odef chain b
@@ -288,13 +285,13 @@
            z12 : odef chain (& (SUP.sup sp1))
            z12 with o≡? (& s) (& (SUP.sup sp1))
            ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋x zc )
-           ... | no ne = z10 {& s} {& (SUP.sup sp1)} (ZChain.chain∋x zc) (SUP.A∋maximal sp1) (c<→o< (subst (λ k → odef A (& k) ) *iso sa) ) (case2 refl ) z13 where
+           ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋x zc ) (SUP.A∋maximal sp1)  (case2 refl ) z13 where
                z13 :  * (& s) < * (& (SUP.sup sp1))
                z13 with SUP.x<sup sp1 (subst (λ k → odef k (& s)) (sym *iso) ( ZChain.chain∋x zc ))
                ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
                ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
            z14 :  f (& (SUP.sup (sp0 f mf zc))) ≡ & (SUP.sup (sp0 f mf zc))
-           z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso) (ZChain.f-next zc z12 z11 ))) (me z12 )
+           z14 with IsStrictTotalOrder.compare (ZChain.f-total zc ) (me (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 ))) (me z12 )
            ... | tri< a ¬b ¬c = ⊥-elim z16 where
                z16 : ⊥
                z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
@@ -303,83 +300,71 @@
            ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b )
            ... | tri> ¬a ¬b c = ⊥-elim z17 where
                z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) <  SUP.sup sp1)
-               z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 z11 ) )
+               z15  = SUP.x<sup sp1 (subst₂ (λ j k → odef j k ) (sym *iso) (sym &iso)  (ZChain.f-next zc z12 ))
                z17 : ⊥
                z17 with z15
                ... | case1 eq = ¬b eq
                ... | case2 lt = ¬a lt
-     z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A sa (cf nmx) (cf-is-≤-monotonic nmx) supO (& A)) → ⊥
+     z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO ) → ⊥
      z04 nmx zc = z01  {* (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 (*)( z03 (cf nmx) (cf-is-≤-monotonic nmx ) zc )))
            (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1))) where
-          sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc 
+          sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc
           c = & (SUP.sup sp1)
-     premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 :  ZChain A sa f mf supO y ) 
+     premax : {x y : Ordinal} → y o< x → ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc0 :  ZChain A sa f mf supO  ) 
         → {a b : Ordinal} (ca : odef (ZChain.chain zc0) a) → (ab : odef A b) → a o< y
         →  Prev< A (ZChain.chain zc0) ab f ∨ (supO (& (ZChain.chain zc0))
              (subst (λ k → k ⊆ A) (sym *iso) (ZChain.chain⊆A zc0))
              (subst IsTotalOrderSet (sym *iso) (ZChain.f-total zc0)) ≡ b)
        → * a < * b → odef (ZChain.chain zc0) b
-     premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab a<y P a<b
+     premax {x} {y} y<x  f mf zc0 {a} {b} ca ab a<y P a<b = ZChain.is-max zc0 ca ab P a<b -- ca ab y P a<b
      -- Union of ZFChain
      UZFChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (B : Ordinal) 
-            → ( (y : Ordinal) → y o< B → ZChain A sa f mf supO y ) → HOD
-     UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B)  ∧ ( (y<b : y o< B) → odef (ZChain.chain (prev y y<b)) y) }
+            → ( (y : Ordinal) → y o< B → (ya : odef A y) → ZChain A ya f mf supO ) → HOD
+     UZFChain f mf B prev = record { od = record { def = λ y → odef A y ∧ (y o< B)  ∧ ( (y<b : y o< B) → (ya : odef A y) → odef (ZChain.chain (prev y y<b ya )) y) }
          ; odmax = & A ; <odmax = z07 }
      -- ZChain is not compatible with the SUP condition
-     ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) → ((y : Ordinal) → y o< x →  ZChain A sa f mf supO y )
-         →  ZChain A sa f mf supO x 
-     ind f mf x prev with Oprev-p x
-     ... | yes op with ODC.∋-p O A (* x)
-     ... | no ¬Ax = zc1 where
-          -- we have previous ordinal and ¬ A ∋ x, use previous Zchain
+     ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) →
+            ((y : Ordinal) → y o< x → (ya : odef A y) → ZChain A ya f mf supO) → (ya : odef A x) → ZChain A ya f mf supO
+     ind f mf x prev ax with Oprev-p x
+     ... | yes op with ODC.∋-p O A (* (Oprev.oprev op))
+     ... | yes apx = zc4 where -- we have previous ordinal and A ∋ op
           px = Oprev.oprev op
-          zc0 : ZChain A sa f mf supO (Oprev.oprev op) 
-          zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
-          zc1 : ZChain A sa f mf supO x 
-          zc1 = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
-             ; f-next = zc20 (ZChain.f-next zc0) ; f-immediate =  ZChain.f-immediate zc0
-             ; ¬chain∋x>z =  λ {a} x<oa → ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa )
-             ; chain∋x  = ZChain.chain∋x zc0 ; is-max = λ za ba a<x → zc20 (λ za a<x → ZChain.is-max zc0 za ba a<x ) za a<x } where
-              zc20 : {P : Ordinal →  Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a)
-                 → {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) →  P a
-              zc20 {P} prev {a} za a<x with trio< a px
-              ... | tri< a₁ ¬b ¬c = prev za a₁ 
-              ... | tri≈ ¬a b ¬c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (subst (λ k → k o< osuc a) b <-osuc ) za )
-              ... | tri> ¬a ¬b c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (ordtrans c <-osuc ) za )
-     ... | yes ax = zc4 where -- we have previous ordinal and A ∋ x
-          px = Oprev.oprev op
-          zc0 : ZChain A sa f mf supO (Oprev.oprev op) 
-          zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc) 
+          apx0 = subst (λ k → odef A k ) &iso apx 
+          zc0 : ZChain A apx0 f mf supO 
+          zc0 = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) apx0
+          ax0 : odef A (& (* x))
+          ax0 = {!!}
           Afx : { x : Ordinal } → A ∋ * x → A ∋ * (f x)
           Afx {x} ax = (subst (λ k → odef A k ) (sym &iso) (proj2 (mf x (subst (λ k → odef A k ) &iso ax))))
           --   x is in the previous chain, use the same
           --   x has some y which y < x ∧ f y ≡ x
           --   x has no y which y < x 
-          zc4 : ZChain A sa f mf supO x
+          zc4 : ZChain A ax f mf supO 
           zc4 with ODC.p∨¬p O ( Prev< A (ZChain.chain zc0) ax f )
           ... | case1 y = zc7 where -- we have previous <
                 chain = ZChain.chain zc0
-                zc7 :  ZChain A sa f mf supO x
+                zc7 :  ZChain A ax f mf supO 
                 zc7 with ODC.∋-p O  (ZChain.chain zc0) (* ( f x ) )
-                ... | yes y = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next =  zc20 (ZChain.f-next zc0)
-                     ; f-immediate =  ZChain.f-immediate zc0 ; ¬chain∋x>z = z22 ; chain∋x  =  ZChain.chain∋x zc0 ; is-max = λ za ba a<x → zc20 (λ za a<x → ZChain.is-max zc0 za ba a<x ) za a<x  }  where -- no extention
+                ... | yes y = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A 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 = {!!} }  where -- no extention
                     z22 : {a : Ordinal} → x o< osuc a → ¬ odef (ZChain.chain zc0) a
-                    z22 {a} x<oa = ZChain.¬chain∋x>z zc0 (ordtrans (subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc ) x<oa )
+                    z22 {a} x<oa = {!!}
                     zc20 : {P : Ordinal →  Set n} → ({a : Ordinal} → odef (ZChain.chain zc0) a → a o< px → P a)
                        → {a : Ordinal} → (za : odef (ZChain.chain zc0) a ) → (a<x : a o< x) →  P a
                     zc20 {P} prev {a} za a<x with trio< a px
                     ... | tri< a₁ ¬b ¬c = prev za a₁
-                    ... | tri≈ ¬a b ¬c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (subst (λ k → k o< osuc a) b <-osuc ) za )
-                    ... | tri> ¬a ¬b c = ⊥-elim ( ZChain.¬chain∋x>z zc0 (ordtrans c <-osuc ) za )
+                    ... | tri≈ ¬a b ¬c = {!!}
+                    ... | tri> ¬a ¬b c = {!!}
                     z21 :  {a : Ordinal} → odef (ZChain.chain zc0) a → a o< x → odef (ZChain.chain zc0) (f a)
                     z21 {a} za a<x with trio< a x
-                    ... | tri< a₁ ¬b ¬c = ZChain.f-next zc0 za {!!} 
+                    ... | tri< a₁ ¬b ¬c = ZChain.f-next zc0 za 
                     ... | tri≈ ¬a b ¬c = {!!}
                     ... | tri> ¬a ¬b c = ⊥-elim ( o<> c a<x )
                 ... | no not = record { chain = zc5 ; chain⊆A =  ⊆-zc5
-                    ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x  = case1 (ZChain.chain∋x zc0) ; ¬chain∋x>z = {!!} ; is-max = {!!} } where
+                    ; f-total = zc6 ; f-next = {!!} ; f-immediate = {!!} ; chain∋x  = case1 {!!} ; ¬chain∋x>z = {!!} ; is-max = {!!} } where
                 --   extend with f x -- cahin ∋ y ∧  chain ∋ f y ∧ x ≡ f ( f y )
                     zc5 : HOD
                     zc5 = record { od = record { def = λ z → odef (ZChain.chain zc0) z ∨ (z ≡ f x) } ; odmax = & A ; <odmax = {!!} }
@@ -387,16 +372,16 @@
                     ⊆-zc5 = record { incl = λ {y} lt → zc15 lt } where
                         zc15 : {z : Ordinal } → ( odef (ZChain.chain zc0) z ∨ (z ≡ f x) ) → odef A z
                         zc15 {z} (case1 zz) = subst (λ k → odef A k ) &iso ( incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain  k ) (sym &iso) zz ) )
-                        zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso ax ) )
+                        zc15 (case2 refl) = proj2 ( mf x (subst (λ k → odef A k ) &iso {!!} ) )
                     IPO = ⊆-IsPartialOrderSet  ⊆-zc5 PO
                     zc8 : { A B x : HOD } → (ax : A ∋ x ) → (P : Prev< A B ax f ) → * (f (& (* (Prev<.y P)))) ≡ x
                     zc8 {A} {B} {x} ax P = subst₂ (λ j k → * ( f j ) ≡ k ) (sym &iso) *iso (sym (cong (*) ( Prev<.x=fy P)))
                     fx=zc :  odef (ZChain.chain zc0) x → Tri  (* (f x) < * x ) (* (f x) ≡ * x) (* x < * (f x) )
-                    fx=zc  c with mf x (subst (λ k → odef A k) &iso  ax )
-                    ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax (Afx ax) (case1 (sym zc13))) zc13 (z01 (Afx ax) ax (case1 zc13)) where
+                    fx=zc  c with mf x (subst (λ k → odef A k) &iso  ax0 )
+                    ... | ⟪ case1 x=fx , afx ⟫ = tri≈ ( z01 ax0 (Afx ax0) (case1 (sym zc13))) zc13 (z01 (Afx ax0) ax0 (case1 zc13)) where
                         zc13 : * (f x) ≡ * x
                         zc13 = subst (λ k → k ≡ * x ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso))) (sym ( x=fx ))
-                    ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax (Afx ax) (case2 zc14)) (λ lt → z01 (Afx ax) ax (case1 lt) zc14) zc14 where
+                    ... | ⟪ case2 x<fx , afx ⟫ = tri> (z01 ax0 (Afx ax0) (case2 zc14)) (λ lt → z01 (Afx ax0) ax0 (case1 lt) zc14) zc14 where
                         zc14 : * x < * (f x)
                         zc14 = subst (λ k → * x < k ) (subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym &iso ))) x<fx
                     cmp : Trichotomous _ _ 
@@ -410,7 +395,7 @@
                     ... | tri< a₁ ¬b ¬c = {!!}
                     ... | tri≈ ¬a b₁ ¬c = subst₂ (λ j k → Tri ( j < k ) (j ≡ k) ( k < j ) ) zc11 zc10 ( fx=zc zc12 ) where
                          zc10 : * x ≡ b
-                         zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax y ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) 
+                         zc10 = subst₂ (λ j k → j ≡ k ) (zc8 ax {!!} ) (zc8 (incl ( ZChain.chain⊆A zc0 ) c) fb) (cong (λ k → * ( f ( & k ))) b₁) 
                          zc11 : * (f x) ≡ a
                          zc11 = subst (λ k → * (f x) ≡ k ) *iso (cong (*) (sym fx))
                          zc12 : odef chain x
@@ -422,15 +407,13 @@
           ... | case2 not with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) {!!} {!!} ) ))
           ... | case1 y = {!!} -- x is sup
           ... | case2 not = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = {!!}
-                     ; f-immediate = {!!} ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} }  -- no extention
-     ind f mf x prev | no ¬ox with trio< (& A) x   --- limit ordinal case
-     ... | tri< a ¬b ¬c = record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0
-              ; f-next = {!!}
-              ; f-immediate = {!!} ; chain∋x  = ZChain.chain∋x zc0 ; is-max = {!!} } where
+                     ; f-immediate = {!!} ; chain∋x  = {!!} ; is-max = {!!} }  -- no extention
+     ... | no noapx = {!!} -- we have previous ordinal but ¬ A ∋ op
+     ind f mf x prev ya | no ¬ox with trio< (& A) x   --- limit ordinal case
+     ... | tri< a ¬b ¬c = {!!} where
           zc0 = prev (& A) a
      ... | tri≈ ¬a b ¬c = {!!}
-     ... | tri> ¬a ¬b c =  record { chain = uzc ; chain⊆A = record { incl = λ {x} lt → proj1 lt } ; f-total = {!!} ; f-next =  {!!}
-              ; f-immediate = {!!} ; chain∋x  = {!!}  ; is-max = {!!} } where
+     ... | tri> ¬a ¬b c =  {!!} where
          uzc : HOD
          uzc = UZFChain f mf x prev
      zorn00 : Maximal A 
@@ -443,32 +426,16 @@
          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  (zorn03 (cf nmx) (cf-is-≤-monotonic nmx))) where
+     ... | yes ¬Maximal = ⊥-elim ( z04 nmx zorn04) where
          -- if we have no maximal, make ZChain, which contradict SUP condition
          nmx : ¬ Maximal A 
          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) ) ⟫
-         record ZChain1 ( A : HOD )  {x : Ordinal} (ax : odef A x) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f )
-                 (sup : (C : Ordinal ) → (* C ⊆ A) → IsTotalOrderSet (* C) → Ordinal) : Set (Level.suc n) where
-           field
-              chain : HOD
-              chain⊆A : chain ⊆ A
-              chain∋x : odef chain x
-              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 ) → (ba : odef A b) 
-                  → Prev< A chain ba f
-                       ∨  (sup (& chain) (subst (λ k → k  ⊆ A) (sym *iso) chain⊆A)  (subst (λ k → IsTotalOrderSet k) (sym *iso) f-total) ≡ b )
-                  → * a < * b  → odef chain b
-         ind4 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (x : Ordinal) →
-            ((y : Ordinal) → y o< x → odef A y ∧ ((ya : odef A y) → ZChain1 A ya f mf supO)) → odef A x ∧ ((ya : odef A x) → ZChain1 A ya f mf supO)
-         ind4 = {!!}
-         zorn04 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → odef A (& s) ∧ ((ya : odef A (& s)) → ZChain1 A ya f mf supO )
-         zorn04 f mf = TransFinite {λ y →  odef A y ∧ ( (ya : odef A y ) → ZChain1 A ya f mf supO ) } (ind4 f mf)  (& s ) 
-         zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → ZChain A sa f mf supO (& A)
-         zorn03 f mf = TransFinite (ind f mf)  (& A) 
+         zorn03 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → (ya : odef A (& s)) → ZChain A ya f mf supO 
+         zorn03 f mf = TransFinite {λ y →  (ya : odef A y ) → ZChain A ya f mf supO  } (ind f mf)  (& s ) 
+         zorn04 : ZChain A (subst (λ k → odef A k ) &iso sa ) (cf nmx) (cf-is-≤-monotonic nmx) supO
+         zorn04 = zorn03 (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso sa )
 
 -- usage (see filter.agda )
 --