# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1671327149 -32400
# Node ID 2fa98e3c0fa36a489eaddab58b6eb9aa2f2f6126
# Parent  9e8cb06f0aff771fcbc5b8edb93e933c0f95496b
order may come from supf-idem

diff -r 9e8cb06f0aff -r 2fa98e3c0fa3 src/zorn.agda
--- a/src/zorn.agda	Sun Dec 18 07:58:00 2022 +0900
+++ b/src/zorn.agda	Sun Dec 18 10:32:29 2022 +0900
@@ -342,15 +342,13 @@
         {y : Ordinal} (ay : odef A y)  ( z : Ordinal ) : Set (Level.suc n) where
    field
       supf :  Ordinal → Ordinal
+      asupf :  {x : Ordinal } → odef A (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
-      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
@@ -400,6 +398,74 @@
    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)
 
+   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≤> (
+             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-max : {a b : Ordinal } → z o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b
+   union-max {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where
+          z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf 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-inject sa<sb )
+          z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf 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-inject ( osucprev (begin
+                 osuc (supf u)  ≡⟨ cong osuc su=u ⟩
+                 osuc u  ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩
+                 z  ≤⟨ z≤sa ⟩
+                 supf a ∎ )) where open o≤-Reasoning O
+
+   sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
+       → IsSUP A (UnionCF A f ay supf b) b  → supf b ≡ b
+   sup=u {b} ab b≤z is-sup = z50 where
+           z48 : supf b o≤ b
+           z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
+           z50 : supf b ≡ b
+           z50 with trio< (supf b) b
+           ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
+                 z47 : b o≤ supf b
+                 z47 = zo≤sz b≤z
+           ... | tri≈ ¬a b ¬c = b
+           ... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb )
+
+   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
+   ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where
+         z46 : odef (UnionCF A f ay supf x) (f (supf x))
+         z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf  z47 )) ⟫ where
+             z47 : supf (supf x) ≡ supf x
+             z47 = supf-idem x≤z  sx≤z
+         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 ⟩                                                                     
+                 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
@@ -439,80 +505,6 @@
    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≤> (
-             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-max : {a b : Ordinal } → z o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b
-   union-max {a} {b} z≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where
-          z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf 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-inject sa<sb )
-          z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf 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-inject ( osucprev (begin
-                 osuc (supf u)  ≡⟨ cong osuc su=u ⟩
-                 osuc u  ≤⟨ osucc (ordtrans<-≤ u<b b≤z ) ⟩
-                 z  ≤⟨ z≤sa ⟩
-                 supf a ∎ )) where open o≤-Reasoning O
-
-   sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
-       → IsSUP A (UnionCF A f ay supf b) b  → supf b ≡ b
-   sup=u {b} ab b≤z is-sup = z50 where
-           z48 : supf b o≤ b
-           z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
-           z50 : supf b ≡ b
-           z50 with trio< (supf b) b
-           ... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
-                 z47 : b o≤ supf b
-                 z47 = zo≤sz b≤z
-           ... | 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
-   ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where
-         z46 : odef (UnionCF A f ay supf x) (f (supf x))
-         z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf  z47 )) ⟫ where
-             z47 : supf (supf x) ≡ supf x
-             z47 = supf-idem x≤z  sx≤z
-         z45 : f (supf x) ≤ supf x
-         z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46
-
-   order0 : {a b w : Ordinal } → b o≤ z → supf a o< supf b → supf a o≤ z → FClosure A f (supf a) w → w ≤ supf b
-   order0 {a} {b} {w} b≤z sa<sb sa≤z fc = 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)  sa≤z)  ⟩
-                 osuc (supf a)  ≤⟨ osucc sa<sb ⟩
-                 supf b ∎ )))) where open o≤-Reasoning O
-
 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
@@ -1118,32 +1110,6 @@
                                 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 ))
@@ -1174,67 +1140,10 @@
                      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 ; order = λ _ s<s → ⊥-elim ( o<¬≡ refl s<s )
+              ; supf-mono = λ _ → o≤-refl 
               ; 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
@@ -1645,64 +1554,6 @@
                    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
      ---