# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1671350177 -32400
# Node ID 125605b5bf47b21cc7a00d0ecc36167ba1995508
# Parent  2fa98e3c0fa36a489eaddab58b6eb9aa2f2f6126
supf-idem is not so easy

diff -r 2fa98e3c0fa3 -r 125605b5bf47 src/zorn.agda
--- a/src/zorn.agda	Sun Dec 18 10:32:29 2022 +0900
+++ b/src/zorn.agda	Sun Dec 18 16:56:17 2022 +0900
@@ -342,13 +342,15 @@
         {y : Ordinal} (ay : odef A y)  ( z : Ordinal ) : Set (Level.suc n) where
    field
       supf :  Ordinal → Ordinal
+
+      cfcs  : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w
+      order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b
+
       asupf :  {x : Ordinal } → odef A (supf x)
-
       supf-mono : {x y : Ordinal } → x o≤  y → supf x o≤ supf y
       zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x
+
       is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x)
-      cfcs  : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w
-      supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z  → supf (supf b) ≡ supf b
 
    chain : HOD
    chain = UnionCF A f ay supf z
@@ -398,6 +400,50 @@
    initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc
    initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc)
 
+   f-total : IsTotalOrderSet chain
+   f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ =
+     subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where
+         fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
+         fc-total with trio< ua ub
+         ... | tri< a₁ ¬b ¬c with ≤-ftrans  (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (s≤fc (supf ub) f mf fcb )
+         ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
+                  ct00 : * (& a) ≡ * (& b)
+                  ct00 = cong (*) eq1
+         ... | case2 a<b =  tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)
+         fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb
+         fc-total | tri> ¬a ¬b c with ≤-ftrans  (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (s≤fc (supf ua) f mf fca )
+         ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
+                  ct00 : * (& a) ≡ * (& b)
+                  ct00 = cong (*) (sym eq1)
+         ... | case2 b<a =  tri> (λ lt → <-irr (case2 b<a ) lt)  (λ eq → <-irr (case1 eq) b<a )  b<a
+   f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where
+      ft01 : (& a) ≤ (& b) → Tri ( a <  b) ( a ≡  b) ( b <  a )
+      ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
+         a=b : a ≡ b
+         a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq)
+      ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)  where
+         a<b : a < b
+         a<b = subst₂ (λ j k → j < k ) *iso *iso lt
+      ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
+      ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb))
+   f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where
+      ft01 : (& b) ≤ (& a) → Tri ( a <  b) ( a ≡  b) ( b <  a )
+      ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
+         a=b : a ≡ b
+         a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq))
+      ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where
+         b<a : b < a
+         b<a = subst₂ (λ j k → j < k ) *iso *iso lt
+      ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
+      ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca))
+   f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ =
+      subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso  (fcn-cmp y f mf fca fcb )
+
+   supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b
+   supf-mono< {a} {b} b≤z sa<sb  with order {a} {b} b≤z sa<sb (init asupf refl)
+   ... | case2 lt = lt
+   ... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb )
+
    supfeq : {a b : Ordinal } → a o≤ z →  b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b
    supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b)
    ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
@@ -436,6 +482,16 @@
            ... | tri≈ ¬a b ¬c = b
            ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb )
 
+   supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z  → supf (supf b) ≡ supf b
+   supf-idem {b} b≤z sfb≤x = z52 where
+       z54 :  {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b)
+       z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc
+       z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x)  fc where
+               u<b : u o< b
+               u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x )
+       z52 : supf (supf b) ≡ supf b
+       z52 = sup=u asupf sfb≤x  record { ax = asupf  ; x≤sup = z54  }
+
    x≤supfx : {x : Ordinal } → x o≤ z → supf x o≤ z → x o≤ supf x
    x≤supfx {x} x≤z sx≤z with x<y∨y≤x (supf x) x
    ... | case2 le = le
@@ -447,64 +503,23 @@
          z45 : f (supf x) ≤ supf x
          z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46
 
-   order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b              
-   order {a} {b} {w} b≤z sa<sb fc with  x<y∨y≤x (supf a) z                                                          
-   ... | case2 z≤sa = ⊥-elim ( o<¬≡ z27 sa<sb ) where               
-         z27 : supf a ≡ supf b               
-         z27 = supfeq (OrdTrans (o<→≤ (supf-inject sa<sb)) b≤z) b≤z ( union-max z≤sa b≤z sa<sb)                                            
-   ... | case1 sa<z = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where                    
-         sa<b : supf a o< b                                                                                          
-         sa<b with x<y∨y≤x (supf a) b                                                                                
-         ... | case1 lt = lt                                                                                         
-         ... | case2 b≤sa = ⊥-elim ( o≤> b≤sa ( supf-inject ( osucprev ( begin                                       
-                 osuc (supf (supf a))  ≡⟨ cong osuc (supf-idem (ordtrans (supf-inject sa<sb) b≤z) (o<→≤ sa<z))  ⟩    
-                 osuc (supf a)  ≤⟨ osucc sa<sb ⟩                                                                     
+   -- order may proved is-minsup and supf-idem, but supf-idem uses order
+   --    Is proving supf-idem is easier than order?
+   --  
+   order0 : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b
+   order0 {a} {b} {w} b≤z sa<sb fc with  x<y∨y≤x (supf a) z
+   ... | case2 z≤sa = ⊥-elim ( o<¬≡ z27 sa<sb ) where
+         z27 : supf a ≡ supf b
+         z27 = supfeq (OrdTrans (o<→≤ (supf-inject sa<sb)) b≤z) b≤z ( union-max z≤sa b≤z sa<sb)
+   ... | case1 sa<z = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where
+         sa<b : supf a o< b
+         sa<b with x<y∨y≤x (supf a) b
+         ... | case1 lt = lt
+         ... | case2 b≤sa = ⊥-elim ( o≤> b≤sa ( supf-inject ( osucprev ( begin
+                 osuc (supf (supf a))  ≡⟨ cong osuc (supf-idem (ordtrans (supf-inject sa<sb) b≤z) (o<→≤ sa<z))  ⟩
+                 osuc (supf a)  ≤⟨ osucc sa<sb ⟩
                  supf b ∎ )))) where open o≤-Reasoning O
 
-   supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b
-   supf-mono< {a} {b} b≤z sa<sb  with order {a} {b} b≤z sa<sb (init asupf refl)
-   ... | case2 lt = lt
-   ... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb )
-
-   f-total : IsTotalOrderSet chain
-   f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ =
-     subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where
-         fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-         fc-total with trio< ua ub
-         ... | tri< a₁ ¬b ¬c with ≤-ftrans  (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (s≤fc (supf ub) f mf fcb )
-         ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
-                  ct00 : * (& a) ≡ * (& b)
-                  ct00 = cong (*) eq1
-         ... | case2 a<b =  tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)
-         fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb
-         fc-total | tri> ¬a ¬b c with ≤-ftrans  (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (s≤fc (supf ua) f mf fca )
-         ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
-                  ct00 : * (& a) ≡ * (& b)
-                  ct00 = cong (*) (sym eq1)
-         ... | case2 b<a =  tri> (λ lt → <-irr (case2 b<a ) lt)  (λ eq → <-irr (case1 eq) b<a )  b<a
-   f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where
-      ft01 : (& a) ≤ (& b) → Tri ( a <  b) ( a ≡  b) ( b <  a )
-      ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
-         a=b : a ≡ b
-         a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq)
-      ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)  where
-         a<b : a < b
-         a<b = subst₂ (λ j k → j < k ) *iso *iso lt
-      ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
-      ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb))
-   f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where
-      ft01 : (& b) ≤ (& a) → Tri ( a <  b) ( a ≡  b) ( b <  a )
-      ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
-         a=b : a ≡ b
-         a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq))
-      ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where
-         b<a : b < a
-         b<a = subst₂ (λ j k → j < k ) *iso *iso lt
-      ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
-      ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca))
-   f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ =
-      subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso  (fcn-cmp y f mf fca fcb )
-
 record ZChain1 ( A : HOD )    ( f : Ordinal → Ordinal )  (mf< : <-monotonic-f A f)
         {y : Ordinal} (ay : odef A y)  (zc : ZChain A f mf< ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where
    supf = ZChain.supf zc
@@ -1110,6 +1125,32 @@
                                 u≤px : u o≤ px
                                 u≤px = ordtrans u<x z≤px
 
+                 supfeq1 : {a b : Ordinal } → a o≤ x →  b o≤ x → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b → supf1 a ≡ supf1 b
+                 supfeq1 {a} {b} a≤z b≤z ua=ub with trio< (supf1 a) (supf1 b)
+                 ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
+                         IsMinSUP.minsup (is-minsup b≤z) asupf1 (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb )
+                 ... | tri≈ ¬a b ¬c = b
+                 ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> (
+                         IsMinSUP.minsup (is-minsup a≤z) asupf1 (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa )
+
+                 union-max1 : {a b : Ordinal } → x o≤ supf1 a → b o≤ x → supf1 a o< supf1 b → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b
+                 union-max1 {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where
+                      z47 : {w : Ordinal } → odef (UnionCF A f ay supf1 a ) w → odef ( UnionCF A f ay supf1 b ) w
+                      z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
+                      z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where
+                          u<b : u o< b
+                          u<b = ordtrans u<a (supf-inject0 supf1-mono sa<sb )
+                      z48 : {w : Ordinal } → odef (UnionCF A f ay supf1 b ) w → odef ( UnionCF A f ay supf1 a ) w
+                      z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
+                      z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where
+                          u<a : u o< a
+                          u<a = supf-inject0 supf1-mono ( osucprev (begin
+                             osuc (supf1 u)  ≡⟨ cong osuc su=u ⟩
+                             osuc u  ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩
+                             x  ≤⟨ z≤sa ⟩
+                             supf1 a ∎ )) where open o≤-Reasoning O
+
+
                  zo≤sz : {z : Ordinal} →  z o≤ x → z o≤ supf1 z
                  zo≤sz {z} z≤x with osuc-≡< z≤x
                  ... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x ))
@@ -1140,10 +1181,67 @@
                      zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39)
                            ( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫  ))
 
+                 order : {a b : Ordinal} {w : Ordinal} →
+                    b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b
+                 order {a} {b} {w} b≤x sa<sb fc = z20 where
+                     a<b : a o< b
+                     a<b = supf-inject0 supf1-mono sa<sb
+                     a≤px : a o≤ px
+                     a≤px with trio< a px
+                     ... | tri< a ¬b ¬c = o<→≤ a
+                     ... | tri≈ ¬a b ¬c = o≤-refl0 b
+                     ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k) (sym (Oprev.oprev=x op))
+                        ( ordtrans<-≤ a<b b≤x) ⟫ ) -- px o< a o< b o≤ x
+                     z20 : w ≤ supf1 b
+                     z20 with trio< b px
+                     ... | tri< b<px ¬b ¬c = ZChain.order zc (o<→≤ b<px) (subst (λ k → k o< supf0 b) (sf1=sf0 (o<→≤ (ordtrans a<b b<px))) sa<sb)
+                          (fcup fc (o<→≤ (ordtrans a<b b<px)))
+                     ... | tri≈ ¬a b=px ¬c = IsMinSUP.x≤sup (ZChain.is-minsup zc (o≤-refl0 b=px)) z26  where
+                          a≤x : a o≤ x
+                          a≤x = o<→≤ (ordtrans ( subst (λ k → a o< k ) b=px a<b ) px<x )
+                          z26 : odef ( UnionCF A f ay supf0 b ) w
+                          z26 with x<y∨y≤x (supf1 a) b
+                          ... | case2 b≤sa = z27 where
+                              z27 : odef (UnionCF A f ay supf0 b) w
+                              z27 with osuc-≡< b≤sa
+                              ... | case2 b<sa = ⊥-elim ( o<¬≡ ( supfeq1 a≤x b≤x
+                                    ( union-max1 x≤sa b≤x (subst (λ k → supf1 a o< k) (sym (sf1=sf0 (o≤-refl0 b=px)))  sa<sb) ))
+                                       (ordtrans<-≤ sa<sb (o≤-refl0 (sym (sf1=sf0 (o≤-refl0 b=px))) ))) where
+                                  x≤sa : x o≤ supf1 a
+                                  x≤sa = subst (λ k → k o≤ supf1 a ) (trans (cong osuc b=px) (Oprev.oprev=x op)) (osucc b<sa )
+                              ... | case1 b=sa = ⊥-elim (o<¬≡ sa=sb sa<sb)  where
+                                  sa=sb : supf1 a ≡ supf0 b
+                                  sa=sb = begin
+                                    supf1 a ≡⟨ sf1=sf0 a≤px ⟩
+                                    supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px (o≤-refl0 (sym (trans (sym b=px) (trans b=sa (sf1=sf0 a≤px) )))))  ⟩
+                                    supf0 (supf0 a) ≡⟨ cong supf0 (sym (sf1=sf0 a≤px )) ⟩
+                                    supf0 (supf1 a) ≡⟨ cong supf0 (sym b=sa) ⟩
+                                    supf0 b ∎ where open ≡-Reasoning
+                          ... | case1 sa<b with cfcs a<b b≤x sa<b fc
+                          ... | ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫
+                          ... | ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = ⟪ ua , ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px)) su=u) (fcup fc u≤px)  ⟫ where
+                              u≤px : u o≤ px
+                              u≤px = o<→≤ ( subst (λ k → u o< k ) b=px u<x )
+                     ... | tri> ¬a ¬b px<b with x<y∨y≤x (supf1 a) b
+                     ... | case1 sa<b = MinSUP.x≤sup sup1 (zc11 ( chain-mono f mf ay supf1 supf1-mono b≤x (cfcs a<b b≤x sa<b fc)))
+                     ... | case2 b≤sa = ⊥-elim ( o<¬≡ z27 sa<sb ) where -- x=b  x o≤ sa   UnionCF a ≡ UnionCF b → supf1 a ≡ supfb b → ⊥
+                          b=x : b ≡ x
+                          b=x with trio< b x
+                          ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , zc-b<x _ a ⟫   ) -- px o< b o< x
+                          ... | tri≈ ¬a b ¬c = b
+                          ... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) -- x o< b ≤ x
+                          a≤x : a o≤ x
+                          a≤x = o<→≤ ( subst (λ k → a o< k ) b=x a<b )
+                          sf1b=sp1 : supf1 b ≡ sp1
+                          sf1b=sp1  = sf1=sp1 (subst (λ k → px o< k) (trans (Oprev.oprev=x op) (sym b=x))  <-osuc)
+                          z27 : supf1 a ≡ sp1
+                          z27 = trans ( supfeq1 a≤x b≤x ( union-max1 (subst (λ k → k o≤ supf1 a) b=x b≤sa)
+                              b≤x (subst (λ k → supf1 a o< k ) (sym sf1b=sp1)  sa<sb )  ) ) sf1b=sp1
+
      ... | no lim with trio< x o∅
      ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )
-     ... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) 
-              ; supf-mono = λ _ → o≤-refl 
+     ... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay)
+              ; supf-mono = λ _ → o≤-refl ; order = λ _ s<s → ⊥-elim ( o<¬≡ refl s<s )
               ; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = λ a<b b≤0 → ⊥-elim ( ¬x<0 (subst (λ k → _ o< k ) x=0 (ordtrans<-≤ a<b b≤0)))    } where
           mf : ≤-monotonic-f A f
           mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
@@ -1153,7 +1251,7 @@
 
           zo≤sz : {z : Ordinal} → z o≤ x → z o≤ MinSUP.sup (ysup f mf ay)
           zo≤sz {z} z≤x with osuc-≡< z≤x
-          ... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z 
+          ... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z
           ... | case2 lt = ⊥-elim ( ¬x<0  (subst (λ k → z o< k ) x=0 lt ) )
 
           is-minsup : {z : Ordinal} → z o≤ x →
@@ -1169,7 +1267,7 @@
                   is02 : {w : Ordinal } →  odef (uchain f mf ay) w → (w ≡ s) ∨ (w << s)
                   is02 fc = sup ⟪ A∋fc _ f mf fc , ch-init fc ⟫
           ... | case2 lt = ⊥-elim ( ¬x<0  (subst (λ k → z o< k ) x=0 lt ) )
-     ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = asupf ; supf-mono = supf-mono ; order = order 
+     ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; asupf = asupf ; supf-mono = supf-mono ; order = order
               ; zo≤sz = zo≤sz   ; is-minsup = is-minsup ; cfcs = cfcs    }  where
 
           mf : ≤-monotonic-f A f
@@ -1554,6 +1652,64 @@
                    z48 = ⟪  proj2 (mf _ (MinSUP.as usup) ) , ic-isup _ (subst (λ k → k o< x) refl spu<x) z50
                         (fsuc _ (init (ZChain.asupf (pzc (ob<x lim spu<x))) z49)) ⟫
 
+          supfeq1 : {a b : Ordinal } → a o≤ x →  b o≤ x → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b → supf1 a ≡ supf1 b
+          supfeq1 {a} {b} a≤z b≤z ua=ub with trio< (supf1 a) (supf1 b)
+          ... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
+                IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb )
+          ... | tri≈ ¬a b ¬c = b
+          ... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> (
+                IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa )
+
+          union-max1 : {a b : Ordinal } → x o≤ supf1 a → b o≤ x → supf1 a o< supf1 b → UnionCF A f ay supf1 a ≡ UnionCF A f ay supf1 b
+          union-max1 {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where
+              z47 : {w : Ordinal } → odef (UnionCF A f ay supf1 a ) w → odef ( UnionCF A f ay supf1 b ) w
+              z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
+              z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where
+                  u<b : u o< b
+                  u<b = ordtrans u<a (supf-inject0 supf-mono sa<sb )
+              z48 : {w : Ordinal } → odef (UnionCF A f ay supf1 b ) w → odef ( UnionCF A f ay supf1 a ) w
+              z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
+              z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where
+                  u<a : u o< a
+                  u<a = supf-inject0 supf-mono ( osucprev (begin
+                     osuc (supf1 u)  ≡⟨ cong osuc su=u ⟩
+                     osuc u  ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩
+                     x  ≤⟨ z≤sa ⟩
+                     supf1 a ∎ )) where open o≤-Reasoning O
+
+          order : {a b w : Ordinal } → b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b
+          order {a} {b} {w} b≤x sa<sb fc with osuc-≡< b≤x
+          ... | case2 b<x = subst (λ k → w ≤ k ) (sym (trans (sf1=sf b<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl ))))  (
+             ZChain.order (pzc b<x) o≤-refl
+                  (subst₂ (λ j k → j o< k ) (trans (sf1=sf a<x) (zeq _ _ (osucc a<b) (o<→≤ <-osuc)))
+                      (trans (sf1=sf b<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl)))  sa<sb)
+                  (subst  (λ k → FClosure A f k w) (trans (sf1=sf a<x) (zeq _ _ (osucc a<b) (o<→≤ <-osuc)) )  fc ) ) where
+               a<b : a o< b
+               a<b = supf-inject0 supf-mono sa<sb
+               a<x : a o< x
+               a<x = ordtrans<-≤ a<b b≤x
+          ... | case1 refl = subst (λ k → w ≤ k ) (sym (sf1=spu refl)) (  zc40 (subst₂ (λ j k → j o< k) (sf1=sf a<x) (sf1=spu refl) sa<sb)
+                   (subst (λ k → FClosure A f k w) (sf1=sf a<x) fc )) where
+               a<x : a o< x
+               a<x = supf-inject0 supf-mono sa<sb
+               zc40 : ZChain.supf (pzc  (ob<x lim a<x )) a o< spu → FClosure A f (ZChain.supf (pzc  (ob<x lim a<x )) a) w → w ≤ spu
+               zc40 sa<sp fc with x<y∨y≤x (supfz a<x) x
+               ... | case1 sa<x = z29 where
+                      z28 : odef (UnionCF A f ay supf1 b) w
+                      z28 = cfcs a<x o≤-refl (subst (λ k → k o< x) (sym (sf1=sf a<x)) sa<x) (subst (λ k → FClosure A f k w ) (sym (sf1=sf a<x)) fc )
+                      z29 : w ≤ spu
+                      z29 with z28
+                      ... | ⟪ aw , ch-init fc ⟫ = MinSUP.x≤sup usup ⟪ aw , ic-init fc ⟫
+                      ... | ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = MinSUP.x≤sup usup ⟪ aw , ic-isup u u<b z30
+                                    (subst (λ k → FClosure A f k w) (sf1=sf u<b) fc) ⟫ where
+                          z30 : supfz u<b o≤ u
+                          z30 = o≤-refl0 ( trans (sym (sf1=sf u<b)) su=u )
+               ... | case2 x≤sa = ⊥-elim ( o<¬≡ z27 sa<sb ) where
+                      z27 : supf1 a ≡ supf1 b
+                      z27 = begin
+                         supf1 a  ≡⟨ ( supfeq1 (o<→≤ a<x) o≤-refl ( union-max1 (subst (λ k → x o≤ k ) (sym (sf1=sf a<x)) x≤sa ) b≤x sa<sb) ) ⟩
+                         supf1 x  ∎ where open ≡-Reasoning
+
      ---
      --- the maximum chain  has fix point of any ≤-monotonic function
      ---