--- a/src/zorn.agda	Mon Aug 15 21:39:17 2022 +0900
+++ b/src/zorn.agda	Tue Aug 16 14:34:54 2022 +0900
@@ -700,7 +700,7 @@
           record xSUP : Set n where
             field
                ax : odef A x
-               not-sup : IsSup A (UnionCF A f mf ay supf0 x) ax
+               is-sup : IsSup A (UnionCF A f mf ay supf0 x) ax
 
           UnionCF⊆ : {z0 z1 : Ordinal} → (z0≤1 : z0 o≤ z1 ) →  (z0≤px :  z0 o≤ px ) → (z1≤x :  z1 o≤ x ) 
                → UnionCF A f mf ay supf0 z0 ⊆' UnionCF A f mf ay supf1 z1 
@@ -738,37 +738,93 @@
           no-extension ¬sp=x = record { supf = supf1 ;  sup = sup
                ; initial = pinit1 ; chain∋init = pcy1 ; sup=u = sup=u ; supf-is-sup = sis ; csupf = csupf
               ;  chain⊆A = λ lt → proj1 lt ;  f-next = pnext1 ;  f-total = ptotal1 }  where
-                 UnionCFR⊆ : {u x : Ordinal} → (a : u o≤ x ) → UnionCF A f mf ay supf1 u ⊆' UnionCF A f mf ay supf0 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
---                         UnionCFR⊆ {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⊆ : {z0 z1 : Ordinal} → z0 o≤ z1 → z0 o≤ x → z1 o≤ px → UnionCF A f mf ay supf1 z0 ⊆' UnionCF A f mf ay supf0 z1 
+                 UnionCFR⊆ {z0} {z1} z0≤1 z0≤x z1≤px ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
+                 UnionCFR⊆ {z0} {z1} z0≤1 z0≤x z1≤px ⟪ 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 = 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 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 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 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) )
+                      ... | 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
+                           ? ≡⟨ ? ⟩
+                           ? ∎ where open ≡-Reasoning
+                         zcsup : xSUP
+                         zcsup = record { ax = subst (λ k → odef A k) zc32 asp ; is-sup = record { x<sup = {!!}  } }
+                      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⊆ {!!}) ( ZChain.sup zc (o<→≤ a) )
-                 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCFR⊆ {!!}) ( ZChain.sup zc (subst (λ k → k o≤ px) (sym b) o≤-refl ))
-                 ... | tri> ¬a ¬b c = {!!}
+                 ... | tri< a ¬b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl z≤x (o<→≤ a)) ( ZChain.sup zc (o<→≤ a) )
+                 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCFR⊆ o≤-refl z≤x (o≤-refl0 b)) ( ZChain.sup zc (subst (λ k → k o≤ px) (sym b) o≤-refl ))
+                 ... | tri> ¬a ¬b c = record { sup = SUP.sup sup1 ; as = SUP.as sup1 ; x<sup = {!!} }
                  sup=u : {b : Ordinal} (ab : odef A b) →
                     b o≤ x → IsSup A (UnionCF A f mf ay supf1 (osuc 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) {!!}
-                 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (subst (λ k → k o≤ px) (sym b) o≤-refl ) {!!}
-                 ... | tri> ¬a ¬b c = {!!}
+                 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 = {!!} }
+                 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (subst (λ k → k o≤ px) (sym b) o≤-refl ) record { x<sup = {!!} }
+                 ... | tri> ¬a ¬b px<b = {!!} 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 ⟫ )
                  csupf : {b : Ordinal} → b o≤ x → odef (UnionCF A f mf ay supf1 b) (supf1 b)
-                 csupf {b} b≤x with trio< b px 
-                 ... | tri< a ¬b ¬c = UnionCF⊆ o≤-refl (o<→≤ a) b≤x ( ZChain.csupf zc (o<→≤ a) )
-                 ... | tri≈ ¬a refl ¬c = UnionCF⊆ o≤-refl o≤-refl b≤x ( ZChain.csupf zc o≤-refl )
-                 ... | tri> ¬a ¬b px<b =  ⟪ {!!} , ch-is-sup b  o≤-refl {!!} {!!} ⟫  where
+                 csupf {b} b≤x with trio< b px | inspect supf1 b
+                 ... | tri< a ¬b ¬c | _ = UnionCF⊆ o≤-refl (o<→≤ a) b≤x ( ZChain.csupf zc (o<→≤ a) )
+                 ... | tri≈ ¬a refl ¬c | _ = UnionCF⊆ o≤-refl o≤-refl b≤x ( ZChain.csupf zc o≤-refl )
+                 ... | tri> ¬a ¬b px<b | record { eq = eq1 } =  
+                    ⟪ SUP.as sup1  , ch-is-sup b  o≤-refl zc31 (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-≡< 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 ⟫ )
+                     zc31 : ChainP A f mf ay supf1 b
+                     zc31 = record { fcy<sup = {!!} ; order = {!!} ; supu=u = {!!} }
                  sis : {z : Ordinal} (z≤x : z o≤ x) → supf1 z ≡ & (SUP.sup (sup z≤x))
-                 sis = {!!}
+                 sis {z} z≤x = {!!}
           zc4 : ZChain A f mf ay x 
           zc4 with ODC.∋-p O A (* x)
           ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip
@@ -791,14 +847,14 @@
              ... | tri> ¬a ¬b c = x
              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 } = ⟪ {!!} , {!!} ⟫
              ... | tri≈ ¬a b ¬c | record { eq = eq1 } = ⟪ {!!} , {!!} ⟫
              ... | tri> ¬a ¬b c | record { eq = eq1 } = {!!} where -- b ≡ x, supf x ≡ sp
                   zc30 : x ≡ b
                   zc30 with trio< x b
-                  ... | tri< a ¬b ¬c = ?
-                  ... | tri≈ ¬a b ¬c = ?
-                  ... | tri> ¬a ¬b c = ?
+                  ... | tri< a ¬b ¬c = {!!}
+                  ... | tri≈ ¬a b ¬c = {!!}
+                  ... | tri> ¬a ¬b c = {!!}
 
           ... | case2 ¬x=sup =  no-extension {!!} -- px is not f y' nor sup of former ZChain from y -- no extention
 
@@ -882,22 +938,22 @@
                  UnionCF⊆ : {u : Ordinal} → (a : u o< x ) → UnionCF A f mf ay supf1 u ⊆' UnionCF A f mf ay (supfu a) x 
                  UnionCF⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
                  UnionCF⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init  au1 refl) ⟫   = ⟪ au , ch-is-sup u1 {!!} {!!} (init {!!} {!!}) ⟫
-                 UnionCF⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ = ? -- with
+                 UnionCF⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ = {!!} -- with
 --                      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) ⟫
-                 UnionCF0⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay psupf0 x 
-                 UnionCF0⊆ {u} u<x ⟪ au , ch-init fc ⟫ = ⟪ au , ch-init fc ⟫ 
-                 UnionCF0⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (init  au1 refl) ⟫   = ⟪ au , ch-is-sup u1 {!!} {!!} (init {!!} {!!}) ⟫
-                 UnionCF0⊆ {u} u<x ⟪ au , ch-is-sup u1 u1≤x u1-is-sup (fsuc xp fcu1) ⟫ = ? -- with
+                 UnionCFR⊆ : {z : Ordinal} → (a : z o≤ x ) → UnionCF A f mf ay supf1 z ⊆' UnionCF A f mf ay psupf0 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
 --                      UnionCF0⊆ {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) ⟫
                  sup : {z : Ordinal} → z o≤ x → SUP A (UnionCF A f mf ay supf1 z)
                  sup {z} z≤x with trio< z x
                  ... | tri< a ¬b ¬c = SUP⊆ (UnionCF⊆ a) (ZChain.sup (pzc (osuc z) {!!}) {!!} )
-                 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCF0⊆ {!!}) usup
-                 ... | tri> ¬a ¬b c = SUP⊆ (UnionCF0⊆ {!!}) usup
+                 ... | tri≈ ¬a b ¬c = SUP⊆ (UnionCFR⊆ {!!}) usup
+                 ... | tri> ¬a ¬b c = SUP⊆ (UnionCFR⊆ {!!}) usup
                  sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup x≤z))
                  sis {z} z≤x with trio< z x
                  ... | tri< a ¬b ¬c = {!!} where