--- a/src/zorn.agda	Sat Apr 30 10:48:23 2022 +0900
+++ b/src/zorn.agda	Sat Apr 30 14:53:04 2022 +0900
@@ -162,6 +162,24 @@
       fc12 : * y < * x
       fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c
 
+fcn-imm : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) 
+    → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → ¬ ( ( * x < * y ) ∧ ( * y < * (f x )) ) 
+fcn-imm {A} s {x} {y} f mf cx cy ⟪ x<y , y<fx ⟫ = {!!} where
+      fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → * y < * (f x ) → (y ≡ x ) ∨ ( * y < * x )
+      fc17 = {!!}
+      fc18 : {x : Ordinal } → (cx : FClosure A f s x) → {!!} -- (y ≡ x ) ∨ ( y ≡ f x )     * x < * y → ¬ ( * y < * ( f x ) )Ljjjj
+      fc18 = {!!}
+      ncx : { x : Ordinal } → (cx : FClosure A f s x) →  (cx1 : FClosure A f s (f x) ) → cx1 ≡ fsuc x cx
+      ncx {x} (init x₁) cx1 = {!!}
+      ncx {.(f x)} (fsuc x cx) cx1 = {!!}
+      fc16 : (x : Ordinal ) → (cx : FClosure A f s x) → (fcn s mf cx  ≡ fcn s mf (fsuc x cx)) ∨ ( suc (fcn s mf cx ) ≡ fcn s mf (fsuc x cx))
+      fc16 x (init sa) with proj1 (mf s sa )
+      ... | case1 _ = case1 refl
+      ... | case2 _ = case2 refl
+      fc16 .(f x) (fsuc x cx ) with proj1 (mf (f x) (A∋fc s f mf (fsuc x cx)) )
+      ... | case1 _ = case1 refl 
+      ... | case2 _ = case2 refl
+
 -- open import Relation.Binary.Properties.Poset as Poset
 
 IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n)
@@ -366,7 +384,7 @@
           ... | case2 ¬fy<x with ODC.p∨¬p O ( x ≡ & ( SUP.sup ( supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) ) ))
           ... | case1 x=sup = -- previous ordinal is a sup of a smaller ZChain
                  record { chain = schain ; chain⊆A = record { incl = A∋schain } ; f-total = scmp ; f-next = scnext 
-                     ; initial = scinit ; f-immediate =  {!!} ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup
+                     ; initial = scinit ; f-immediate =  simm ; chain∋x  = case1 (ZChain.chain∋x zc0) ; is-max = {!!} } where -- x is sup
                 sup0 = supP ( ZChain.chain zc0 ) (ZChain.chain⊆A zc0 ) (ZChain.f-total zc0) 
                 sp = SUP.sup sup0 
                 chain = ZChain.chain zc0
@@ -406,10 +424,31 @@
                 scinit :  {x : Ordinal} → odef schain x → * y ≤ * x
                 scinit {x} (case1 zx) = ZChain.initial zc0 zx
                 scinit {x} (case2 sx) with  (s≤fc (& sp) f mf sx ) |  SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) ( ZChain.chain∋x zc0 ) )
-                ... | case1 sp=x | case1 y=sp = case1 {!!}
-                ... | case1 sp=x | case2 y<sp = case2 {!!}
-                ... | case2 sp<x | case1 y=sp = case2 {!!}
-                ... | case2 sp<x | case2 y<sp = case2 {!!}
+                ... | case1 sp=x | case1 y=sp = case1 (trans y=sp (subst (λ k → k ≡ * x ) *iso sp=x ) )
+                ... | case1 sp=x | case2 y<sp = case2 (subst (λ k → * y < k ) (trans (sym *iso) sp=x) y<sp )
+                ... | case2 sp<x | case1 y=sp = case2 (subst (λ k → k < * x ) (trans *iso (sym y=sp )) sp<x )
+                ... | case2 sp<x | case2 y<sp = case2 (ptrans y<sp (subst (λ k → k < * x ) *iso sp<x) )
+                A∋za : {a : Ordinal } → odef chain a → odef A a
+                A∋za za = (subst (λ k → odef A k ) &iso (incl (ZChain.chain⊆A zc0) (subst (λ k → odef chain k ) (sym &iso) za) ) )
+                za<sup :  {a : Ordinal } → odef chain a → ( * a ≡ sp ) ∨  ( * a < sp )
+                za<sup za =  SUP.x<sup sup0 (subst (λ k → odef chain k ) (sym &iso) za )
+                simm : {a b : Ordinal}  → odef schain a → odef schain b → ¬ (* a < * b) ∧ (* b < * (f a))
+                simm {a} {b} (case1 za) (case1 zb) = ZChain.f-immediate zc0 za zb
+                simm {a} {b} (case1 za) (case2 sb) p with  proj1 (mf a (A∋za za) )
+                ... | case1 eq = <-irr (case2  (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) 
+                ... | case2 a<fa with za<sup  ( ZChain.f-next zc0 za ) | s≤fc (& sp) f mf sb
+                ... | case1 fa=sp | case1 sp=b = <-irr (case1 (trans fa=sp (trans (sym *iso) sp=b )) ) ( proj2 p )
+                ... | case2 fa<sp | case1 sp=b = <-irr (case2 fa<sp) (subst (λ k → k < * (f a) ) (trans (sym sp=b) *iso) (proj2 p ) )
+                ... | case1 fa=sp | case2 sp<b = <-irr (case2 (proj2 p )) (subst  (λ k → k < * b) (sym fa=sp) (subst (λ k → k < * b ) *iso sp<b ) )
+                ... | case2 fa<sp | case2 sp<b = <-irr (case2 (proj2 p )) (ptrans fa<sp (subst (λ k → k < * b ) *iso sp<b ) )
+                simm {a} {b} (case2 sa) (case1 zb) p with  proj1 (mf a ( subst (λ k → odef A k) &iso ( A∋schain (case2 (subst (λ k → FClosure A f (& sp) k ) (sym &iso) sa) )) ) )
+                ... | case1 eq = <-irr (case2  (subst (λ k → * b < k ) (sym eq) (proj2 p))) (proj1 p) 
+                ... | case2 b<fb with  s≤fc (& sp) f mf sa | za<sup zb
+                ... | case1 sp=a | case1 b=sp = <-irr (case1 (trans b=sp (trans (sym *iso) sp=a )) ) (proj1 p )
+                ... | case1 sp=a | case2 b<sp = <-irr (case2 (subst (λ k → * b < k ) (trans (sym *iso) sp=a)  b<sp ) ) (proj1 p )
+                ... | case2 sp<a | case1 b=sp = <-irr (case2 (subst ( λ  k → k < * a ) (trans *iso (sym b=sp)) sp<a  )) (proj1 p )
+                ... | case2 sp<a | case2 b<sp = <-irr (case2 (ptrans b<sp (subst (λ k → k < * a) *iso sp<a ))) (proj1 p )
+                simm {a} {b} (case2 sa) (case2 sb) p = {!!}
           ... | case2 ¬x=sup =  -- x is not f y' nor sup of former ZChain from y
                    record { chain = ZChain.chain zc0 ; chain⊆A = ZChain.chain⊆A zc0 ; f-total = ZChain.f-total zc0 ; f-next = ZChain.f-next zc0
                      ; initial = ZChain.initial zc0 ; f-immediate = ZChain.f-immediate zc0 ; chain∋x  =  ZChain.chain∋x zc0 ; is-max = {!!} }  where -- no extention