Members/kono/Proof/ZF-in-agda: src/zorn.agda comparison

comparison src/zorn.agda @ 966:39c680188738

...

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Sat, 05 Nov 2022 13:21:42 +0900
parents	1c1c6a6ed4fa
children	cd0ef83189c5

comparison

equal deleted inserted replaced

-:1c1c6a6ed4fa
+:39c680188738
 {-# OPTIONS --allow-unsolved-metas #-}
 open import Level hiding ( suc ; zero )
 open import Ordinals
 open import Relation.Binary
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality
 import OD
 module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where
 --
 -- Zorn-lemma : { A : HOD }
 --     → o∅ o< & A
 --     → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B   ) -- SUP condition
 --     → Maximal A
 --
 open import zf
 open import logic
 -- open import partfunc {n} O
 open import Relation.Nullary
 open import Data.Empty
 import BAlgbra
 open import Data.Nat hiding ( _<_ ; _≤_ )
 open import Data.Nat.Properties
 open import nat
 open inOrdinal O
 open OD O
 open OD.OD
 --
 -- Partial Order on HOD ( possibly limited in A )
 --
 _<<_ : (x y : Ordinal ) → Set n
 x << y = * x < * y
 _<=_ : (x y : Ordinal ) → Set n    -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain
 x <= y = (x ≡ y ) ∨ ( * x < * y )
 POO : IsStrictPartialOrder _≡_ _<<_
 POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans }
 ; trans = IsStrictPartialOrder.trans PO
 ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y
 ; <-resp-≈ =  record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } }
 _≤_ : (x y : HOD) → Set (Level.suc n)
 x ≤ y = ( x ≡ y ) ∨ ( x < y )
 ≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z
 ≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl
 ≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z
 ≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y
 ≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z )
 <=-trans : {x y z : Ordinal } →  x <=  y →  y <=  z →  x <=  z
 <=-trans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl
 <=-trans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z
 <=-trans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y
 <=-trans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z )
 ftrans<=-< : {x y z : Ordinal } →  x <=  y →  y << z →  x <<  z
 ftrans<=-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq))  y<z
 ftrans<=-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z
 <=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y
 <=to≤ (case1 eq) = case1 (cong (*) eq)
 <=to≤ (case2 lt) = case2 lt
 ≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y
 ≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso  (cong (&) eq) )
 ≤to<= (case2 lt) = case2 lt
 <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
 <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO   (sym a=b) b<a
 ptrans =  IsStrictPartialOrder.trans PO
 open _==_
 open _⊆_
 -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A
 --     → ({x : HOD} → A ∋ x →  ({y : HOD} → A ∋  y → y < x → P y ) → P x) → P x
 -- <-TransFinite = ?
 --
 -- Closure of ≤-monotonic function f has total order
 A∋fc {A} s f mf (init as refl ) = as
 A∋fc {A} s f mf (fsuc y fcy) = proj2 (mf y ( A∋fc {A} s  f mf fcy ) )
 A∋fcs : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A s
 A∋fcs {A} s f mf (init as refl) = as
 A∋fcs {A} s f mf (fsuc y fcy) = A∋fcs {A} s f mf fcy
 s≤fc : {A : HOD} (s : Ordinal ) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → * s ≤ * y
 s≤fc {A} s {.s} f mf (init x refl ) = case1 refl
 s≤fc {A} s {.(f x)} f mf (fsuc x fcy) with proj1 (mf x (A∋fc s f mf fcy ) )
 ... | case1 x=fx = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy )
 ... | case2 x<fx with s≤fc {A} s f mf fcy
 ... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx )
 ... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx )
 fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ
 fcn s mf (init as refl) = zero
 fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p))
 ... | case1 eq = fcn s mf p
 ... | case2 y<fy = suc (fcn s mf p )
 fcn-inject : {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 ) → fcn s mf cx  ≡ fcn s mf cy → * x ≡ * y
 fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where
 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y  ) { j : ℕ } →  ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq )
 fc06 {x} {y} refl {j} not = fc08 not where
 fc08 :  {j : ℕ} → ¬ suc j ≡ 0
 fc08 ()
 fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x
 fc07 {x} (init as refl) eq = refl
 fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) )
 ... | case1 x=fx = subst (λ k → * s ≡  k ) x=fx ( fc07 cx eq )
 fc04 = fc00 i j (cong pred i=j) cx1 cy (cong pred i=x1) (cong pred j=y)
 ... | case2 x<fx | case1 y=fy = subst (λ k → * (f x) ≡ k ) y=fy (fc03 y cy j=y) where
 fc03 : (y1 : Ordinal) → (cy1 : FClosure A f s y1 ) →  suc j ≡ fcn s mf cy1 → * (f x)  ≡ * y1
 fc03 y1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x)
 fc03 .(f y1) (fsuc y1 cy1) j=y1 with proj1 (mf y1 (A∋fc s f mf cy1 ) )
 ... | case1 eq = trans ( fc03  y1 cy1 j=y1 ) eq
 ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where
 fc05 : * x ≡ * y1
 fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1)
 ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y)))
 fcn-< : {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 ) → fcn s mf cx Data.Nat.< fcn s mf cy  → * x < * y
 fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where
 fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y  ) { j : ℕ } →  ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq )
 fc06 {x} {y} refl {j} not = fc08 not where
 fc08 :  {j : ℕ} → ¬ suc j ≡ 0
 fc08 ()
 fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y
 fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x)
 fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) )
 ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i)  )
 ... | case2 y<fy with <-cmp (fcn s mf cx ) i
 ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c )
 ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy
 ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where
 fc03 :  suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy
 fc03 eq = cong pred eq
 fc02 :  * x < * y1
 fc02 =  fc01 i cx cy (fc03 i=y ) a
 fcn-cmp : {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 ) → Tri (* x < * y) (* x ≡ * y) (* y < * x )
 fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy )
 ... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
 fc11 : * x < * y
 fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a
 ... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where
 fc10 : * x ≡ * y
 fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b
 ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12  where
 fc12 : * y < * x
 fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c
 record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal )   : Set n where
 field
 ax : odef A x
 y : Ordinal
 ay : odef B y
 x=fy :  x ≡ f y
 record IsSUP (A B : HOD) {x : Ordinal } (xa : odef A x)     : Set n where
 field
 x≤sup   : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x )
 record IsMinSUP (A B : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (xa : odef A x)     : Set n where
 field
 x≤sup   : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x )
 minsup : { sup1 : Ordinal } → odef A sup1
 →  ( {z : Ordinal  } → odef B z → (z ≡ sup1 ) ∨ (z << sup1 ))  → x o≤ sup1
 not-hp : ¬ ( HasPrev A B f x )
 record SUP ( A B : HOD )  : Set (Level.suc n) where
 field
 --
 --  sup and its fclosure is in a chain HOD
 --    chain HOD is sorted by sup as Ordinal and <-ordered
 --    whole chain is a union of separated Chain
 --    minimum index is sup of y not ϕ
 --
 record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f)  {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where
 field
 fcy<sup  : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u )
 order    : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u )
 supu=u   : supf u ≡ u
 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-is-sup  : (u : Ordinal) {z : Ordinal }  (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u )
+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 )
 ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z
 --
 --         f (f ( ... (sup y))) f (f ( ... (sup z1)))
 --        /          |         /             |
 chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f)  {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal )
 {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a )
 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
+... | 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
+ct00 = trans (cong (*) eq) eq1
+... | 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
+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
+... | 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
+ct00 = sym (trans (cong (*) eq) eq1 )
+... | 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
+ct00 : * b < * (supf ua)
+ct00 = lt
+ct01 : * b < * a
+ct01 with s≤fc (supf ua) f mf fca
+... | case1 eq =  subst (λ k → * b < k ) eq ct00
+... | case2 lt =  IsStrictPartialOrder.trans POO ct00 lt
 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub
 ... | tri< a₁ ¬b ¬c with ChainP.order supb  (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁)  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
 ct00 = trans (cong (*) eq) eq1
 ... | case2 lt =  tri< ct02  (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt)  where
 ct02 : * a < * b
 ct02 = subst (λ k → * k < * b ) (sym eq) lt
 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02  (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt)  where
 ct03 : * a < * (supf ub)
 ct03 = lt
 ct02 : * a < * b
 ct02 with s≤fc (supf ub) f mf fcb
 ... | case1 eq =  subst (λ k → * a < k ) eq ct03
 ... | case2 lt =  IsStrictPartialOrder.trans POO ct03 lt
 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x  supb fcb) | tri≈ ¬a  eq ¬c
 = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb )
 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) 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
 ct00 = sym (trans (cong (*) eq) eq1)
 ... | case2 lt =  tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02    where
 ct02 : * b < * a
 ct02 = subst (λ k → * k < * a ) (sym eq) lt
 ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04    where
 ct05 : * b < * (supf ua)
 ct05 = lt
 ct04 : * b < * a
 ct04 with s≤fc (supf ua) f mf fca
 ... | case1 eq =  subst (λ k → * b < k ) eq ct05
 ... | case2 lt =  IsStrictPartialOrder.trans POO ct05 lt
 ∈∧P→o< :  {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A
 ∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
 -- Union of supf z which o< x
 --
 UnionCF : ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y )
 ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD
 UnionCF A f mf ay supf x
 = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
 supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } →  x o≤  y  → supf x o≤ supf y )
 → supf x o< supf y → x o<  y
 supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y
 ... | tri< a ¬b ¬c = a
 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy )
 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) )
 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy )
 record MinSUP ( A B : HOD )  : Set n where
 field
 sup : Ordinal
 asm : odef A sup
 x≤sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup )
 minsup : { sup1 : Ordinal } → odef A sup1
 →  ( {x : Ordinal  } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 ))  → sup o≤ sup1
 z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A
 z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
 M→S  : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y}  { x : Ordinal }
 →  (supf : Ordinal → Ordinal )
 →  MinSUP A (UnionCF A f mf ay supf x)
 → SUP A (UnionCF A f mf ay supf x)
 M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms)
 ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x≤sup = ms00 } where
 msup = MinSUP.sup ms
 ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup)
 ms00 {z} uz with MinSUP.x≤sup ms uz
 ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq))
 ... | case2 lt = case2 (subst₂ (λ j k →  j < k ) *iso refl lt )
 chain-mono : {A : HOD}  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f )  {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal )
 (supf-mono : {x y : Ordinal } →  x o≤  y  → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b
 → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c
+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  ⟫
 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
 field
 supf :  Ordinal → Ordinal
 sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
 → IsSUP A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) f b ) → supf b ≡ b
 asupf :  {x : Ordinal } → odef A (supf x)
 supf-mono : {x y : Ordinal } → x o≤  y → supf x o≤ supf y
 supf-< : {x y : Ordinal } → supf x o< supf  y → supf x << supf y
 supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z
-chain∋init : odef (UnionCF A f mf ay supf z) y
+minsup : {x : Ordinal } → x o≤ z  → MinSUP A (UnionCF A f mf ay supf x)
-minsup : {x : Ordinal } → x o≤ z  → MinSUP A (UnionCF A f mf ay supf x)
 supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z )
 csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain
 chain : HOD
 chain = UnionCF A f mf ay supf z
 chain⊆A : chain ⊆' A
 chain⊆A = λ lt → proj1 lt
 sup : {x : Ordinal } → x o≤ z  → SUP A (UnionCF A f mf ay supf x)
 sup {x} x≤z = M→S supf (minsup x≤z)
 s=ms : {x : Ordinal } → (x≤z : x o≤ z ) → & (SUP.sup (sup x≤z)) ≡ MinSUP.sup (minsup x≤z)
 s=ms {x} x≤z = &iso
+chain∋init : odef chain y
+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 ) ⟫
 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
 zc7 : y <= supf u
 zc7 = ChainP.fcy<sup is-sup (init ay refl)
 f-total : IsTotalOrderSet chain
 f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where
 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
 uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb))
 supf-<= : {x y : Ordinal } → supf x <= supf  y → supf x o≤ supf y
 supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy
 supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y)
 ... | tri< a ¬b ¬c = o<→≤ a
 ... | tri≈ ¬a b ¬c = o≤-refl0 b
 ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) )
 supf-inject : {x y : Ordinal } → supf x o< supf y → x o<  y
 supf-inject {x} {y} sx<sy with trio< x y
 ... | tri< a ¬b ¬c = a
 ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy )
 ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) )
 ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy )
 ... | case2 lt = ⊥-elim ( o<> sx<sy lt )
 fcy<sup  : {u w : Ordinal } → u o≤ z  → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf
-fcy<sup  {u} {w} u≤z fc with chain∋init
+fcy<sup  {u} {w} u≤z fc with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc)
-... | ⟪ ay1 , ch-is-sup uy uy<x is-sup fcy ⟫ = <=-trans (ChainP.fcy<sup is-sup fc) ?
+, ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫
+... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso  (trans eq (sym (supf-is-minsup u≤z ) ) ))
--- with MinSUP.x≤sup (minsup u≤z) ⟪ subst (λ k → odef A k ) (sym &iso) (A∋fc {A} y f mf fc)
+... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt )
---     , ? ⟫ --ch-init (subst (λ k → FClosure A f y k) (sym &iso) fc ) ⟫
--- ... | case1 eq = case1 (subst (λ k → k ≡ supf u ) &iso  (trans eq (sym (supf-is-minsup u≤z ) ) ))
+-- ordering is not proved here but in ZChain1
--- ... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt )
+IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp
--- ordering is not proved here but in ZChain1
-IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp
 → ({y : Ordinal} → odef (UnionCF A f mf ay supf x) y → (y ≡ sp ) ∨ (y << sp ))
 → ( {a : Ordinal } → a << f a )
 → ¬ ( HasPrev A (UnionCF A f mf ay supf x) f sp )
 IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<-irr ( <=to≤  fsp≤sp) sp<fsp ) where
 sp<fsp : sp << f sp
 sp<fsp = <-mono-f
 pr = HasPrev.y hp
 im00 : f (f pr) <= sp
 im00 = is-sup ( f-next (f-next (HasPrev.ay hp)))
 fsp≤sp : f sp <=  sp
 fsp≤sp = subst (λ k → f k <= sp ) (sym (HasPrev.x=fy hp)) im00
-record ZChain1 ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f)
+UChain⊆ : { supf1 : Ordinal → Ordinal }
+→ ( { x : Ordinal } → x o< z → supf x ≡ supf1 x)
+→ ( { 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⊆ {supf1} eq<x lex ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
+UChain⊆ {supf1} 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-inject 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 ) )
+fc1 : FClosure A f (supf1 u) x
+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))
+uc02 :  supf1 u o≤ supf1 s
+uc02 = begin
+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 ⟩
+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)
+(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)  }
+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
 field
 is-max :  {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z  → (ab : odef A b)
 → HasPrev A (UnionCF A f mf ay supf z) f b ∨  IsSUP A (UnionCF A f mf ay supf b) ab
 → * a < * b  → odef ((UnionCF A f mf ay supf z)) b
 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)
 record Maximal ( A : HOD )  : Set (Level.suc n) where
 field
 maximal : HOD
 as : A ∋ maximal
 ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x       -- A is Partial, use negative
-Zorn-lemma : { A : HOD }
+init-uchain : (A : HOD)  ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y )
-→ o∅ o< & A
+{ supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y
+init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl)   ⟫
+Zorn-lemma : { A : HOD }
+→ o∅ o< & A
 → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B   ) -- SUP condition
 → Maximal A
 Zorn-lemma {A}  0<A supP = zorn00 where
 <-irr0 : {a b : HOD} → A ∋ a → A ∋ b  → (a ≡ b ) ∨ (a < b ) → b < a → ⊥
 <-irr0 {a} {b} A∋a A∋b = <-irr
 z07 :   {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A
 z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p )))
 s : HOD
 s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))
 as : A ∋ * ( & s  )
 as =  subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A ))  )
 as0 : odef A  (& s  )
 as0 =  subst (λ k → odef A k ) &iso as
 s<A : & s o< & A
 s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as )
 HasMaximal : HOD
 HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) →  odef A m → ¬ (* x < * m)) }  ; odmax = & A ; <odmax = z07 }
 no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) →  odef A m →  odef A x ∧ (¬ (* x < * m) )) →  ⊥
 no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ )
 Gtx : { x : HOD} → A ∋ x → HOD
 Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 }
 z08  : ¬ Maximal A →  HasMaximal =h= od∅
 z08 nmx  = record { eq→  = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt)
 ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← =  λ {y} lt → ⊥-elim ( ¬x<0 lt )}
 x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m))
 x-is-maximal nmx {x} ax nogt m am  = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) ,  ¬x<m  ⟫ where
 ¬x<m :  ¬ (* x < * m)
 ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫  (≡o∅→=od∅ nogt)
 minsupP :  ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B
 minsupP B B⊆A total = m02 where
 xsup : (sup : Ordinal ) → Set n
 xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup )
 ∀-imply-or :  {A : Ordinal  → Set n } {B : Set n }
 → ((x : Ordinal ) → A x ∨ B) →  ((x : Ordinal ) → A x) ∨ B
 s1 = & (SUP.sup S)
 m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 )
 m05 {w} bw with SUP.x≤sup S {* w} (subst (λ k → odef B k) (sym &iso) bw )
 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) )
 ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt )
 m02 : MinSUP A B
 m02 = dont-or (m00 (& A)) m03
 -- Uncountable ascending chain by axiom of choice
 cf : ¬ Maximal A → Ordinal → Ordinal
 cf  nmx x with ODC.∋-p O A (* x)
 -- maximality of chain
 --
 --     supf is fixed for z ≡ & A , we can prove order and is-max
 --
 SZ1 : ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f)
 {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal)   → ZChain1 A f mf ay zc x
 SZ1 f mf {y} ay zc x = zc1 x where
 chain-mono1 :  {a b c : Ordinal} → a o≤ b
 → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c
 chain-mono1  {a} {b} {c} a≤b = chain-mono f mf ay (ZChain.supf zc) (ZChain.supf-mono zc) a≤b
 is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a → (ab : odef A b)
 → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) f b
 → * 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))  ⟫
 supf = ZChain.supf zc
 csupf-fc : {b s z1 : Ordinal} → b o< & A → supf s o< supf b → FClosure A f (supf s) z1 → UnionCF A f mf ay supf b ∋ * z1
 csupf-fc {b} {s} {z1} b<z ss<sb (init x refl ) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc05  where
 s<z = ordtrans s<b b<z
 zc04 : odef (UnionCF A f mf ay supf (& A))  (supf s)
 zc04 = ZChain.csupf zc (z09 (ZChain.asupf zc))
 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
 zc07 : FClosure A f (supf u) (supf s)   -- supf u ≤ supf s  → supf u o≤ supf s
 zc07 = fc
 zc06 : supf u ≡ u
 zc06 = ChainP.supu=u is-sup
 zc08 : supf u o< supf b
 zc08 = ordtrans≤-< (ZChain.supf-<= zc (≤to<= ( s≤fc _ f mf fc ))) ss<sb
 csupf-fc {b} {s} {z1} b<z ss≤sb (fsuc x fc) = subst (λ k → odef (UnionCF A f mf ay supf b) k ) (sym &iso) zc04  where
 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-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf _ as) , ch-is-sup u u<x is-sup (fsuc _ 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) ⟫
 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) ) )
 zc00 = MinSUP.x≤sup (ZChain.minsup zc (o<→≤  b<z) ) (subst (λ k →  odef (UnionCF A f mf ay (ZChain.supf zc) b) k ) &iso (csupf-fc b<z ss<sb fc ))
 -- supf (supf b) ≡ supf b
 is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a →
 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) →
 HasPrev A (UnionCF A f mf ay supf x) f b  ∨ IsSUP A (UnionCF A f mf ay supf b) ab →
 * a < * b → odef (UnionCF A f mf ay supf x) b
 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
 b<A : b o< & A
 b<A = z09 ab
 b<x : b o< x
 b<x  = ZChain.supf-inject zc sb<sx
 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b
 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay =
 chain-mono1 (o<→≤ b<x) (HasPrev.ay  nhp) ; x=fy = HasPrev.x=fy nhp } )
 m05 : ZChain.supf zc b ≡ b
 m05 =  ZChain.sup=u zc ab (o<→≤ (z09 ab) ) ⟪ record { x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz  }  , m04 ⟫
 m08 : {z : Ordinal} → (fcz : FClosure A f y z ) → z <= ZChain.supf zc b
 m08 {z} fcz = ZChain.fcy<sup zc (o<→≤ b<A) fcz
 m09 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b
 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b
 m09 {s} {z} s<b fcz = order b<A s<b fcz
 m06 : ChainP A f mf ay supf b
 m06 = record {  fcy<sup = m08  ; order = m09 ; supu=u = m05 }
 ... | no lim = record { is-max = is-max ; order = order }  where
 is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a →
 ZChain.supf zc b o< ZChain.supf zc x → (ab : odef A b) →
 HasPrev A (UnionCF A f mf ay supf x) f b  ∨ IsSUP A (UnionCF A f mf ay supf b) ab →
 * a < * b → odef (UnionCF A f mf ay supf x) b
 is-max {a} {b} ua sb<sx ab P a<b with ODC.or-exclude O P
 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
+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 )
-= ⟪ ab , ch-is-sup b sb<sx m06 (subst (λ k → FClosure A f k b) (sym m05) (init ab refl))  ⟫ where
+... | 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
 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
 b<x  = ZChain.supf-inject zc sb<sx
 m07 : {z : Ordinal} → FClosure A f y z → z <= ZChain.supf zc b
 m07 {z} fc = ZChain.fcy<sup zc (o<→≤ m09) fc
 m08 : {s z1 : Ordinal} → ZChain.supf zc s o< ZChain.supf zc b
 → FClosure A f (ZChain.supf zc s) z1 → z1 <= ZChain.supf zc b
 m08 {s} {z1} s<b fc = order m09 s<b fc
 m04 : ¬ HasPrev A (UnionCF A f mf ay supf b) f b
 m04 nhp = proj1 is-sup ( record { ax = HasPrev.ax nhp ; y = HasPrev.y nhp ; ay =
 chain-mono1 (o<→≤ b<x) (HasPrev.ay  nhp)
 ; x=fy = HasPrev.x=fy nhp } )
 m05 : ZChain.supf zc b ≡ b
 m05 = ZChain.sup=u zc ab (o<→≤  m09) ⟪ record { x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫    -- ZChain on x
 m06 : ChainP A f mf ay supf b
 m06 = record { fcy<sup = m07 ;  order = m08 ; supu=u = m05 }
 uchain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → HOD
 uchain f mf {y} ay = record { od = record { def = λ x → FClosure A f y x } ; odmax = & A ; <odmax =
 λ {z} cz → subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (A∋fc y f mf cz ))) }
 utotal : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y)
 → IsTotalOrderSet (uchain f mf ay)
 utotal f mf {y} ay {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where
 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
 uz01 = fcn-cmp y f mf ca cb
 ysup : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y)
 →  MinSUP A (uchain f mf ay)
 ysup f mf {y} ay = minsupP (uchain f mf ay) (λ lt → A∋fc y f mf lt)  (utotal f mf ay)
-UChainPreserve : {x z : Ordinal } { f : Ordinal → Ordinal } → {mf : ≤-monotonic-f A f } {y : Ordinal} {ay : odef A y}
-→ { supf0 supf1 : Ordinal → Ordinal }
-→ ( ( z : Ordinal ) → z o< x → supf0 z ≡ supf1 z)
-→ UnionCF A f mf ay supf0 x ≡ UnionCF A f mf ay supf1 x
-UChainPreserve {x} {f} {mf} {y} {ay} {supf0} {supf1} <x→eq = ?
 SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B
 SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; as = SUP.as sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt)    }
 record xSUP (B : HOD) (f : Ordinal → Ordinal ) (x : Ordinal) : Set n where
 --
 -- create all ZChains under o< x
 --
 ind : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal)
 → ((z : Ordinal) → z o< x → ZChain A f mf ay z) → ZChain A f mf ay x
 ind f mf {y} ay x prev with Oprev-p x
 ... | yes op = zc41 where
 --
 -- we have previous ordinal to use induction
 --
 px = Oprev.oprev op
 zc : ZChain A f mf ay (Oprev.oprev op)
 zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc )
 px<x : px o< x
 px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc
 opx=x : osuc px ≡ x
 opx=x = Oprev.oprev=x op
 zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px
 zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt
 supf0 = ZChain.supf zc
 pchain  : HOD
 pchain   = UnionCF A f mf ay supf0 px
 supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b
 supf-mono = ZChain.supf-mono zc
 zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x )
 zc04 {b} b≤x with trio< b px
 ... | tri< a ¬b ¬c = case1 (o<→≤ a)
 ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b)
 ... | tri> ¬a ¬b px<b with osuc-≡< b≤x
 ... | case1 eq = case2 eq
 ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x  ⟫ )
 --
 -- find the next value of supf
 --
 pchainpx : HOD
 pchainpx = record { od = record { def = λ z →  (odef A z ∧ UChain A f mf ay supf0 px z )
 ∨ FClosure A f (supf0 px) z  } ; odmax = & A ; <odmax = zc00 } where
 zc00 : {z : Ordinal } → (odef A z ∧ UChain A f mf ay supf0 px z ) ∨ FClosure A f (supf0 px) z → z o< & A
 zc00 {z} (case1 lt) = z07 lt
 zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf fc )
 zc02 : { a b : Ordinal } → odef A a ∧ UChain A f mf ay supf0 px a → FClosure A f (supf0 px) b → a <= b
 zc02 {a} {b} ca fb = zc05 fb where
 zc06 : MinSUP.sup (ZChain.minsup zc o≤-refl) ≡ supf0 px
 zc06 = trans (sym ( ZChain.supf-is-minsup zc o≤-refl )) refl
 zc05 : {b : Ordinal } → FClosure A f (supf0 px) b → a <= b
 zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc (supf0 px) f mf fb ))
 ... | case1 eq = subst (λ k → a <= k ) (subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) eq)) (zc05 fb)
 ... | case2 lt = <=-trans (zc05 fb) (case2 lt)
 zc05 (init b1 refl) with MinSUP.x≤sup (ZChain.minsup zc o≤-refl)
 (subst (λ k → odef A k ∧ UChain A f mf ay supf0 px k) (sym &iso) ca )
 ... | case1 eq = case1 (subst₂ (λ j k → j ≡ k ) &iso zc06 eq )
 ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso (cong (*) zc06) lt )
 ptotal : IsTotalOrderSet pchainpx
 ptotal (case1 a) (case1 b) =  subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso
 (chain-total A f mf ay supf0 (proj2 a) (proj2 b))
 ptotal {a0} {b0} (case1 a) (case2 b) with zc02 a b
 ... | case1 eq = tri≈ (<-irr (case1 (sym eq1))) eq1 (<-irr (case1 eq1)) where
 eq1 : a0 ≡ b0
 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq )
 ... | case2 lt = tri< lt1 (λ eq → <-irr (case1 (sym eq)) lt1) (<-irr (case2 lt1)) where
 eq1 = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq )
 ... | case2 lt = tri> (<-irr (case2 lt1)) (λ eq → <-irr (case1 eq) lt1) lt1  where
 lt1 : a0 < b0
 lt1 = subst₂ (λ j k → j < k ) *iso *iso lt
 ptotal (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp (supf0 px) f mf a b)
 pcha : pchainpx ⊆' A
 pcha (case1 lt) = proj1 lt
 pcha (case2 fc) = A∋fc _ f mf fc
 sup1 : MinSUP A pchainpx
 sup1 = minsupP pchainpx pcha ptotal
 sp1 = MinSUP.sup sup1
 --
 --     supf0 px o≤ sp1
 --
 zc41 : ZChain A f mf ay x
 zc41 with MinSUP.x≤sup sup1 (case2 (init (ZChain.asupf zc {px}) refl ))
 zc41 | (case2 sfpx<sp1) =  record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf1-mono ; supf-< = ?
 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = csupf1    }  where
 --  supf0 px is included by the chain of sp1
 --  ( UnionCF A f mf ay supf0 px ∪ FClosure (supf0 px) ) ≡ UnionCF supf1 x
 --  supf1 x ≡ sp1, which is not included now
 supf1 : Ordinal → Ordinal
 supf1 z with trio< z px
 ... | tri< a ¬b ¬c = supf0 z
 ... | tri≈ ¬a b ¬c = supf0 z
 ... | tri> ¬a ¬b c = sp1
 sf1=sf0 : {z : Ordinal } → z o≤ px → supf1 z ≡ supf0 z
 sf1=sf0 {z} z≤px with trio< z px
 ... | tri< a ¬b ¬c = refl
 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z )
 ... | tri> ¬a ¬b c = refl
 asupf1 : {z : Ordinal } → odef A (supf1 z)
 asupf1 {z} with trio< z px
 ... | tri< a ¬b ¬c = ZChain.asupf zc
 ... | tri≈ ¬a b ¬c = ZChain.asupf zc
 ... | tri> ¬a ¬b c = MinSUP.asm sup1
 supf1-mono : {a b : Ordinal } → a o≤ b → supf1 a o≤ supf1 b
 supf1-mono {a} {b} a≤b with trio< b px
 ... | tri< a ¬b ¬c =  subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (o<→≤ (ordtrans≤-< a≤b a)))) refl ( supf-mono a≤b )
 ... | tri≈ ¬a b ¬c =  subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (subst (λ k → a o≤ k) b a≤b))) refl ( supf-mono a≤b )
 supf1-mono {a} {b} a≤b | tri> ¬a ¬b c with trio< a px
 ... | tri< a<px ¬b ¬c = zc19 where
 zc21 : MinSUP A (UnionCF A f mf ay supf0 a)
 zc21 = ZChain.minsup zc (o<→≤ a<px)
 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1)
 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o<→≤ a<px) ux ) )
 zc19 : supf0 a o≤ sp1
 zc19 = subst (λ k → k o≤ sp1) (sym (ZChain.supf-is-minsup zc  (o<→≤ a<px))) ( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 )
 ... | tri≈ ¬a b ¬c = zc18 where
 zc21 : MinSUP A (UnionCF A f mf ay supf0 a)
 zc21 = ZChain.minsup zc (o≤-refl0 b)
 zc20 : MinSUP.sup zc21 ≡ supf0 a
 zc20 = sym (ZChain.supf-is-minsup zc (o≤-refl0 b))
 zc24 : {x₁ : Ordinal} → odef (UnionCF A f mf ay supf0 a) x₁ → (x₁ ≡ sp1) ∨ (x₁ << sp1)
 zc24 {x₁} ux = MinSUP.x≤sup sup1 (case1 (chain-mono f mf ay supf0 (ZChain.supf-mono zc) (o≤-refl0 b) ux ) )
 zc18 : supf0 a o≤ sp1
 zc18 = subst (λ k → k o≤ sp1) zc20( MinSUP.minsup zc21 (MinSUP.asm sup1) zc24 )
 ... | tri> ¬a ¬b c = o≤-refl
 fcup : {u z : Ordinal } → FClosure A f (supf1 u) z → u o≤ px → FClosure A f (supf0 u) z
 fcup {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sf1=sf0 u≤px) fc
 fcpu : {u z : Ordinal } → FClosure A f (supf0 u) z → u o≤ px →  FClosure A f (supf1 u) z
 fcpu {u} {z} fc u≤px = subst (λ k → FClosure A f k z ) (sym (sf1=sf0 u≤px)) fc
 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≤px : u o≤ px
 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-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-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-... | case2 fc = case2 (fsuc _ fc)
+... | 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
 ; 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
 asp0 : odef A (supf0 u)
 asp0 = ZChain.asupf zc
 zc17 :  {s : Ordinal} {z1 : Ordinal} → supf0 s o< supf0 u →
 FClosure A f (supf0 s) z1 → (z1 ≡ supf0 u) ∨ (z1 << supf0 u)
 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) ((sf1=sf0 u≤px)) ( ChainP.order is-sup
 (subst₂ (λ j k → j o< k ) (sym (sf1=sf0 zc18)) (sym (sf1=sf0 u≤px)) ss<spx) (fcpu fc zc18) ) where
 zc18 : s o≤ px
 zc18 = ordtrans (ZChain.supf-inject zc ss<spx) u≤px
 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  ⟫ )
 zc12 : {z : Ordinal} → odef pchainpx z → odef (UnionCF A f mf ay supf1 x) z
+zc12 {z} (case1 ⟪ az , ch-init fc ⟫ ) = ⟪ az , ch-init fc ⟫
 zc12 {z} (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 : 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) )
 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-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-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 ) with trio< u px | inspect supf1 u
 ... | tri< a ¬b ¬c | _ = ⟪ asp , ch-is-sup u
 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)
 zc17 {s} {z1} ss<su fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k)) (sym (sf1=sf0 (o<→≤ u<px))) ( ChainP.order is-sup
 (subst₂ (λ j k → j o< k ) (sf1=sf0 s≤px) (sf1=sf0 (o<→≤ u<px)) ss<su) (fcup fc s≤px) ) where
 s≤px : s o≤ px -- ss<su : supf1 s o< supf1 u
 s≤px = ordtrans ( supf-inject0 supf1-mono ss<su ) (o<→≤ u<px)
 zc14 : FClosure A f (supf1 u) (supf0 u)
 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
 ; 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 )
 zc18 : supf1 px ≡ px
 zc18 = begin
 --- supf0 px o< sp1
 zc20 : (supf0 px ≡ px ) ∨ ( supf0 px o< px )
 zc20 = ?
 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-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-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 ) with  zc20
 ... | case1 sfpx=px =  ⟪ asp , ch-is-sup px zc18
 record {fcy<sup = zc13 ; order = zc17 ; supu=u = zc15 } zc14 ⟫ where
 zc15 : supf1 px ≡ px
 zc15 = trans (sf1=sf0 o≤-refl ) (sfpx=px)
 zc18 : supf1 px o< supf1 x
 zc18 = ?
 zc14 = init (subst (λ k → odef A k) (sym (sf1=sf0 o≤-refl)) asp) (sf1=sf0 o≤-refl)
 zc13 : {z : Ordinal } → FClosure A f y z → (z ≡ supf1 px ) ∨ ( z << supf1 px )
 zc13 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 o≤-refl)) ( ZChain.fcy<sup zc o≤-refl fc )
 zc17 :  {s : Ordinal} {z1 : Ordinal} → supf1 s o< supf1 px →
 FClosure A f (supf1 s) z1 → (z1 ≡ supf1 px) ∨ (z1 << supf1 px)
 zc17 {s} {z1} ss<spx fc = subst (λ k → (z1 ≡ k) ∨ (z1 << k )) mins-is-spx
 (MinSUP.x≤sup mins (csupf17 (fcup fc (o<→≤ s<px) )) ) where
 mins : MinSUP A (UnionCF A f mf ay supf0 px)
 mins = ZChain.minsup zc o≤-refl
 mins-is-spx : MinSUP.sup mins ≡ supf1 px
 mins-is-spx = trans (sym ( ZChain.supf-is-minsup zc o≤-refl ) ) (sym (sf1=sf0 o≤-refl ))
 s<px : s o< px
 s<px = supf-inject0 supf1-mono ss<spx
 sf<px : supf0 s o< px
 sf<px = subst₂ (λ j k → j o< k ) (sf1=sf0 (o<→≤ s<px)) (trans (sf1=sf0 o≤-refl) (sfpx=px)) ss<spx
 csupf17 : {z1 : Ordinal } → FClosure A f (supf0 s) z1 → odef (UnionCF A f mf ay supf0 px) z1
 csupf17 (init as refl ) = ZChain.csupf zc sf<px
 csupf17 (fsuc x fc) = ZChain.f-next zc (csupf17 fc)
 ... | case2 sfp<px  with ZChain.csupf zc sfp<px --  odef (UnionCF A f mf ay supf0 px) (supf0 px)
-... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18
+... | ⟪ ua1 , ch-init fc₁ ⟫ = ⟪ ua1 , ch-init fc₁ ⟫
+... | ⟪ ua1 , ch-is-sup u u<x is-sup fc₁ ⟫ = ⟪ ua1 , ch-is-sup u zc18
 record { fcy<sup = z10 ; order = z11 ; supu=u = z12 } (fcpu fc₁ ? ) ⟫  where
 z10 :  {z : Ordinal } → FClosure A f y z → (z ≡ supf1 u) ∨ ( z << supf1 u )
 z10 {z} fc = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? )) (ChainP.fcy<sup is-sup fc)
 z11  : {s z1 : Ordinal} → (lt : supf1 s o< supf1 u ) → FClosure A f (supf1 s ) z1
 → (z1 ≡ supf1 u ) ∨ ( z1 << supf1 u )
 z11 {s} {z} lt fc  = subst (λ k → (z ≡ k) ∨ ( z << k )) (sym (sf1=sf0 ? ))
 (ChainP.order is-sup lt0 (fcup fc s≤px )) where
 s<u : s o< u
 s<u = supf-inject0 supf1-mono lt
 s≤px : s o≤ px
 s≤px = ordtrans s<u ? -- (o<→≤ u<x)
 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where
 field
 tsup : MinSUP A (UnionCF A f mf ay supf1 z)
 tsup=sup : supf1 z ≡ MinSUP.sup tsup
 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x
 sup {z} z≤x with trio< z px
 ... | tri< a ¬b ¬c = record { tsup = record { sup = MinSUP.sup m  ; asm = MinSUP.asm m
 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o<→≤ a) ) (ZChain.supf-is-minsup zc (o<→≤ a)) } where
 m = ZChain.minsup zc (o<→≤ a)
 ms00 :  {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m)
 ms00 {x} ux = MinSUP.x≤sup m ?
 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} →
 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2
 ms01 {sup2} us P = MinSUP.minsup m ? ?
 ... | tri≈ ¬a b ¬c = record { tsup = record { sup = MinSUP.sup m  ; asm = MinSUP.asm m
 ; x≤sup = ms00 ; minsup = ms01 } ; tsup=sup = trans (sf1=sf0 (o≤-refl0 b) ) (ZChain.supf-is-minsup zc (o≤-refl0 b))  } where
 m = ZChain.minsup zc (o≤-refl0 b)
 ms00 :  {x : Ordinal} → odef (UnionCF A f mf ay supf1 z) x → (x ≡ MinSUP.sup m) ∨ (x << MinSUP.sup m)
 ms00 {x} ux = MinSUP.x≤sup m ?
 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} →
 ms00 {x} ux = MinSUP.x≤sup m ?
 ms01 : {sup2 : Ordinal} → odef A sup2 → ({x : Ordinal} →
 odef (UnionCF A f mf ay supf1 z) x → (x ≡ sup2) ∨ (x << sup2)) → MinSUP.sup m o≤ sup2
 ms01 {sup2} us P = MinSUP.minsup m ? ?
 csupf1 : {z1 : Ordinal } → supf1 z1 o< x → odef (UnionCF A f mf ay supf1 x) (supf1 z1)
 csupf1 {z1} sz1<x = csupf2 where
 --  z1 o< px → supf1 z1 ≡ supf0 z1
 --  z1 ≡ px , supf0 px o< px   .. px o< z1, x o≤ z1 ...  supf1 z1 ≡ sp1
 --  z1 ≡ px , supf0 px ≡  px
 psz1≤px :  supf1 z1 o≤ px
 psz1≤px =  subst (λ k → supf1 z1 o< k ) (sym opx=x) sz1<x
 csupf2 :  odef (UnionCF A f mf ay supf1 x) (supf1 z1)
 csupf2 with  trio< z1 px | inspect supf1 z1
 csupf2 | tri< a ¬b ¬c | record { eq = eq1 } with osuc-≡< psz1≤px
 ... | case2 lt  = zc12 (case1 cs03)  where
 cs03 : odef (UnionCF A f mf ay supf0 px) (supf0 z1)
 cs03 = ZChain.csupf zc (subst (λ k → k o< px) (sf1=sf0 (o<→≤ a)) lt )
 ... | case1 sfz=px with osuc-≡< ( supf1-mono (o<→≤ a) )
 ... | case1 sfz=sfpx =  zc12 (case2 (init (ZChain.asupf zc) cs04 )) where
 supu=u : supf1 (supf1 z1) ≡ supf1 z1
 supu=u = trans (cong supf1 sfz=px) (sym sfz=sfpx)
 cs04 : supf0 px ≡ supf0 z1
 cs04 = begin
 supf0 px ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩
 supf1 px ≡⟨ sym sfz=sfpx ⟩
 supf1 z1 ≡⟨ sf1=sf0 (o<→≤ a)  ⟩
 supf0 z1 ∎ where open ≡-Reasoning
 cs05 : px o< supf0 px
 cs05 = subst₂ ( λ j k → j o< k ) sfz=px (sf1=sf0 o≤-refl ) sfz<sfpx
 cs06 : supf0 px o< osuc px
 cs06 = subst (λ k → supf0 px o< k ) (sym opx=x) ?
 csupf2 | tri≈ ¬a b ¬c | record { eq = eq1 } =  zc12 (case2 (init (ZChain.asupf zc) (cong supf0 (sym b))))
 csupf2 | tri> ¬a ¬b px<z1 | record { eq = eq1 } = ?
 -- ⊥-elim ( ¬p<x<op ⟪ px<z1 , subst (λ k → z1 o< k) (sym opx=x) z1<x ⟫ )
 zc41 | (case1 sfp=sp1 ) = record { supf = supf0 ; sup=u = ? ; asupf = ? ; supf-mono = ? ; supf-< = ?
 ; supfmax = ? ; minsup = ? ; supf-is-minsup = ? ; csupf = ?    }  where
 --  supf0 px not is included by the chain
 --     supf1 x ≡ supf0 px because of supfmax
 supf1 : Ordinal → Ordinal
 supf1 z with trio< z px
 ... | tri< a ¬b ¬c = supf0 z
 ... | tri≈ ¬a b ¬c = supf0 px
 ... | tri> ¬a ¬b c = supf0 px
 sf1=sf0 : {z : Ordinal } → z o< px → supf1 z ≡ supf0 z
 zc17 : {z : Ordinal } → supf0 z o≤ supf0 px
 zc17 = ? -- px o< z, px o< supf0 px
 supf-mono1 : {z w : Ordinal } → z o≤ w → supf1 z o≤ supf1 w
 supf-mono1 {z} {w} z≤w with trio< w px
 ... | tri< a ¬b ¬c = subst₂ (λ j k → j o≤ k ) (sym (sf1=sf0 (ordtrans≤-< z≤w a))) refl ( supf-mono z≤w )
 ... | tri≈ ¬a refl ¬c with trio< z px
 ... | tri< a ¬b ¬c = zc17
 ... | tri≈ ¬a refl ¬c = o≤-refl
 ... | tri> ¬a ¬b c = o≤-refl
 pchain1 : HOD
 pchain1 = UnionCF A f mf ay supf1 x
 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 u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ?  ?  ⟫
 zc111 : {z : Ordinal} → z o< px → OD.def (od pchain1) z → OD.def (od pchain) z
+zc111 {z} z<px ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
 zc111 {z} z<px ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ = ⟪ az , ch-is-sup u1 ? ?  ?  ⟫
 zc11 : (¬ xSUP (UnionCF A f mf ay supf0 px) f x ) ∨  (HasPrev A pchain f x )
 → {z : Ordinal} → OD.def (od pchain1) z → OD.def (od pchain) z ∨ ( (supf0 px ≡ px) ∧ FClosure A f px z )
-zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px
+zc11 P {z} ⟪ az , ch-init fc ⟫ = case1 ⟪ az , ch-init fc ⟫
-... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc  ⟫
+zc11 P {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ with trio< u1 px
+... | tri< u1<px ¬b ¬c = case1 ⟪ az , ch-is-sup u1 ? ? fc  ⟫
 ... | tri≈ ¬a eq ¬c  = case2 ⟪ subst (λ k → supf0 k ≡ k) eq s1u=u , subst (λ k → FClosure A f k z) zc12 ? ⟫ where
 s1u=u : supf0 u1 ≡ u1
 s1u=u = ? -- ChainP.supu=u u1-is-sup
 zc12 : supf0 u1 ≡ px
 zc12 = trans s1u=u eq
 zc11 (case1 ¬sp=x) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = ⊥-elim (¬sp=x zcsup) where
 eq : u1 ≡ x
 eq with trio< u1 x
 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ )
 ... | tri≈ ¬a b ¬c = b
 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? )
 s1u=x : supf0 u1 ≡ x
 s1u=x = trans ? eq
 zc13 : osuc px o< osuc u1
 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) ? ))
 ... | case1 eq1 = ⊥-elim ( o<¬≡ zc14 ? ) where
 zc14 : u ≡ osuc px
 zc14 = begin
 u ≡⟨ sym ( ChainP.supu=u is-sup) ⟩
 supf0 u ≡⟨ ? ⟩
 supf0 u1 ≡⟨ s1u=x ⟩
 x ≡⟨ sym (Oprev.oprev=x op) ⟩
 osuc px ∎ where open ≡-Reasoning
 ... | case2 lt = subst (λ k → (w ≡ k) ∨ (w << k)) s1u=x ?
 zc12 : supf0 x ≡ u1
 zc12 = subst  (λ k → supf0 k ≡ u1) eq ?
 zcsup : xSUP (UnionCF A f mf ay supf0 px) f x
 zcsup = record { ax =  subst (λ k → odef A k) (trans zc12 eq) (ZChain.asupf zc)
 ; is-sup = record { x≤sup = x≤sup ; minsup = ? } }
 zc11 (case2 hp) {z} ⟪ az , ch-is-sup u1 u1<x u1-is-sup fc ⟫ | tri> ¬a ¬b px<u = case1 ? where
 eq : u1 ≡ x
 eq with trio< u1 x
 ... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<u , subst (λ k → u1 o< k ) (sym (Oprev.oprev=x op )) a ⟫ )
 ... | tri≈ ¬a b ¬c = b
 ... | tri> ¬a ¬b c = ⊥-elim ( o<> u1<x ? )
 zc20 : {z : Ordinal} → FClosure A f (supf0 u1) z → OD.def (od pchain) z
 zc20 {.(f w)} (fsuc w fc) = ZChain.f-next zc (zc20 fc)
 record STMP {z : Ordinal} (z≤x : z o≤ x ) : Set (Level.suc n) where
 field
 tsup : MinSUP A (UnionCF A f mf ay supf1 z)
 tsup=sup : supf1 z ≡ MinSUP.sup tsup
 sup : {z : Ordinal} → (z≤x : z o≤ x ) → STMP z≤x
 sup {z} z≤x with trio< z px
 ... | tri< a ¬b ¬c = ? -- jrecord { tsup = ZChain.minsup zc (o<→≤ a)  ; tsup=sup = ZChain.supf-is-minsup zc (o<→≤ a) }
 ... | tri≈ ¬a b ¬c = ? -- record { tsup = ZChain.minsup zc (o≤-refl0 b)  ; tsup=sup = ZChain.supf-is-minsup zc (o≤-refl0 b) }
 ... | tri> ¬a ¬b px<z = zc35 where
 zc30 : z ≡ x
 zc30 with osuc-≡< z≤x
 ... | case1 eq = eq
 ... | case2 z<x = ⊥-elim (¬p<x<op ⟪ px<z , subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ⟫ )
 zc32 = ZChain.sup zc o≤-refl
 zc34 : ¬ (supf0 px ≡ px) → {w : HOD} → UnionCF A f mf ay supf0 z ∋ w → (w ≡ SUP.sup zc32) ∨ (w < SUP.sup zc32)
 zc34 ne {w} lt with zc11 ? ⟪ proj1 lt , ? ⟫
 ... | case1 lt = SUP.x≤sup zc32 lt
 ... | case2 ⟪ spx=px  , fc ⟫ = ⊥-elim ( ne spx=px )
 zc33 : supf0 z ≡ & (SUP.sup zc32)
 zc33 = ? -- trans (sym (supfx (o≤-refl0 (sym zc30)))) ( ZChain.supf-is-minsup zc o≤-refl  )
 zc36 : ¬ (supf0 px ≡ px) → STMP z≤x
 zc36 ne = ? -- record { tsup = record { sup = SUP.sup zc32 ; as = SUP.as zc32 ; x≤sup = zc34 ne } ; tsup=sup = zc33  }
 zc35 : STMP z≤x
 zc35 with trio< (supf0 px) px
 ... | tri< a ¬b ¬c = zc36 ¬b
 ... | tri> ¬a ¬b c = zc36 ¬b
 ... | tri≈ ¬a b ¬c = record { tsup = ? ; tsup=sup = ?  } where
 zc37 : MinSUP A (UnionCF A f mf ay supf0 z)
 zc37 = record { sup = ? ; asm = ? ; x≤sup = ? }
 sup=u : {b : Ordinal} (ab : odef A b) →
 b o≤ x → IsMinSUP A (UnionCF A f mf ay supf0 b) supf0 ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf0 b) f b ) → supf0 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) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫
 ... | tri≈ ¬a b ¬c = ZChain.sup=u zc ab (o≤-refl0 b) ⟪ record { x≤sup = λ lt → IsMinSUP.x≤sup (proj1 is-sup) lt } , proj2 is-sup ⟫
 ... | tri> ¬a ¬b px<b = zc31 ? 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 ⟫ )
 zcsup : xSUP (UnionCF A f mf ay supf0 px) supf0 x
 zcsup with zc30
 ... | refl = record { ax = ab ; is-sup = record { x≤sup = λ {w} lt →
 IsMinSUP.x≤sup (proj1 is-sup) ? ; minsup = ? } }
 zc31 : ( (¬ xSUP (UnionCF A f mf ay supf0 px) supf0 x ) ∨  HasPrev A (UnionCF A f mf ay supf0 px) f x ) → supf0 b ≡ b
 zc31 (case1 ¬sp=x) with zc30
 ... | refl = ⊥-elim (¬sp=x zcsup )
 zc31 (case2 hasPrev ) with zc30
 ... | refl = ⊥-elim ( proj2 is-sup record { ax = HasPrev.ax hasPrev ; y = HasPrev.y hasPrev
 ; ay = ? ; x=fy = HasPrev.x=fy hasPrev } )
 ... | no lim = zc5 where
 pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z
 pzc z z<x = prev z z<x
 pchain : HOD
 pchain = UnionCF A f mf ay supf0 x
 ptotal0 : IsTotalOrderSet pchain
 ptotal0 {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where
 uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
 uz01 = chain-total A f mf ay supf0 ( (proj2 ca)) ( (proj2 cb))
 usup : MinSUP A pchain
 usup = minsupP pchain (λ lt → proj1 lt) ptotal0
 spu = MinSUP.sup usup
 supf1 : Ordinal → Ordinal
 pchain1 : HOD
 pchain1 = UnionCF A f mf ay supf1 x
 is-max-hp : (supf : Ordinal → Ordinal) (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay supf x) a →
 b o< x → (ab : odef A b) →
 HasPrev A (UnionCF A f mf ay supf x) f b  →
 * 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-is-sup u u<x is-sup fc ⟫ = ? -- ⟪ ab ,
+... | ⟪ 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 supf x k )
 --     (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 = ?
 no-extension : ¬ ( xSUP (UnionCF A f mf ay supf0 x) supf0 x ) → ZChain A f mf ay x
 no-extension ¬sp=x  = ? where -- record { supf = supf1  ; sup=u = sup=u
 -- ; sup = sup ; supf-is-sup = sis ; supf-mono = {!!} ; asupf = ? } where
 supfu : {u : Ordinal } → ( a : u o< x ) → (z : Ordinal) → Ordinal
 supfu {u} a z = ZChain.supf (pzc (osuc u) (ob<x lim a)) z
 pchain0=1 : pchain ≡ pchain1
 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
 zc12 : {z : Ordinal} →  FClosure A f (supf0 u) z → odef pchain1 z
 zc12 (fsuc x fc) with zc12 fc
-... | ⟪ 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-init fc₁ ⟫ = ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-zc12 (init asu su=z ) = ⟪ subst (λ k → odef A k) su=z asu , ch-is-sup u ? ? (init ? ? ) ⟫
+... | ⟪ 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
 zc13 : {z : Ordinal} →  FClosure A f (supf1 u) z → odef pchain z
 zc13 (fsuc x fc) with zc13 fc
-... | ⟪ 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-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) ⟫
 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 )
 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) {!!}) {!!} )
 ... | tri≈ ¬a b ¬c = {!!}
 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x ))
 sis : {z : Ordinal} (x≤z : z o≤ x) → supf1 z ≡ & (SUP.sup (sup {!!}))
 sis {z} z≤x with trio< z x
 ... | tri< a ¬b ¬c = {!!} where
 zc8 = ZChain.supf-is-minsup (pzc z a) {!!}
 ... | tri≈ ¬a b ¬c = {!!}
 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x ))
 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f mf ay supf1 b) f ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf1 b) f b ) → supf1 b ≡ b
 sup=u {z} ab z≤x is-sup with trio< z x
 ... | tri< a ¬b ¬c = ? -- ZChain.sup=u (pzc (osuc b)  (ob<x lim a))  ab {!!} record { x≤sup = {!!} }
 ... | tri≈ ¬a b ¬c = {!!} -- ZChain.sup=u (pzc (osuc ?)  ?)  ab {!!} record { x≤sup = {!!} }
 ... | tri> ¬a ¬b x<z = ⊥-elim (o<¬≡ refl (ordtrans<-≤ x<z z≤x ))
 zc5 : ZChain A f mf ay x
 zc5 with ODC.∋-p O A (* x)
 ... | no noax = no-extension {!!} -- ¬ A ∋ p, just skip
 ... | yes ax with ODC.p∨¬p O ( HasPrev A pchain f x  )
 -- we have to check adding x preserve is-max ZChain A y f mf x
 ... | case1 pr = no-extension {!!}
 ... | case2 ¬fy<x with ODC.p∨¬p O (IsMinSUP A pchain f ax )
 ... | case1 is-sup = ? -- record { supf = supf1  ; sup=u = {!!}
 -- ; sup = {!!} ; supf-is-sup = {!!} ; supf-mono = {!!};  asupf = {!!}  } -- where -- x is a sup of (zc ?)
 ... | case2 ¬x=sup =  no-extension {!!} -- x is not f y' nor sup of former ZChain from y -- no extention
 ---
 --- the maximum chain  has fix point of any ≤-monotonic function
 ---
 SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf ay x
 SZ f mf {y} ay x = TransFinite {λ z → ZChain A f mf ay z  } (λ x → ind f mf ay x   ) x
 msp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y)
 → (zc : ZChain A f mf ay x )
 → MinSUP A (UnionCF A f mf ay (ZChain.supf zc) x)
 msp0 f mf {x} ay zc = minsupP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc)
 fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f )  (zc : ZChain A f mf as0 (& A) )
 → (sp1 : MinSUP A (ZChain.chain zc))
 → (ssp<as :  ZChain.supf zc (MinSUP.sup sp1) o< ZChain.supf zc (& A))
 → f (MinSUP.sup sp1)  ≡ MinSUP.sup sp1
 fixpoint f mf zc sp1 ssp<as = z14 where
 chain = ZChain.chain zc
 supf = ZChain.supf zc
 sp : Ordinal
 sp = MinSUP.sup sp1
 asp : odef A sp
 asp = MinSUP.asm sp1
 z10 :  {a b : Ordinal } → (ca : odef chain a ) → supf b o< supf (& A) → (ab : odef A b )
 →  HasPrev A chain f b  ∨  IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) b) ab
 → * a < * b  → odef chain b
 z10 = ZChain1.is-max (SZ1 f mf as0 zc (& A) )
 z22 : sp o< & A
 z22 = z09 asp
 ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc )
 ... | no ne = ZChain1.is-max (SZ1 f mf as0 zc (& A)) {& s} {sp} ( ZChain.chain∋init zc ) ssp<as asp (case2 z19 ) z13 where
 z13 :  * (& s) < * sp
 z13 with MinSUP.x≤sup sp1 ( ZChain.chain∋init zc )
 ... | case1 eq = ⊥-elim ( ne eq )
 ... | case2 lt = lt
 z19 : IsSUP A (UnionCF A f mf as0 (ZChain.supf zc) sp) asp
 z19 = record {   x≤sup = z20 }  where
 z20 : {y : Ordinal} → odef (UnionCF A f mf as0 (ZChain.supf zc) sp) y → (y ≡ sp) ∨ (y << sp)
 z20 {y} zy with MinSUP.x≤sup sp1
 (subst (λ k → odef chain k ) (sym &iso) (chain-mono f mf as0 supf (ZChain.supf-mono zc) (o<→≤ z22)  zy ))
 ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p )
 ... | case2 y<p = case2 (subst (λ k → * k < _ ) &iso y<p )
 z14 :  f sp ≡ sp
 z14 with ZChain.f-total zc (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) (subst (λ k → odef chain k) (sym &iso) z12 )
 ... | tri< a ¬b ¬c = ⊥-elim z16 where
 z16 : ⊥
 z16 with proj1 (mf (( MinSUP.sup sp1)) ( MinSUP.asm sp1 ))
 ... | case1 eq = ⊥-elim (¬b (sym eq) )
 ... | case2 lt = ⊥-elim (¬c lt )
 ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b )
 ... | tri> ¬a ¬b c = ⊥-elim z17 where
 z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) )
 z15  = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 )
 z17 : ⊥
 z17 with z15
 -- ZChain forces fix point on any ≤-monotonic function (fixpoint)
 -- ¬ Maximal create cf which is a <-monotonic function by axiom of choice. This contradicts fix point of ZChain
 --
 z04 :  (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)) → ⊥
 z04 nmx zc = <-irr0  {* (cf nmx c)} {* c}
 (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.asm  msp1 ))))
 (subst (λ k → odef A k) (sym &iso) (MinSUP.asm msp1) )
 (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc msp1  ss<sa ))) -- x ≡ f x ̄
 (proj1 (cf-is-<-monotonic nmx c (MinSUP.asm msp1 ))) where          -- x < f x
 supf = ZChain.supf zc
 msp1 : MinSUP A (ZChain.chain zc)
 msp1 = msp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc
 c : Ordinal
 c = MinSUP.sup msp1
 mc = c
 mc<A : mc o< & A
 mc<A =  ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫
 c=mc : c ≡ mc
 c=mc = refl
 z20 : mc << cf nmx mc
 z20 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1) )
 asc : odef A (supf mc)
 asc = ZChain.asupf zc
 spd : MinSUP A (uchain (cf nmx)  (cf-is-≤-monotonic nmx) asc )
 spd = ysup (cf nmx)  (cf-is-≤-monotonic nmx) asc
 d = MinSUP.sup spd
 d<A : d o< & A
 d<A =  ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫
 msup : MinSUP  A (UnionCF A (cf nmx)  (cf-is-≤-monotonic nmx) as0 supf d)
 msup = ZChain.minsup zc (o<→≤ d<A)
 sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) )
 sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A)
 sc<<d : {mc : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx)  (cf-is-≤-monotonic nmx) asc ))
 → supf mc << MinSUP.sup spd
 sc<<d {mc} asc spd = z25 where
 d1 :  Ordinal
 d1 = MinSUP.sup spd -- supf d1 ≡ d
 z24 : (supf mc ≡ d1) ∨ ( supf mc << d1 )
 --  supf mc ≡ d1
 z32 : ((cf nmx (supf mc)) ≡ d1) ∨ ( (cf nmx (supf mc)) << d1 )
 z32 = MinSUP.x≤sup spd (fsuc _ (init asc refl))
 z29 :  (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1)
 z29  with z32
 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1)  )
 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt)
 fsc<<d : {mc z : Ordinal } → (asc : odef A (supf mc)) → (spd : MinSUP A (uchain (cf nmx)  (cf-is-≤-monotonic nmx) asc ))
 → (fc : FClosure A (cf nmx) (supf mc) z) → z << MinSUP.sup spd
 fsc<<d {mc} {z} asc spd fc = z25 where
 d1 :  Ordinal
 d1 = MinSUP.sup spd -- supf d1 ≡ d
 z24 : (z ≡ d1) ∨ ( z << d1 )
 --  supf mc ≡ d1
 z32 : ((cf nmx z) ≡ d1) ∨ ( (cf nmx z) << d1 )
 z32 = MinSUP.x≤sup spd (fsuc _ fc)
 z29 :  (* (cf nmx d1) ≡ * d1) ∨ (* (cf nmx d1) < * d1)
 z29  with z32
 ... | case1 eq1 = case1 (cong (*) (trans (cong (cf nmx) (sym eq)) eq1)  )
 ... | case2 lt = case2 (subst (λ k → * k < * d1 ) (cong (cf nmx) eq) lt)
 smc<<d : supf mc << d
 smc<<d = sc<<d asc spd
 sz<<c : {z : Ordinal } → z o< & A → supf z <= mc
 sz<<c z<A = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc)  ))
 sc=c : supf mc ≡ mc
 sc=c = ZChain.sup=u zc (MinSUP.asm msp1) (o<→≤ (∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫ )) ⟪ is-sup , not-hasprev ⟫ where
 not-hasprev :  ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) mc) (cf nmx) mc
+not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where
+z30 : * mc < * (cf nmx mc)
+z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1))
+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
 z30 : * mc < * (cf nmx mc)
 z30 = proj1 (cf-is-<-monotonic nmx mc (MinSUP.asm msp1))
 z31 : ( supf mc ≡  mc ) ∨ ( * (supf mc) < * mc )
 z31 = MinSUP.x≤sup msp1 (ZChain.csupf zc (z09 (ZChain.asupf zc)  ))
 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 ) }
 not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) (cf nmx) d
+not-hasprev record { ax = ax ; y = y ; ay = ⟪ ua1 , ch-init fc ⟫ ; x=fy = x=fy } = ⊥-elim ( <-irr z31 z30 ) where
+z30 : * d < * (cf nmx d)
+z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd))
+z32 : ( cf nmx (cf nmx y)  ≡  supf mc ) ∨ ( * (cf nmx (cf nmx y)) < * (supf mc) )
+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
 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) )
 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)) ⟫
 z51 : supf u o≤ supf mc
 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57 where
 z56 : u ≡ mc → supf u ≡  supf mc
 z56 eq = cong supf eq
 z57 : u << mc → supf u o≤ supf mc
 z57 lt = ZChain.supf-<= zc (case2 z58) where
 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d
 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c)  lt
 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 ))
 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
 z47 {mc} {d1} {asc} spd (case1 eq) smc<d = subst (λ k → k < * d1 ) (sym (cong (*) eq))  smc<d
 z47 {mc} {d1} {asc} spd (case2 lt) smc<d = IsStrictPartialOrder.trans PO lt smc<d
 z30 : * d < * (cf nmx d)
 z30 = proj1 (cf-is-<-monotonic nmx d (MinSUP.asm spd))
 z46 : (* (cf nmx d) ≡ * d) ∨ (* (cf nmx d) < * d)
 z46 = or (osuc-≡< z51) z55 z54 where
 is-sup = record { x≤sup = z22 } where
 z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd)
 z23 lt = MinSUP.x≤sup spd lt
 z22 :  {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y →
 (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd)
+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)
 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) )
 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)) ⟫
 z56 : u ≡ mc → supf u ≡  supf mc
 z56 eq = cong supf eq
 z57 : u << mc → supf u o≤ supf mc
 z57 lt = ZChain.supf-<= zc (case2 z58) where
 z58 : supf u << supf mc -- supf u o< supf d -- supf u << supf d
 z58 = subst₂ ( λ j k → j << k ) (sym (ChainP.supu=u is-sup1)) (sym sc=c)  lt
 z51 : supf u o≤ supf mc
 z51 = or z52 (λ le → o≤-refl0 (z56 le) ) z57
 z60 : u o< supf mc → (a ≡ d ) ∨ ( * a < * d )
 z60 u<smc = case2 ( ftrans<=-< (ZChain1.order (SZ1 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc (& A)) mc<A
 (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
 --     supf u o< spuf c → order
 ... | case2 lt = lt
 sms<sa : supf mc o< supf (& A)
 sms<sa with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ MinSUP.asm msp1 , lift true ⟫) ))
 ... | case2 lt = lt
 ... | case1 eq = ⊥-elim ( o<¬≡ eq ( ordtrans<-≤ (sc<sd (subst (λ k → supf mc << k ) (sym sd=d) (sc<<d {mc} asc spd)) )
 ( ZChain.supf-mono zc (o<→≤ d<A ))))
 ss<sa : supf c o< supf (& A)
 ss<sa = subst (λ k → supf k o< supf (& A)) (sym c=mc) sms<sa
 zorn00 : Maximal A
 zorn00 with is-o∅ ( & HasMaximal )  -- we have no Level (suc n) LEM
 ... | no not = record { maximal = ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x  = zorn02 } where
 -- yes we have the maximal
 zorn03 :  odef HasMaximal ( & ( ODC.minimal O HasMaximal  (λ eq → not (=od∅→≡o∅ eq)) ) )
 zorn03 =  ODC.x∋minimal  O HasMaximal  (λ eq → not (=od∅→≡o∅ eq))   -- Axiom of choice
 zorn01 :  A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq))
 zorn01  = proj1  zorn03
 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x)
 zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x )
 ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A) )) where
 -- if we have no maximal, make ZChain, which contradict SUP condition
 nmx : ¬ Maximal A
 nmx mx =  ∅< {HasMaximal} zc5 ( ≡o∅→=od∅  ¬Maximal ) where
 zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) →  odef A y → ¬ (* (& (Maximal.maximal mx)) < * y))
 zc5 = ⟪  Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫
 -- usage (see filter.agda )
 --
 -- _⊆'_ : ( A B : HOD ) → Set n
 -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x
 -- MaximumSubset : {L P : HOD}
 --        → o∅ o< & L →  o∅ o< & P → P ⊆ L
 --        → IsPartialOrderSet P _⊆'_
 --        → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ )
 --        → Maximal P (_⊆'_)
 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP  = Zorn-lemma {P} {_⊆'_} 0<P PO SP

Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison src/zorn.agda @ 966:39c680188738