# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1661218427 -32400
# Node ID 3fa321cbc337f8dd0bccd3273935f9535062177d
# Parent  e61cbf28ec3121fbafb8e106ed6bfd902cc9fd1d
... dead end

diff -r e61cbf28ec31 -r 3fa321cbc337 src/zorn.agda
--- a/src/zorn.agda	Mon Aug 22 22:07:02 2022 +0900
+++ b/src/zorn.agda	Tue Aug 23 10:33:47 2022 +0900
@@ -649,6 +649,11 @@
      SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B
      SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt)    }
 
+     record xSUP (B : HOD) (x : Ordinal) : Set n where
+        field
+           ax : odef A x
+           is-sup : IsSup A B ax
+
      --
      -- create all ZChains under o< x
      --
@@ -690,166 +695,47 @@
 
           supf0 = ZChain.supf zc
 
-          sup1 : SUP A (UnionCF A f mf ay supf0 px)
-          sup1 = supP pchain pchain⊆A ptotal
-          sp1 = & (SUP.sup sup1 )
-          supf1 : Ordinal → Ordinal
-          supf1 z with trio< z px
-          ... | tri< a ¬b ¬c = ZChain.supf zc z
-          ... | tri≈ ¬a b ¬c = sp1 
-          ... | tri> ¬a ¬b c = sp1 
-
-          pchain1 : HOD
-          pchain1  = UnionCF A f mf ay supf1 x
-          pcy1 : odef pchain1 y
-          pcy1 = ⟪ ay , ch-init (init ay refl)    ⟫
-          pinit1 :  {y₁ : Ordinal} → odef pchain1 y₁ → * y ≤ * y₁
-          pinit1 {a} ⟪ aa , ua ⟫  with  ua
-          ... | ch-init fc = s≤fc y f mf fc
-          ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc)  where
-               zc7 : y <= supf1 u 
-               zc7 = ChainP.fcy<sup is-sup (init ay refl)
-          pnext1 : {a : Ordinal} → odef pchain1 a → odef pchain1 (f a)
-          pnext1 {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc)  ⟫
-          pnext1 {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫
-          ptotal1 : IsTotalOrderSet pchain1
-          ptotal1 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where 
-               uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-               uz01 = chain-total A f mf ay supf1 ( (proj2 ca)) ( (proj2 cb)) 
-
-          zc64 : {z : Ordinal } → supf0 z o< supf0 px → odef (UnionCF A f mf ay supf0 px) (supf0 z)
-          zc64 {z} sz<spx = zc73 where
-               z<px = ZChain.supf-inject zc sz<spx
-               zc70 : odef (UnionCF A f mf ay supf0 (supf0 z) ) (supf0 z)
-               zc70 = ZChain.csupf zc (o<→≤ z<px )
-               zc73 : odef (UnionCF A f mf ay supf0 px ) (supf0 z)
-               zc73 with osuc-≡< (ZChain.supf-mono zc (o<→≤ z<px))
-               ... | case1 eq2 = ⊥-elim ( o<¬≡ eq2 sz<spx )
-               ... | case2 lt = subst (λ k → odef (UnionCF A f mf ay supf0 px ) k ) &iso ( ZChain.csupf-fc zc  o≤-refl lt (init (proj1 zc70) refl) )
-
-          supf1<sp1 : {z : Ordinal } → supf1 z o≤ sp1
-          supf1<sp1 {z} = ? where
-               zc50 : supf0 px ≡ sp1
-               zc50 = ? -- ZChain.sup=u zc ? ? ?
-               zc53 : SUP A ( UnionCF A f mf ay supf0 px )
-               zc53 = ZChain.sup zc o≤-refl 
-               zc52 : supf0 px ≡ ?
-               zc52 = ? -- ZChain.sup=u zc ? ? ?
-               zc51 : supf0 sp1 ≡ sp1
-               zc51 = ZChain.sup=u zc ? ? ?
-
           -- if previous chain satisfies maximality, we caan reuse it
           --
           --  supf0 px is sup of UnionCF px , supf0 x is sup of UnionCF x 
 
-          record xSUP : Set n where
-            field
-               ax : odef A x
-               is-sup : IsSup A (UnionCF A f mf ay supf0 px) ax
-
-          UnionCF⊆ : {z0 z1 : Ordinal} → (z0≤1 : z0 o≤ z1 ) →  (z0≤px :  z0 o< px ) → UnionCF A f mf ay supf0 z0 ⊆' UnionCF A f mf ay supf1 z1 
-          UnionCF⊆ {z0} {z1} z0≤1 z0<px ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
-          UnionCF⊆ {z0} {z1} z0≤1 z0<px ⟪ au , ch-is-sup u1 {w} u1≤x u1-is-sup fc ⟫   = zc60 fc where
-              zc60 : {w : Ordinal } → FClosure A f (supf0 u1) w → odef (UnionCF A f mf ay supf1 z1 ) w
-              zc60 (init asp refl) with trio< u1 px | inspect supf1 u1
-              ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc  , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) 
-                record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 (ChainP.supu=u u1-is-sup)   } (init (subst (λ k → odef A k ) (sym eq1) asp) eq1 )  ⟫ where
-                  fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 )
-                  fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) ( ChainP.fcy<sup u1-is-sup fc )
-                  order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 →
-                    (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1)
-                  order {s} {z2} s<u1 fc with trio< s px 
-                  ... | tri< a ¬b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup zc61 fc ) where
-                      zc61 : supf0 s o< supf0 u1
-                      zc61 = subst (λ k → supf0 s o< k ) eq1 s<u1
-                  ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> supf1<sp1 s<u1 )
-                  ... | tri> ¬a ¬b px<s = ⊥-elim ( o<¬≡ refl (ordtrans px<s {!!} ))  --  px o< s < u1 < px
-              ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc  , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) 
-                record { fcy<sup = fcy<sup ; order = order ; supu=u = trans eq1 {!!}   } (init (subst (λ k → odef A k ) (sym eq1) {!!} ) {!!} )  ⟫ where
-                  fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf1 u1) ∨ (z << supf1 u1 )
-                  fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) (sym eq1) {!!} -- ( ChainP.fcy<sup u1-is-sup fc )
-                  order : {s : Ordinal} {z2 : Ordinal} → supf1 s o< supf1 u1 → FClosure A f (supf1 s) z2 →
-                    (z2 ≡ supf1 u1) ∨ (z2 << supf1 u1)
-                  order {s} {z2} s<u1 fc with trio< s px 
-                  ... | tri< a ¬b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) {!!}
-                  ... | tri≈ ¬a b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) {!!} -- ( ChainP.order u1-is-sup s<u1 fc )
-                  ... | tri> ¬a ¬b px<s = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k) b {!!} ) ))  --  px o< s < u1 = px
-              ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with osuc-≡< (OrdTrans u1≤x (o<→≤ z0<px))
-              ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) 
-              ... | case2 lt = ⊥-elim ( o<> lt px<u1 )
-              zc60 (fsuc w1 fc) with zc60 fc
-              ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫ 
-              ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ 
-          no-extension : ¬ xSUP → ZChain A f mf ay x
-          no-extension ¬sp=x = record { supf = supf1 ;  sup = sup ; supf-mono = {!!}
-               ; initial = pinit1 ; chain∋init = pcy1 ; sup=u = sup=u ; supf-is-sup = sis ; csupf = {!!}
-              ;  chain⊆A = λ lt → proj1 lt ;  f-next = pnext1 ;  f-total = ptotal1 }  where
-                 UnionCFR⊆ : {z0 z1 : Ordinal} → z0 o≤ z1 → z0 o< x → UnionCF A f mf ay supf1 z0 ⊆' UnionCF A f mf ay supf0 z1 
-                 UnionCFR⊆ {z0} {z1} z0≤1 z0<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
-                 UnionCFR⊆ {z0} {z1} z0≤1 z0<x ⟪ au , ch-is-sup u1 {w} u1≤x u1-is-sup fc ⟫   = zc60 fc where
-                      zc60 : {w : Ordinal } → FClosure A f (supf1 u1) w → odef (UnionCF A f mf ay supf0 z1 ) w
-                      zc60 {w} (init asp refl) with trio< u1 px | inspect supf1 u1
-                      ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc  , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) 
-                        record { fcy<sup = fcy<sup ; order = {!!} ; supu=u = trans (sym eq1) (ChainP.supu=u u1-is-sup)   } (init asp refl )  ⟫ where
-                          fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 )
-                          fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) eq1 ( ChainP.fcy<sup u1-is-sup fc )
-                          order : {s : Ordinal} {z2 : Ordinal} → s o< u1 → FClosure A f (supf0 s) z2 →
-                            (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1)
-                          order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s
-                          ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) fc ))
-                          ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) {!!} ))
-                          ... | tri> ¬a ¬b px<s | record { eq = eq2 } = ⊥-elim ( o<¬≡ refl (ordtrans px<s (ordtrans s<u1 a) ))  --  px o< s < u1 < px
-                      ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ A∋fcs _ f mf fc  , ch-is-sup u1 (OrdTrans u1≤x z0≤1 ) 
-                        record { fcy<sup = fcy<sup ; order = {!!} ; supu=u = trans (sym {!!} ) (ChainP.supu=u u1-is-sup)   } (init {!!} {!!} )  ⟫ where
-                          fcy<sup : {z : Ordinal} → FClosure A f y z → (z ≡ supf0 u1) ∨ (z << supf0 u1 )
-                          fcy<sup {z} fc = subst ( λ k → (z ≡ k) ∨ (z << k )) {!!} ( ChainP.fcy<sup u1-is-sup fc )
-                          order : {s : Ordinal} {z2 : Ordinal} → s o< u1 → FClosure A f (supf0 s) z2 →
-                            (z2 ≡ supf0 u1) ∨ (z2 << supf0 u1)
-                          order {s} {z2} s<u1 fc with trio< s px | inspect supf1 s
-                          ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) {!!}( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) fc ))
-                          ... | tri≈ ¬a b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) {!!} ( ChainP.order u1-is-sup {!!} (subst (λ k → FClosure A f k z2) (sym eq2) {!!} ))
-                          ... | tri> ¬a ¬b px<s | _ = ⊥-elim ( o<¬≡ refl (ordtrans px<s (subst (λ k → s o< k) b s<u1 ) ))  --  px o< s < u1 = px
-                      ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with trio< z0 px
-                      ... | tri< a ¬b ¬c with osuc-≡< (OrdTrans u1≤x (o<→≤ a) )
-                      ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) 
-                      ... | case2 lt = ⊥-elim ( o<> lt px<u1 )
-                      zc60 (init asp refl) | tri> ¬a ¬b px<u1 | record { eq = eq1} | tri≈ ¬a' b ¬c with osuc-≡< (OrdTrans u1≤x (o≤-refl0 b) )
-                      ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) 
-                      ... | case2 lt = ⊥-elim ( o<> lt px<u1 )
-                      zc60 (init asp refl) | tri> ¬a ¬b px<u1 | record { eq = eq1} | tri> ¬a' ¬b' px<z0 = ⊥-elim (¬p<x<op ⟪ px<z0 , subst (λ k → z0 o< k ) (sym (Oprev.oprev=x op)) z0<x  ⟫ )
-                      zc60 (fsuc w1 fc) with zc60 fc
-                      ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫ 
-                      ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫ 
-                 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z)
+          no-extension : (¬ xSUP (UnionCF A f mf ay supf0 px) x ) ∨ HasPrev A pchain ? f → ZChain A f mf ay x
+          no-extension ¬sp=x = record { supf = supf0 ;  sup = ? ; supf-mono = {!!}
+               ; initial = ? ; chain∋init = ? ; sup=u = ? ; supf-is-sup = ? ; csupf = {!!}
+              ;  chain⊆A = λ lt → proj1 lt ;  f-next = ? ;  f-total = ? }  where
+                 pchain0=1 : ?
+                 pchain0=1 = ?
+                 sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf0 z)
                  sup {z} z≤x with trio< z px
-                 ... | tri< a ¬b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl ?  ) ( ZChain.sup zc (o<→≤   a) ) 
-                 ... | tri≈ ¬a b ¬c = record { sup = SUP.sup sup1 ; as = SUP.as sup1 ; x<sup = zc61 } where
-                     zc61 : {w : HOD} → UnionCF A f mf ay supf1 z ∋ w → (w ≡ SUP.sup sup1) ∨ (w < SUP.sup sup1)
+                 ... | tri< a ¬b ¬c = SUP⊆ ? ( ZChain.sup zc (o<→≤   a) ) 
+                 ... | tri≈ ¬a b ¬c = record { sup = SUP.sup ? ; as = SUP.as ? ; x<sup = ? } where
+                     zc61 : {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup ?) ∨ (w < SUP.sup ? )
                      zc61 {w} lt = ? -- SUP.x<sup sup1 (UnionCFR⊆ (o<→≤ z<x) z<x  lt )
                  ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ? ⟫ )
                  sup=u : {b : Ordinal} (ab : odef A b) →
-                    b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b
+                    b o≤ x → IsSup A (UnionCF A f mf ay supf0 b) ab → supf0 b ≡ b
                  sup=u {b} ab b≤x is-sup with trio< b px
-                 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl  a lt) } 
+                 ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup ? } 
                  ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u zc ab (o≤-refl0 b) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl (o≤-refl0 b) lt) } 
-                 ... | tri> ¬a ¬b px<b = ⊥-elim (¬sp=x zcsup ) where
+                 ... | tri> ¬a ¬b px<b = ? where -- ⊥-elim (¬sp=x zcsup ) where
                      zc30 : x ≡ b
                      zc30 with osuc-≡< b≤x
                      ... | case1 eq = sym (eq)
                      ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ )
-                     zcsup : xSUP
-                     zcsup with zc30
-                     ... | refl = record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ (pxo≤x op) {!!} lt) } } 
-                 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf1 b) (supf1 b)
-                 csupf {b} b<x with trio< b px | inspect supf1 b
-                 ... | tri< a ¬b ¬c | _ = UnionCF⊆ o≤-refl a {!!}
+                     zcsup : ?
+                     zcsup = ? -- with zc30
+                     -- ... | refl = case1 record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup ? } } 
+                 csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf0 b) (supf0 b)
+                 csupf {b} b<x with trio< b px | inspect supf0 b
+                 ... | tri< a ¬b ¬c | _ = ? -- UnionCF⊆ o≤-refl a {!!}
                  ... | tri≈ ¬a refl ¬c | _ = {!!} -- UnionCF⊆ o≤-refl o≤-refl ( ZChain.csupf zc o≤-refl )
                  ... | tri> ¬a ¬b px<b | record { eq = eq1 } = {!!} --  UnionCF⊆ (o<→≤ px<b) o≤-refl ( ZChain.csupf zc o≤-refl ) 
-                 sis : {z : Ordinal} (z≤x : z o≤ x) → supf1 z ≡ & (SUP.sup (sup z≤x))
+                 sis : {z : Ordinal} (z≤x : z o≤ x) → supf0 z ≡ & (SUP.sup (sup z≤x))
                  sis {z} z<x with trio< z px
                  ... | tri< a ¬b ¬c = ZChain.supf-is-sup zc (o<→≤ a )
                  ... | tri≈ ¬a b ¬c = {!!}
                  ... | tri> ¬a ¬b px<z = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) ?  ⟫ )
+
           zc4 : ZChain A f mf ay x 
           zc4 with ODC.∋-p O A (* x)
           ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip
@@ -887,36 +773,19 @@
 
           pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z
           pzc z z<x = prev z z<x
-          ysp =  & (SUP.sup (ysup f mf ay))
 
-          record SupE ( z : Ordinal ) : Set n where
+          record Usupf : Set n where
              field
-                z<x : z o< x
-                z=supfz : z ≡  ZChain.supf (pzc z z<x) z
-
-          psupf0 : (z : Ordinal) → Ordinal
-          psupf0 z with trio< z x
-          ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z
-          ... | tri≈ ¬a b ¬c = ysp 
-          ... | tri> ¬a ¬b c = ysp 
-
-          pchain0 : HOD
-          pchain0 = UnionCF A f mf ay psupf0 x
-
-          ptotal0 : IsTotalOrderSet pchain0
-          ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where 
-               uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-               uz01 = chain-total A f mf ay psupf0 ( (proj2 ca)) ( (proj2 cb)) 
-
-          usup : SUP A pchain0
-          usup = supP pchain0 (λ lt → proj1 lt) ptotal0
-          spu = & (SUP.sup usup)
+                umax : Ordinal → Ordinal
+                umax<x : {z : Ordinal } → umax z o< x
+             supf : Ordinal → Ordinal
+             supf z = ZChain.supf (pzc (osuc (umax z)) (ob<x lim umax<x )) z
+             field
+                supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y 
 
           supf1 : Ordinal → Ordinal
-          supf1 z with trio< z x 
-          ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z
-          ... | tri≈ ¬a b ¬c = spu 
-          ... | tri> ¬a ¬b c = spu
+          supf1 z = ?
+          -- ... | tri< a ¬b ¬c = ZChain.supf (pzc (osuc z) (ob<x lim a)) z
 
           pchain : HOD
           pchain = UnionCF A f mf ay supf1 x
@@ -949,12 +818,7 @@
                      subst (λ k → UChain A f mf ay supf x k )
                           (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc))  ⟫ 
 
-          record xSUP : Set n where
-            field
-               ax : odef A x
-               not-sup : IsSup A (UnionCF A f mf ay psupf0 x) ax
-
-          no-extension : ¬ xSUP → ZChain A f mf ay x
+          no-extension : ¬ ( xSUP (UnionCF A f mf ay supf1 x) x ) ∨ HasPrev A pchain ? f → ZChain A f mf ay x
           no-extension ¬sp=x  = record { initial = pinit ; chain∋init = pcy  ; supf = supf1  ; sup=u = sup=u
                ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!}
                ; csupf = {!!} ; chain⊆A = pchain⊆A ;  f-next = pnext ;  f-total = ptotal } where
@@ -967,7 +831,7 @@
 --                      UnionCF⊆ {u} u<x ⟪ A∋fc _ f mf fcu1 , ch-is-sup u1 u1≤x u1-is-sup fcu1 ⟫ 
 --                 ... | ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf _ aa ) , ch-init (fsuc _ fc) ⟫
 --                 ... | ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪  proj2 ( mf _ aa ) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫
-                 UnionCFR⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay psupf0 x 
+                 UnionCFR⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay supf1 x 
                  UnionCFR⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
                  UnionCFR⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init  au1 refl) ⟫   = ⟪ au , ch-is-sup u1 {!!} {!!} (init {!!} {!!}) ⟫
                  UnionCFR⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ = {!!} -- with
@@ -987,12 +851,12 @@
                  ... | tri> ¬a ¬b c = {!!}
                  sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSup A (UnionCF A f mf ay supf1 b) ab → supf1 b ≡ b
                  sup=u {b} ab b<x is-sup with trio< b x
-                 ... | tri< a ¬b ¬c = ZChain.sup=u (pzc (osuc b)  (ob<x lim a))  ab {!!} record { x<sup = {!!} }
+                 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b)  (ob<x lim a))  ab {!!} record { x<sup = {!!} }
                  ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?)  ?)  ab {!!} record { x<sup = {!!} }
                  ... | tri> ¬a ¬b c = {!!}
                  csupf : {z : Ordinal} → z o≤ x → odef (UnionCF A f mf ay supf1 z) (supf1 z)
                  csupf {z} z≤x with trio< z x
-                 ... | tri< a ¬b ¬c = zc9 where
+                 ... | tri< a ¬b ¬c = ?  where
                      zc9 : odef (UnionCF A f mf ay supf1 z)  (ZChain.supf  (pzc (osuc z) (ob<x lim a)) z)
                      zc9 = {!!}
                      zc8 : odef (UnionCF A f mf ay (supfu a) z)  (ZChain.supf  (pzc (osuc z) (ob<x lim a)) z)