# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1667937656 -32400
# Node ID 6229017a61767a11d8b7269fe453a4e6cf512fda
# Parent  94357ced682dcfa9b4082668c4540e67c00fe7dc
chain is now u≤x again

diff -r 94357ced682d -r 6229017a6176 src/zorn.agda
--- a/src/zorn.agda	Wed Nov 09 03:14:40 2022 +0900
+++ b/src/zorn.agda	Wed Nov 09 05:00:56 2022 +0900
@@ -273,15 +273,15 @@
 data UChain  ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y )
        (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where
     ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z
-    ch-is-sup  : (u : Ordinal) {z : Ordinal }  (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u )
+    ch-is-sup  : (u : Ordinal) {z : Ordinal }  (u≤x : supf u o≤ supf x) ( is-sup : ChainP A f mf ay supf u )
         ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z
 
 --
 --         f (f ( ... (sup y))) f (f ( ... (sup z1)))
 --        /          |         /             |
 --       /           |        /              |
---    sup y   <       sup z1          <    sup z2
---           o<                      o<
+--    sup y   ≤       sup z1          ≤    sup z2
+--           o≤                      o≤
 -- data UChain is total
 
 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f)  {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal )
@@ -289,7 +289,7 @@
 chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where
      ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a)
      ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb
-     ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) with ChainP.fcy<sup supb fca
+     ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca
      ... | case1 eq with 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
@@ -297,14 +297,14 @@
      ... | case2 lt = tri< ct01  (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt)  where
           ct01 : * a < * b
           ct01 = subst (λ k → * k < * b ) (sym eq) lt
-     ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u<x supb fcb) | case2 lt = tri< ct01  (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt)  where
+     ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) | case2 lt = tri< ct01  (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt)  where
           ct00 : * a < * (supf ub)
           ct00 = lt
           ct01 : * a < * b
           ct01 with s≤fc (supf ub) f mf fcb
           ... | case1 eq =  subst (λ k → * a < k ) eq ct00
           ... | case2 lt =  IsStrictPartialOrder.trans POO ct00 lt
-     ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb
+     ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb
      ... | case1 eq with 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
@@ -312,7 +312,7 @@
      ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01    where
           ct01 : * b < * a
           ct01 = subst (λ k → * k < * a ) (sym eq) lt
-     ct-ind xa xb {a} {b} (ch-is-sup ua u<x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01    where
+     ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01    where
           ct00 : * b < * (supf ua)
           ct00 = lt
           ct01 : * b < * a
@@ -402,7 +402,7 @@
 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc  ⟫ =
         ⟪ ua , ch-init fc  ⟫
 chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa ,  ch-is-sup ua ua<x is-sup fc  ⟫ =
-        ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc  ⟫
+        ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x ? ) is-sup fc  ⟫
 
 record ZChain ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f)
         {y : Ordinal} (ay : odef A y)  ( z : Ordinal ) : Set (Level.suc n) where
@@ -435,11 +435,11 @@
    chain∋init = ⟪ ay , ch-init (init ay refl)    ⟫
    f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a)
    f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc)  ⟫
-   f-next {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 ) ⟫
+   f-next {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 ) ⟫
    initial : {z : Ordinal } → odef chain z → * y ≤ * z
    initial {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
+   ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc)  where
         zc7 : y <= supf u
         zc7 = ChainP.fcy<sup is-sup (init ay refl)
    f-total : IsTotalOrderSet chain
@@ -490,35 +490,35 @@
         → ( { x : Ordinal } → z o≤ x → supf z o≤ supf1 x)
         → UnionCF A f mf ay supf z ⊆' UnionCF A f mf ay supf1 z
 UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ 
-UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-is-sup u {x} u<x is-sup fc ⟫ = ⟪ az , ch-is-sup u u<x1 cp1 fc1 ⟫  where
-          u<x0 : u o< z
-          u<x0 = supf-inject0 supf-mono u<x
-          u<x1 : supf1 u o< supf1 z
-          u<x1 = subst (λ k → k o< supf1 z ) (eq<x u<x0) (ordtrans<-≤ u<x (lex o≤-refl ) )
+UChain⊆ A f mf {z} {y} ay {supf} {supf1} supf-mono eq<x lex ⟪ az , ch-is-sup u {x} u≤x is-sup fc ⟫ = ⟪ az , ch-is-sup u ? cp1 fc1 ⟫  where
+          u≤x0 : u o< z
+          u≤x0 = supf-inject0 supf-mono ? 
+          u≤x1 : supf1 u o< supf1 z
+          u≤x1 = subst (λ k → k o< supf1 z ) (eq<x u≤x0) (ordtrans<-≤ u≤x ? ) -- (lex o≤-refl ) )
           fc1 : FClosure A f (supf1 u) x
-          fc1 = subst (λ k → FClosure A f k x ) (eq<x u<x0) fc
+          fc1 = subst (λ k → FClosure A f k x ) (eq<x u≤x0) fc
           uc01 : {s : Ordinal } →  supf1 s o< supf1 u → s o< z
           uc01 {s} s<u with trio< s z
           ... | tri< a ¬b ¬c = a
-          ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02  s<u ) where -- (supf-mono (o<→≤ u<x0))
+          ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> uc02  s<u ) where -- (supf-mono (o<→≤ u≤x0))
                uc02 :  supf1 u o≤ supf1 s
                uc02 = begin
-                 supf1 u  <⟨ u<x1 ⟩
+                 supf1 u  <⟨ u≤x1 ⟩
                  supf1 z  ≡⟨ cong supf1 (sym b) ⟩
                  supf1 s ∎  where open o≤-Reasoning O
           ... | tri> ¬a ¬b c = ⊥-elim ( o≤> uc03 s<u ) where
                uc03 :  supf1 u o≤ supf1 s
                uc03 = begin
-                 supf1 u  ≡⟨ sym (eq<x u<x0) ⟩
-                 supf u  <⟨ u<x ⟩
+                 supf1 u  ≡⟨ sym (eq<x u≤x0) ⟩
+                 supf u  <⟨ ? ⟩
                  supf z  ≤⟨ lex (o<→≤ c) ⟩
                  supf1 s ∎  where open o≤-Reasoning O
           cp1 : ChainP A f mf ay supf1 u
-          cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) (ChainP.fcy<sup is-sup fc )  
-               ; order =  λ {s} {z} s<u fc  → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u<x0) 
-                  (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u<x0)) s<u)  
+          cp1 = record { fcy<sup = λ {z} fc → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u≤x0) (ChainP.fcy<sup is-sup fc )  
+               ; order =  λ {s} {z} s<u fc  → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (eq<x u≤x0) 
+                  (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (sym (eq<x (uc01 s<u) )) (sym (eq<x u≤x0)) s<u)  
                     (subst (λ k → FClosure A f k z ) (sym (eq<x (uc01 s<u) )) fc ))
-               ; supu=u = trans (sym (eq<x u<x0)) (ChainP.supu=u is-sup)  }
+               ; supu=u = trans (sym (eq<x u≤x0)) (ChainP.supu=u is-sup)  }
 
 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
@@ -654,8 +654,8 @@
             → * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
         is-max-hp x {a} {b} ua ab has-prev a<b with HasPrev.ay has-prev
         ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫
-        ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k )
-                      (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x is-sup (fsuc _ fc))  ⟫
+        ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ ab , subst (λ k → UChain A f mf ay (ZChain.supf zc) x k )
+                      (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc))  ⟫
 
         supf = ZChain.supf zc
 
@@ -670,7 +670,7 @@
                 zc05 : odef (UnionCF A f mf ay supf b)  (supf s)
                 zc05 with zc04
                 ... | ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc  ⟫
-                ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ as , ch-is-sup u zc08 is-sup fc ⟫ where
+                ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ as , ch-is-sup u ? is-sup fc ⟫ where
                     zc07 : FClosure A f (supf u) (supf s)   -- supf u ≤ supf s  → supf u o≤ supf s
                     zc07 = fc
                     zc06 : supf u ≡ u
@@ -681,7 +681,7 @@
                 zc04 : odef (UnionCF A f mf ay supf b) (f x)
                 zc04 with subst (λ k → odef (UnionCF A f mf ay supf b) k ) &iso (csupf-fc b<z ss≤sb fc  )
                 ... | ⟪ as , ch-init fc ⟫ = ⟪ proj2 (mf _ as) , ch-init (fsuc _ fc) ⟫
-                ... | ⟪ as , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ fc) ⟫
+                ... | ⟪ as , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u≤x is-sup (fsuc _ fc) ⟫
         order : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → (z1 ≡ supf b) ∨ (z1 << supf b)
         order {b} {s} {z1} b<z ss<sb fc = zc04 where
           zc00 : ( z1  ≡  MinSUP.sup (ZChain.minsup zc (o<→≤ b<z) )) ∨ ( z1  << MinSUP.sup ( ZChain.minsup zc (o<→≤ b<z) ) )
@@ -705,7 +705,7 @@
            is-max {a} {b} ua b<x ab P a<b with ODC.or-exclude O P
            is-max {a} {b} ua b<x ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
            is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup
-             = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫   where
+             = ⟪ ab , ch-is-sup b ? m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl)) ⟫   where
               b<A : b o< & A
               b<A = z09 ab
               b<x : b o< x
@@ -731,7 +731,7 @@
            is-max {a} {b} ua sb<sx ab P a<b | case1 has-prev = is-max-hp x {a} {b} ua ab has-prev a<b
            is-max {a} {b} ua sb<sx ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (init-uchain A f mf ay )
            ... | case1 b=y = ⟪ subst (λ k → odef A k ) b=y ay , ch-init (subst (λ k → FClosure A f y k ) b=y (init ay refl ))  ⟫
-           ... | case2 y<b = ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl))  ⟫ where
+           ... | case2 y<b = ⟪ ab , ch-is-sup b ? m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl))  ⟫ where
               m09 : b o< & A
               m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
               b<x : b o< x
@@ -968,17 +968,17 @@
                  zc11 : {z : Ordinal} → odef (UnionCF A f mf ay supf1 x) z → odef pchainpx z
                  zc11 {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫
                  zc11 {z} ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ = zc21 fc where
-                    u<x :  u o< x
-                    u<x =  supf-inject0 supf1-mono su<sx
+                    u≤x :  u o< x
+                    u≤x =  supf-inject0 supf1-mono ? -- su<sx
                     u≤px : u o≤ px
-                    u≤px = zc-b<x _ u<x
+                    u≤px = zc-b<x _ u≤x
                     zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1
                     zc21 {z1} (fsuc z2 fc ) with zc21 fc
                     ... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-                    ... | case1 ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫
+                    ... | case1 ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫
                     ... | case2 fc = case2 (fsuc _ fc)
                     zc21 (init asp refl ) with trio< (supf0 u) (supf0 px) | inspect supf1 u
-                    ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u a record {fcy<sup = zc13 ; order = zc17
+                    ... | tri< a ¬b ¬c | _ = case1 ⟪ asp , ch-is-sup u ? record {fcy<sup = zc13 ; order = zc17
                          ; supu=u = trans (sym (sf1=sf0 (o<→≤ u<px))) (ChainP.supu=u is-sup) } (init asp0 (sym (sf1=sf0 (o<→≤ u<px))) ) ⟫ where
                         u<px :  u o< px
                         u<px =  ZChain.supf-inject zc a
@@ -993,41 +993,41 @@
                         zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf0 u) ∨ ( z << supf0 u )
                         zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sf1=sf0 (o<→≤ u<px)) ( ChainP.fcy<sup is-sup fc )
                     ... | tri≈ ¬a b ¬c | _ = case2 (init (subst (λ k → odef A k) b (ZChain.asupf zc) ) (sym (trans (sf1=sf0 u≤px) b )))
-                    ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x  ⟫ )
+                    ... | tri> ¬a ¬b c | _ = ⊥-elim ( ¬p<x<op ⟪ ZChain.supf-inject zc c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u≤x  ⟫ )
 
                  zc35 : {z : Ordinal} → ¬ (supf0 px ≡ px) → odef (UnionCF A f mf ay supf1 x) z  → odef (UnionCF A f mf ay supf0 x) z
                  zc35 {z} ne ⟪ ua1 , ch-init fc ⟫ = ⟪ ua1 , ch-init fc ⟫ 
-                 zc35 {z} ne ⟪ ua1 , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( ZChain.supf-mono zc (o<→≤ (supf-inject0 supf1-mono u<x)) )
+                 zc35 {z} ne ⟪ ua1 , ch-is-sup u u≤x is-sup fc ⟫ with osuc-≡< ( ZChain.supf-mono zc (o<→≤ (supf-inject0 supf1-mono ?)) )
                  ... | case1 eq = zc34 where
-                      u<x0 : u o≤ px
-                      u<x0 = subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) (supf-inject0 supf1-mono u<x )
+                      u≤x0 : u o≤ px
+                      u≤x0 = subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) (supf-inject0 supf1-mono ? )
                       zc34 : odef (UnionCF A f mf ay supf0 x) z
                       zc34 with trio< (supf0 px) px
                       ... | tri< sf0px<px ¬b ¬c = cp3 ( subst ( λ k → FClosure A f k z) 
-                          (trans (trans (sf1=sf0 u<x0) eq) (ZChain.supfmax zc px<x) ) fc) where 
+                          (trans (trans (sf1=sf0 u≤x0) eq) (ZChain.supfmax zc px<x) ) fc) where 
                               cp3 : {z : Ordinal } → FClosure A f (supf0 px) z → odef (UnionCF A f mf ay supf0 x) z
                               cp3 (init uz refl) = chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ px<x) (ZChain.csupf zc sf0px<px)
                               cp3 (fsuc uz fc) = ZChain.f-next zc (cp3 fc)
                       ... | tri≈ ¬a b ¬c = ⊥-elim (ne b)
                       ... | tri> ¬a ¬b px<sf0px = ⊥-elim (¬p<x<op ⟪ cp5 , cp4 ⟫ ) where
                            cp4 : u o< osuc px
-                           cp4 = subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) ( supf-inject0 supf1-mono u<x  )
+                           cp4 = subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) ( supf-inject0 supf1-mono ?  )
                            cp5 : px o< u
                            cp5 = subst (λ k → px o< k ) ( begin
                              supf0 px ≡⟨ sym (ZChain.supfmax zc px<x)  ⟩
                              supf0 x  ≡⟨ sym eq  ⟩
-                             supf0 u  ≡⟨ sym (sf1=sf0 u<x0)  ⟩
+                             supf0 u  ≡⟨ sym (sf1=sf0 u≤x0)  ⟩
                              supf1 u  ≡⟨ ChainP.supu=u is-sup  ⟩
                              u ∎  ) px<sf0px  where open ≡-Reasoning 
-                 ... | case2 u<x1 = ⟪ ua1 , ch-is-sup u u<x1 cp1 fc1 ⟫ where
-                      u<x0 : u o< x
-                      u<x0 = supf-inject0 supf1-mono u<x -- supf0 u o≤  px ≡ supf0 px → supf0 u ≡ supf0 px ∨ u o< px
+                 ... | case2 u≤x1 = ⟪ ua1 , ch-is-sup u ? cp1 fc1 ⟫ where
+                      u≤x0 : u o< x
+                      u≤x0 = supf-inject0 supf1-mono ? -- supf0 u o≤  px ≡ supf0 px → supf0 u ≡ supf0 px ∨ u o< px
                       sf0u=sf1u : supf0 u ≡ supf1 u
-                      sf0u=sf1u = sym (sf1=sf0 (subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x0 ))
+                      sf0u=sf1u = sym (sf1=sf0 (subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u≤x0 ))
                       cp2 : {s : Ordinal } → supf0 s o< supf0 u → supf0 s ≡ supf1 s
                       cp2 {s} ss<su = sym ( sf1=sf0 ( begin
                              s  <⟨ ZChain.supf-inject zc ss<su ⟩
-                             u  ≤⟨ subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x0  ⟩
+                             u  ≤⟨ subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u≤x0  ⟩
                              px ∎ )) where open o≤-Reasoning O
                       fc1 : FClosure A f (supf0 u) z
                       fc1 = subst (λ k → FClosure A f k z ) (sym sf0u=sf1u) fc
@@ -1036,7 +1036,7 @@
                            ; order =  λ {s} {z} s<u fc  → subst (λ k → (z ≡ k) ∨ ( z << k ) ) (sym sf0u=sf1u)
                               (ChainP.order is-sup (subst₂ (λ j k → j o< k ) (cp2 s<u) sf0u=sf1u s<u)  
                                 (subst (λ k → FClosure A f k z ) (cp2 s<u) fc ))
-                           ; supu=u = trans (sym (sf1=sf0 (subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x0 ))) (ChainP.supu=u is-sup)  }
+                           ; supu=u = trans (sym (sf1=sf0 (subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u≤x0 ))) (ChainP.supu=u is-sup)  }
 
                  zc33 : {z : Ordinal} → ¬ (supf0 px o< sp1) → odef (UnionCF A f mf ay supf1 x) z  → odef (UnionCF A f mf ay supf0 x) z
                  zc33 {z} lt = UChain⊆ A f mf ay supf1-mono (λ lt → sym (sf=eq lt)) sf≤0  where
@@ -1063,20 +1063,20 @@
                  zc12 {z} sfpx=px sfpx<sp (case1 ⟪ az , ch-init fc ⟫ ) = ⟪ az , ch-init fc ⟫
                  zc12 {z} sfpx=px sfpx<sp (case1 ⟪ az , ch-is-sup u su<sx is-sup fc ⟫ ) = zc21 fc where
                         u<px :  u o< px
-                        u<px = ZChain.supf-inject zc su<sx
-                        u<x : u o< x
-                        u<x = ordtrans u<px px<x
+                        u<px = ZChain.supf-inject zc ? -- su<sx
+                        u≤x : u o< x
+                        u≤x = ordtrans u<px px<x
                         u≤px : u o≤ px
                         u≤px = o<→≤ u<px
                         s1u<s1x : supf1 u o< supf1 x
-                        s1u<s1x = ordtrans<-≤ (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 u≤px )) (sym (sf1=sf0 o≤-refl)) su<sx) (supf1-mono (o<→≤ px<x) )
+                        s1u<s1x = ordtrans<-≤ (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 u≤px )) (sym (sf1=sf0 o≤-refl)) ?) (supf1-mono (o<→≤ px<x) )
                         zc21 : {z1 : Ordinal } → FClosure A f (supf0 u) z1 → odef (UnionCF A f mf ay supf1 x) z1
                         zc21 {z1} (fsuc z2 fc ) with zc21 fc
                         ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-                        ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫
+                        ... | ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫
                         zc21 (init asp refl ) with trio< u px | inspect supf1 u
                         ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u
-                             s1u<s1x
+                             ? -- s1u<s1x
                                 record {fcy<sup = zc13 ; order = zc17 ; supu=u = trans (sf1=sf0 u≤px ) (ChainP.supu=u is-sup) } zc14 ⟫  where
                             zc17 :  {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 u →
                                   FClosure A f (supf1 s) z1 → (z1 ≡ supf1 u) ∨ (z1 << supf1 u)
@@ -1088,7 +1088,7 @@
                             zc14 = init (subst (λ k → odef A k ) (sym (sf1=sf0 u≤px)) asp) (sf1=sf0 u≤px)
                             zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u )
                             zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 u≤px )) ( ChainP.fcy<sup is-sup fc )
-                        ... | tri≈ ¬a b ¬c | _ = ⟪ asp , ch-is-sup px (subst (λ k → supf1 k o< supf1 x) b s1u<s1x)  record { fcy<sup = zc13
+                        ... | tri≈ ¬a b ¬c | _ = ⟪ asp , ch-is-sup px ?  record { fcy<sup = zc13
                               ; order = zc17 ; supu=u = zc18   } (init asupf1 (trans (sf1=sf0 o≤-refl ) (cong supf0 (sym b)))  ) ⟫ where
                             zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px) ∨ ( z << supf1 px )
                             zc13 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (trans (cong supf0 b) (sym (sf1=sf0 o≤-refl))) (ChainP.fcy<sup is-sup fc )
@@ -1113,8 +1113,8 @@
                         zc21 : {z1 : Ordinal } → FClosure A f (supf0 px) z1 → odef (UnionCF A f mf ay supf1 x) z1
                         zc21 {z1} (fsuc z2 fc ) with zc21 fc
                         ... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-                        ... | ⟪ ua1 , ch-is-sup u u<x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u<x u1-is-sup (fsuc _ fc₁) ⟫
-                        zc21 (init asp refl ) =  ⟪ asp , ch-is-sup px zc18
+                        ... | ⟪ ua1 , ch-is-sup u u≤x u1-is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x u1-is-sup (fsuc _ fc₁) ⟫
+                        zc21 (init asp refl ) =  ⟪ asp , ch-is-sup px ? 
                                      record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where
                               zc15 : supf1 px ≡ px
                               zc15 = trans (sf1=sf0 o≤-refl ) (sfpx=px)
@@ -1179,6 +1179,7 @@
                          (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ px<x) 
                          (ZChain.csupf zc (subst (λ k → k o< px) (sf1=sf0 z≤px) s0z<px))))
                  -- px o< z1 , px o≤ supf1 z1  -->   px o≤ sp1 o< x -- sp1 ≡ px--> odef (UnionCF A f mf ay supf1 x) sp1
+                 --      supf1 px ≡ sp1 o< supf1 x
 
                  csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1)
                  csupf1 {z1} sz1<x = csupf2 where
@@ -1320,7 +1321,7 @@
                         zc13 = o≤-refl0 ( trans (Oprev.oprev=x op) (sym eq ) )
                         x≤sup : {w : Ordinal} → odef (UnionCF A f mf ay supf0 px) w → (w ≡ x) ∨ (w << x)
                         x≤sup {w} ⟪ az , ch-init {w} fc ⟫ = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ?
-                        x≤sup {w} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u<x) ? ))
+                        x≤sup {w} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ with osuc-≡< ( supf-mono (ordtrans (o<→≤ u≤x) ? ))
                         ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where
                                  zc14 : u ≡ osuc px
                                  zc14 = begin
@@ -1444,9 +1445,9 @@
                 * a < * b → odef (UnionCF A f mf ay supf x) b
           is-max-hp supf x {a} {b} ua b<x ab has-prev a<b with HasPrev.ay has-prev
           ... | ⟪ ab0 , ch-init fc ⟫ = ⟪ ab , ch-init ( subst (λ k → FClosure A f y k) (sym (HasPrev.x=fy has-prev)) (fsuc _ fc )) ⟫
-          ... | ⟪ ab0 , ch-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab ,
+          ... | ⟪ ab0 , ch-is-sup u u≤x is-sup fc ⟫ = ? -- ⟪ ab ,
                      -- 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))  ⟫
+                     --     (sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u≤x is-sup (fsuc _ fc))  ⟫
 
           zc70 : HasPrev A pchain f x  → ¬ xSUP pchain f x
           zc70 pr xsup = ?
@@ -1460,23 +1461,23 @@
                  pchain0=1 =  ==→o≡ record { eq→ = zc10 ; eq← = zc11 } where
                      zc10 :  {z : Ordinal} → OD.def (od pchain) z → OD.def (od pchain1) z
                      zc10 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-                     zc10 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc12 fc where
+                     zc10 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc12 fc where
                           zc12 : {z : Ordinal} →  FClosure A f (supf0 u) z → odef pchain1 z
                           zc12 (fsuc x fc) with zc12 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) ⟫
+                          ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫
                           zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫
                      zc11 :  {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z
                      zc11 {z} ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-                     zc11 {z} ⟪ az , ch-is-sup u u<x is-sup fc ⟫ = zc13 fc where
+                     zc11 {z} ⟪ az , ch-is-sup u u≤x is-sup fc ⟫ = zc13 fc where
                           zc13 : {z : Ordinal} →  FClosure A f (supf1 u) z → odef pchain z
                           zc13 (fsuc x fc) with zc13 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) ⟫
+                          ... | ⟪ ua1 , ch-is-sup u u≤x is-sup fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-is-sup u u≤x is-sup (fsuc _ fc₁) ⟫
                           zc13 (init asu su=z ) with trio< u x
                           ... | tri< a ¬b ¬c = ⟪ ? , ch-is-sup u ? ? (init ? ? ) ⟫
                           ... | tri≈ ¬a b ¬c = ?
-                          ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u<x c )
+                          ... | tri> ¬a ¬b c = ? -- ⊥-elim ( o≤> u≤x c )
                  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⊆ ? (ZChain.sup (pzc (osuc z) {!!}) {!!} )
@@ -1674,7 +1675,7 @@
                    z31 : ( * (cf nmx mc)  ≡  * mc ) ∨ ( * (cf nmx mc) < * mc )
                    z31 = <=to≤ ( MinSUP.x≤sup msp1 (subst (λ k →  odef (ZChain.chain zc) (cf nmx k)) (sym x=fy)
                        ⟪ proj2  (cf-is-≤-monotonic nmx _ (proj2 (cf-is-≤-monotonic nmx _ ua1 ) )) , ch-init (fsuc _ (fsuc _ fc))  ⟫ ))
-              not-hasprev record { ax = ax ; y = y ; ay =   ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z48 z32 ) where
+              not-hasprev record { ax = ax ; y = y ; ay =   ⟪ ua1 , ch-is-sup u u≤x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z48 z32 ) where
                    z30 : * mc < * (cf nmx mc)
                    z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1))
                    z31 : ( supf mc ≡  mc ) ∨ ( * (supf mc) < * mc )
@@ -1682,7 +1683,7 @@
                    z32 : * (supf mc) < * (cf nmx (cf nmx y))
                    z32 = ftrans<=-< z31 (subst (λ k → * mc < * k ) (cong (cf nmx) x=fy) z30 )
                    z48 : ( * (cf nmx (cf nmx y)) ≡ * (supf mc)) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) )
-                   z48 = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A u<x (fsuc _ ( fsuc  _ fc )))
+                   z48 = <=to≤ (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A ? (fsuc _ ( fsuc  _ fc )))
               is-sup :  IsSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc)  (MinSUP.asm msp1)
               is-sup = record { x≤sup = λ zy → MinSUP.x≤sup msp1 (chain-mono (cf nmx) (cf-is-≤-monotonic nmx) as0 supf (ZChain.supf-mono zc) (o<→≤ mc<A) zy ) }
 
@@ -1695,16 +1696,16 @@
                 z32 = ZChain.fcy<sup zc (o<→≤ mc<A) (fsuc _ (fsuc _ fc))
                 z31 : ( * (cf nmx d)  ≡  * d ) ∨ ( * (cf nmx d) < * d )
                 z31 = case2 ( subst (λ k → * (cf nmx k) < * d ) (sym x=fy) ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) ))
-          not-hasprev record { ax = ax ; y = y ; ay =   ⟪ ua1 , ch-is-sup u u<x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z46 z30 ) where
+          not-hasprev record { ax = ax ; y = y ; ay =   ⟪ ua1 , ch-is-sup u u≤x is-sup1 fc ⟫; x=fy = x=fy } = ⊥-elim ( <-irr z46 z30 ) where
                 z45 : (* (cf nmx (cf nmx y)) ≡ * d) ∨ (* (cf nmx (cf nmx y)) < * d) → (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d)
                 z45 p = subst (λ k → (* (cf nmx k) ≡ * d) ∨ (* (cf nmx k) < * d)) (sym x=fy) p
                 z48 : supf mc << d
                 z48 = sc<<d {mc} asc spd
                 z53 : supf u o< supf (& A)
-                z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) )
+                z53 = ordtrans<-≤ u≤x ?
                 z52 : ( u ≡ mc ) ∨  ( u << mc )
                 z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc)
-                        , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc)  (ChainP.supu=u is-sup1)) ⟫
+                        , ch-is-sup u ? is-sup1 (init (ZChain.asupf zc)  (ChainP.supu=u is-sup1)) ⟫
                 z51 : supf u o≤ supf mc
                 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 where
                     z56 : u ≡ mc → supf u ≡  supf mc
@@ -1716,7 +1717,7 @@
                 z49 : supf u o< supf mc → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) )
                 z49 su<smc = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A su<smc (fsuc _ ( fsuc  _ fc ))
                 z50 : (cf nmx (cf nmx y) ≡ supf d) ∨ (* (cf nmx (cf nmx y)) < * (supf d) )
-                z50 = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) d<A u<x (fsuc _ ( fsuc  _ fc ))
+                z50 = ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) d<A ? (fsuc _ ( fsuc  _ fc ))
                 z47 : {mc d1 : Ordinal } {asc : odef A (supf mc)} (spd : MinSUP A (uchain (cf nmx)  (cf-is-≤-monotonic nmx) asc ))
                     → (cf nmx (cf nmx y) ≡ supf mc) ∨ (* (cf nmx (cf nmx y)) < * (supf mc) ) → supf mc << d1
                     →  * (cf nmx (cf nmx y)) < * d1
@@ -1743,13 +1744,13 @@
                 z22 {a} ⟪ aa , ch-init fc ⟫ = case2 ( ( ftrans<=-< z32 ( sc<<d {mc} asc spd ) )) where
                     z32 : ( a  ≡  supf mc ) ∨ ( * a < * (supf mc) )
                     z32 = ZChain.fcy<sup zc (o<→≤ mc<A) fc
-                z22 {a} ⟪ aa , ch-is-sup u u<x is-sup1 fc ⟫ = tri u (supf mc)
+                z22 {a} ⟪ aa , ch-is-sup u u≤x is-sup1 fc ⟫ = tri u (supf mc)
                        z60 z61 ( λ sc<u → ⊥-elim ( o≤> ( subst (λ k → k o≤ supf mc) (ChainP.supu=u is-sup1) z51) sc<u )) where
                     z53 : supf u o< supf (& A)
-                    z53 = ordtrans<-≤ u<x (ZChain.supf-mono zc (o<→≤ d<A) )
+                    z53 = ordtrans<-≤ u≤x ?
                     z52 : ( u ≡ mc ) ∨  ( u << mc )
                     z52 = MinSUP.x≤sup msp1 ⟪ subst (λ k → odef A k ) (ChainP.supu=u is-sup1) (A∋fcs _ _ (cf-is-≤-monotonic nmx) fc)
-                           , ch-is-sup u z53 is-sup1 (init (ZChain.asupf zc)  (ChainP.supu=u is-sup1)) ⟫
+                           , ch-is-sup u ? is-sup1 (init (ZChain.asupf zc)  (ChainP.supu=u is-sup1)) ⟫
                     z56 : u ≡ mc → supf u ≡  supf mc
                     z56 eq = cong supf eq
                     z57 : u << mc → supf u o≤ supf mc
@@ -1763,7 +1764,7 @@
                         (subst (λ k → k o< supf mc) (sym (ChainP.supu=u is-sup1))  u<smc)  fc ) smc<<d )
                     z61 : u ≡ supf mc → (a ≡ d ) ∨ ( * a < * d )
                     z61 u=sc = case2 (fsc<<d {mc} asc spd (subst (λ k → FClosure A (cf nmx) k a) (trans (ChainP.supu=u is-sup1) u=sc) fc ) )
-                    -- u<x    : ZChain.supf zc u o< ZChain.supf zc d
+                    -- u≤x    : ZChain.supf zc u o< ZChain.supf zc d
                     --     supf u o< supf c → order
 
           sd=d : supf d ≡ d