# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1660790909 -32400
# Node ID c97cc257374b87a5e6a97d7c8c570a5726aa278b
# Parent  22676639125fe372a04d99a3091e9d8ef0dfd976
...

diff -r 22676639125f -r c97cc257374b src/zorn.agda
--- a/src/zorn.agda	Wed Aug 17 15:51:47 2022 +0900
+++ b/src/zorn.agda	Thu Aug 18 11:48:29 2022 +0900
@@ -285,8 +285,8 @@
 
       sup : {x : Ordinal } → x o< z  → SUP A (UnionCF A f mf ay supf x)
       sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z  → IsSup A (UnionCF A f mf ay supf b) ab → supf b ≡ b 
-      supf-is-sup : {x : Ordinal } → (x<z : x o< z) → supf x ≡ & (SUP.sup (sup x<z) )
-      csupf : {b : Ordinal } → b o< z → odef (UnionCF A f mf ay supf b) (supf b) 
+      supf-is-sup : {x : Ordinal } → (x≤z : x o< z) → supf x ≡ & (SUP.sup (sup x≤z) )
+      csupf : {b : Ordinal } → b o≤ z → odef (UnionCF A f mf ay supf b) (supf b) 
 
    -- ordering is proved here for totality and sup
 
@@ -303,7 +303,7 @@
             s<z : s o< z
             s<z = ordtrans s<b b<z
             zc03 : odef (UnionCF A f mf ay supf b)  (supf s)
-            zc03 with csupf s<z 
+            zc03 with csupf (o<→≤ s<z) 
             ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc  ⟫ 
             ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u (ordtrans u≤x (osucc s<b)) is-sup fc ⟫ 
       zc01  (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04  where
@@ -672,8 +672,8 @@
           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
+          ... | tri≈ ¬a b ¬c = ZChain.supf zc z
+          ... | tri> ¬a ¬b c = ZChain.supf zc px
 
           pchain1 : HOD
           pchain1  = UnionCF A f mf ay supf1 x
@@ -702,12 +702,12 @@
                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
+          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 ) 
+              ... | 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 )
@@ -715,21 +715,21 @@
                     (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 s<u1 fc )
-                  ... | tri≈ ¬a b ¬c = ? -- subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup s<u1 fc )
+                  ... | 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 (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 = order ; supu=u = trans eq1 ?   } ?  ⟫ where
+              ... | 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 )
+                  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} → s o< 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 s<u1 fc )
-                  ... | tri≈ ¬a b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ? -- ( ChainP.order u1-is-sup s<u1 fc )
+                  ... | tri< a ¬b ¬c = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) (sym eq1) ( ChainP.order u1-is-sup s<u1 fc )
+                  ... | 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 s<u1 ) ))  --  px o< s < u1 = px
-              ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } = ? -- with osuc-≡< (OrdTrans u1≤x z0≤px)
-              -- ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) px<u1 ) 
-              -- ... | case2 lt = ⊥-elim ( o<> lt px<u1 )
+              ... | tri> ¬a ¬b px<u1 | record { eq = eq1 } with osuc-≡< (OrdTrans u1≤x 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) ⟫ 
@@ -750,70 +750,41 @@
                             (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 s<u1 (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 s<u1 (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 s<u1 (subst (λ k → FClosure A f k z2) (sym eq2) fc ))
                           ... | 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 = order ; supu=u = trans (sym ? ) (ChainP.supu=u u1-is-sup)   } (init ? ? )  ⟫ where
+                        record { fcy<sup = fcy<sup ; order = 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 )) ? ( ChainP.fcy<sup u1-is-sup fc )
+                          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) ) ? ( ChainP.order u1-is-sup s<u1 (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 s<u1 (subst (λ k → FClosure A f k z2) (sym eq2) ? ))
+                          ... | tri< a ¬b ¬c | record { eq = eq2 } = subst (λ k → (z2 ≡ k) ∨ (z2 << k) ) eq1 ( ChainP.order u1-is-sup s<u1 (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 s<u1 (subst (λ k → FClosure A f k z2) (sym eq2) fc ))
                           ... | 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 = eq2} | tri≈ ¬a' b ¬c with osuc-≡< (OrdTrans u1≤x (o≤-refl0 b) )
+                      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 = eq2} | tri> ¬a' ¬b' px<z0 = ⊥-elim ( ¬sp=x zcsup ) where
-                         zc30 : x ≡ z0
-                         zc30 = ? -- with osuc-≡< z0≤x
-                         -- ... | case1 eq = sym (eq)
-                         -- ... | case2 z0<x = ⊥-elim (¬p<x<op ⟪ px<z0 , subst (λ k → z0 o< k ) (sym (Oprev.oprev=x op)) z0<x ⟫ )
-                         zc31 : x ≡ u1 
-                         zc31 with trio< x u1 
-                         ... | tri≈ ¬a b ¬c = b
-                         ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ px<u1 , subst (λ k → u1 o< k) (sym (Oprev.oprev=x op)) c ⟫ )
-                         zc31 | tri< a ¬b ¬c with osuc-≡< (subst (λ k → u1 o≤ k ) (sym zc30) u1≤x ) -- px<u1 u1≤x,
-                         ... | case1 u1=x = ⊥-elim ( ¬b (sym u1=x) )
-                         ... | case2 u1<x = ⊥-elim ( o<> u1<x a )
-                         zc33 : supf1 u1 ≡ u1   -- u1 ≡ supf1 u1 ≡ supf1 x ≡ sp1
-                         zc33 = ChainP.supu=u u1-is-sup
-                         zc32 : sp1 ≡ x 
-                         zc32 = begin
-                           sp1 ≡⟨ sym eq2 ⟩
-                           supf1 u1 ≡⟨ zc33 ⟩
-                           u1 ≡⟨ sym zc31 ⟩
-                           x ∎ where open ≡-Reasoning
-                         zc34 : {z : Ordinal} → odef (UnionCF A f mf ay supf0 px) z → (z ≡ x) ∨ (z << x)
-                         zc34 {z} lt with SUP.x<sup sup1 (subst (λ k → odef (UnionCF A f mf ay supf0 x) k ) (sym &iso) (chain-mono f mf ay supf0 (pxo≤x op) lt ) )
-                         ... | case1 eq = case1 ( begin
-                           z ≡⟨ sym &iso ⟩
-                           & (* z) ≡⟨ cong (&) eq ⟩
-                           sp1 ≡⟨ zc32 ⟩
-                           x ∎ ) where open ≡-Reasoning
-                         ... | case2 lt = case2 ( subst (λ k → * z < k ) (trans (sym *iso) (cong (*) zc32 )) lt )
-                         zcsup : xSUP
-                         zcsup = record { ax = subst (λ k → odef A k) zc32 asp ; is-sup = record { x<sup = zc34 } }
+                      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)
                  sup {z} z<x with trio< z px
-                 ... | tri< a ¬b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl ? ) ( ZChain.sup zc  a ) 
+                 ... | tri< a ¬b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl z<x  ) ( ZChain.sup zc  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)
-                     zc61 {w} lt = SUP.x<sup sup1 (UnionCFR⊆ (o<→≤ z<x) ?  lt )
+                     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)) z<x ⟫ )
                  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 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 ? lt) } 
-                 ... | 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 ¬c = ZChain.sup=u zc ab (o<→≤ a) record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ o≤-refl (o<→≤ a) lt) } 
+                 ... | 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
                      zc30 : x ≡ b
                      zc30 with osuc-≡< b≤x
@@ -821,22 +792,12 @@
                      ... | 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)
+                     ... | refl = record { ax = ab ; is-sup = record { x<sup = λ lt → IsSup.x<sup is-sup (UnionCF⊆ (pxo≤x op) o≤-refl 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 ( ZChain.csupf zc  a  )
-                 ... | tri≈ ¬a b ¬c | _ = ? -- UnionCF⊆ o≤-refl o≤-refl ( ZChain.csupf zc ? )
-                 ... | tri> ¬a ¬b px<b | record { eq = eq1 } =  
-                    ⟪ SUP.as sup1  , ch-is-sup ?  ? ? (subst (λ k → FClosure A f k sp1) (sym eq1) (init (SUP.as sup1) refl))  ⟫  
-                         where
-                     --   px< b ≤ x
-                     -- b ≡ x, supf x ≡ sp1 , ¬ x ≡ sp1
-                     zc30 : x ≡ b
-                     zc30 = ? -- with osuc-≡< ?
-                     -- ... | 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 ⟫ )
-                     zc31 : ChainP A f mf ay supf1 b
-                     zc31 = record { fcy<sup = {!!} ; order = {!!} ; supu=u = {!!} }
+                 ... | tri< a ¬b ¬c | _ = UnionCF⊆ o≤-refl (o<→≤ a) ( ZChain.csupf zc (o<→≤ 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} z≤x = {!!}
           zc4 : ZChain A f mf ay x 
@@ -859,7 +820,7 @@
              ... | tri< a ¬b ¬c = ZChain.supf zc z
              ... | tri≈ ¬a b ¬c = x
              ... | tri> ¬a ¬b c = x
-             csupf : {b : Ordinal} → b o< x → odef (UnionCF A f mf ay psupf1 b) (psupf1 b)
+             csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay psupf1 b) (psupf1 b)
              csupf {b} b≤x with trio< b px | inspect psupf1 b
              ... | tri< a ¬b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫
              ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫
@@ -979,7 +940,7 @@
                  ... | 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 : 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
                      zc9 : odef (UnionCF A f mf ay supf1 z)  (ZChain.supf  (pzc (osuc z) (ob<x lim a)) z)