{-# 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
open ODAxiom odAxiom
import OrdUtil
import ODUtil
open Ordinals.Ordinals  O
open Ordinals.IsOrdinals isOrdinal
open Ordinals.IsNext isNext
open OrdUtil O
open ODUtil O


import ODC

open _∧_
open _∨_
open Bool

open HOD

--
-- Partial Order on HOD ( possibly limited in A )
--

_<<_ : (x y : Ordinal ) → Set n    -- Set n order
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 )

<-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
<-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO   refl
          (IsStrictPartialOrder.trans PO     b<a a<b)

ptrans =  IsStrictPartialOrder.trans PO

open _==_
open _⊆_

--
-- Closure of ≤-monotonic function f has total order
--

≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set (Level.suc n)
≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧  odef A (f x )

data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where
   init : odef A s → FClosure A f s s
   fsuc : (x : Ordinal) ( p :  FClosure A f s x ) → FClosure A f s (f x)

A∋fc : {A : HOD} (s : Ordinal) {y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) → (fcy : FClosure A f s y ) → odef A y
A∋fc {A} s f mf (init as) = 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) = 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) = 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) = 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
     fc00 :  (i j : ℕ ) → i ≡ j  →  {x y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → i ≡ fcn s mf cx  → j ≡ fcn s mf cy → * x ≡ * y
     fc00 zero zero refl (init _) (init x₁) i=x i=y = refl
     fc00 zero zero refl (init as) (fsuc y cy) i=x i=y with proj1 (mf y (A∋fc s f mf cy ) )
     ... | case1 y=fy = subst (λ k → * s ≡ k ) y=fy ( fc00 zero zero refl (init as) cy i=x i=y )
     fc00 zero zero refl (fsuc x cx) (init as) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) )
     ... | case1 x=fx = subst (λ k → k ≡ * s ) x=fx ( fc00 zero zero refl cx (init as) i=x i=y )
     fc00 zero zero refl (fsuc x cx) (fsuc y cy) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) )
     ... | case1 x=fx  | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 zero zero refl cx cy  i=x i=y )
     fc00 (suc i) (suc j) i=j {.(f x)} {.(f y)} (fsuc x cx) (fsuc y cy) i=x j=y with proj1 (mf x (A∋fc s f mf cx ) ) | proj1 (mf y (A∋fc s f mf cy ) )
     ... | case1 x=fx | case1 y=fy = subst₂ (λ j k → j ≡ k ) x=fx y=fy ( fc00 (suc i) (suc j) i=j cx cy  i=x j=y )
     ... | case1 x=fx | case2 y<fy = subst (λ k → k ≡ * (f y)) x=fx (fc02 x cx i=x) where
          fc02 : (x1 : Ordinal) → (cx1 : FClosure A f s x1 ) →  suc i ≡ fcn s mf cx1 → * x1 ≡ * (f y)
          fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) )
          ... | case1 eq = trans (sym eq) ( fc02  x1 cx1 i=x1 )  -- derefence while f x ≡ x
          ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc04) where
               fc04 : * x1 ≡ * y
               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 .(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
     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) {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


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 ⟫ = fc21 where
      fc20 : fcn s mf cy Data.Nat.< suc (fcn s mf cx) → (fcn s mf cy ≡ fcn s mf cx) ∨ ( fcn s mf cy Data.Nat.< fcn s mf cx )
      fc20 y<sx with <-cmp ( fcn s mf cy ) (fcn s mf cx )
      ... | tri< a ¬b ¬c = case2 a
      ... | tri≈ ¬a b ¬c = case1 b
      ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> y<sx (s≤s c))
      fc17 : {x y : Ordinal } → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → suc (fcn s mf cx) ≡ fcn s mf cy → * (f x ) ≡ * y
      fc17 {x} {y} cx cy sx=y = fc18 (fcn s mf cy) cx cy refl sx=y  where
          fc18 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → suc (fcn s mf cx) ≡  i → * (f x) ≡ * y
          fc18 (suc i) {y} cx (fsuc y1 cy)  i=y sx=i with proj1 (mf y1 (A∋fc s f mf cy ) )
          ... | case1 y=fy = subst (λ k → * (f x) ≡  k ) y=fy ( fc18 (suc i) {y1} cx cy i=y sx=i)  -- dereference
          ... | case2 y<fy = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k) ) ) fc19) where
               fc19 : * x ≡ * y1
               fc19 = fcn-inject s mf cx cy (cong pred ( trans sx=i i=y ))
      fc21 :  ⊥
      fc21 with <-cmp (suc ( fcn s mf cx )) (fcn s mf cy )
      ... | tri< a ¬b ¬c = <-irr (case2 y<fx) (fc22 a) where -- suc ncx < ncy
           cxx :  FClosure A f s (f x)
           cxx = fsuc x cx 
           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 as) with proj1 (mf s as )
           ... | 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
           fc22 : (suc ( fcn s mf cx ))  Data.Nat.< (fcn s mf cy ) → * (f x) < * y
           fc22 a with fc16 x cx
           ... | case1 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq (<-trans a<sa a))
           ... | case2 eq = fcn-< s mf cxx cy (subst (λ k → k Data.Nat.< fcn s mf cy ) eq a )
      ... | tri≈ ¬a b ¬c = <-irr (case1 (fc17 cx cy b)) y<fx
      ... | tri> ¬a ¬b c with fc20 c -- ncy < suc ncx
      ... | case1 y=x = <-irr (case1 ( fcn-inject s mf cy cx  y=x ))  x<y
      ... | case2 y<x = <-irr (case2 x<y) (fcn-< s mf cy cx y<x ) 

fc-conv : (A : HOD ) (f : Ordinal → Ordinal) {b u : Ordinal } 
      →  {p0 p1 : Ordinal → Ordinal}
      →  p0 u ≡ p1 u
      →  FClosure A f (p0 u) b → FClosure A f (p1 u) b 
fc-conv A f {.(p0 u)} {u}  {p0} {p1} p0u=p1u (init ap0u) = subst (λ k → FClosure A f (p1 u) k) (sym p0u=p1u) 
       ( init (subst (λ k → odef A k) p0u=p1u ap0u ))
fc-conv A f {_} {u}  {p0} {p1} p0u=p1u (fsuc z fc) = fsuc z (fc-conv A f {_} {u}  {p0} {p1} p0u=p1u fc)

-- open import Relation.Binary.Properties.Poset as Poset

IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n)
IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b)  → Tri (a < b) (a ≡ b) (b < a )

⊆-IsTotalOrderSet : { A B : HOD } →  B ⊆ A  → IsTotalOrderSet A → IsTotalOrderSet B
⊆-IsTotalOrderSet {A} {B} B⊆A T  ax ay = T (incl B⊆A ax) (incl B⊆A ay)

_⊆'_ : ( A B : HOD ) → Set n
_⊆'_ A B = {x : Ordinal } → odef A x → odef B x

--
-- inductive maxmum tree from x
-- tree structure
--

record HasPrev (A B : HOD) {x : Ordinal } (xa : odef A x)  ( f : Ordinal → Ordinal )  : Set n where
   field
      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 SUP ( A B : HOD )  : Set (Level.suc n) where
   field
      sup : HOD
      A∋maximal : A ∋ sup
      x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup )   -- B is Total, use positive

--
--  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 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 z : Ordinal) : Set n where
   field
      fcy<sup  : {z : Ordinal } → FClosure A f y z → z << supf u
      csupz    : FClosure A f (supf u) z 
      order    : {sup1 z1 : Ordinal} → (lt : sup1 o< u ) → FClosure A f (supf sup1 ) z1 → z1 << supf u
      y<s      : y << supf u  -- not a initial chain
      supfu=u  : supf u ≡ u 

-- Union of supf z which o< x
--

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 : u o< x ) ( is-sup : ChainP A f mf ay supf u z) 
        ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z

∈∧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 )))

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 }

record ZChain ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) 
        {init : Ordinal} (ay : odef A init)  ( z : Ordinal ) : Set (Level.suc n) where
   field
      supf :  Ordinal → Ordinal 
   chain : HOD
   chain = UnionCF A f mf ay supf z
   field
      chain⊆A : chain ⊆' A
      chain∋init : odef chain init
      initial : {y : Ordinal } → odef chain y → * init ≤ * y
      f-next : {a : Ordinal } → odef chain a → odef chain (f a)
      f-total : IsTotalOrderSet chain
      chain∋supf : {z : Ordinal } → odef chain (supf z)
      sup=u : {b : Ordinal} → {ab : odef A b} → b o< z → IsSup A chain ab → supf b ≡ b 
      order : {b sup1 z1 : Ordinal} → b o< z → sup1 o< b → FClosure A f (supf sup1) z1 → z1 << supf b

record ZChain1 ( A : HOD )    ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) 
        {init : Ordinal} (ay : odef A init)  (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where
   field
      chain-mono2 : { a b c : Ordinal }  → 
          a o≤ b → b o≤ z → odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c
      is-max :  {a b : Ordinal } → (ca : odef (UnionCF A f mf ay (ZChain.supf zc) z) a ) →  b o< z  → (ab : odef A b) 
          → HasPrev A (UnionCF A f mf ay (ZChain.supf zc) z) ab f ∨  IsSup A (UnionCF A f mf ay (ZChain.supf zc) z) ab  
          → * a < * b  → odef ((UnionCF A f mf ay (ZChain.supf zc) z)) b
      fcy<sup  : {u w : Ordinal } → u o< z  → FClosure A f init w → w << (ZChain.supf zc) u
      sup=u : {b : Ordinal} → {ab : odef A b} → b o< z  → IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab → (ZChain.supf zc) b ≡ b 
      order : {b sup1 z1 : Ordinal} → b o< z → sup1 o< b → FClosure A f ((ZChain.supf zc) sup1) z1 → z1 << (ZChain.supf zc) b

record Maximal ( A : HOD )  : Set (Level.suc n) where
   field
      maximal : HOD
      A∋maximal : A ∋ maximal
      ¬maximal<x : {x : HOD} → A ∋ x  → ¬ maximal < x       -- A is Partial, use negative

-- 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 )
   {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 ub<x supb fcb) = tri< ct01  (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt)  where
          ct00 : * a < * (supf ub)
          ct00 = ChainP.fcy<sup supb fca
          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 ua<x supa fca) (ch-init fcb)= tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01    where
          ct00 : * b < * (supf ua)
          ct00 = ChainP.fcy<sup supa fcb
          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 = tri< ct02  (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt)  where
          ct03 : * a < * (supf ub)
          ct03 = ChainP.order supb  a₁ (ChainP.csupz supa)
          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
     ... | tri≈ ¬a  refl ¬c = fcn-cmp (supf ua) f mf fca fcb
     ... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04    where
          ct05 : * b < * (supf ua)
          ct05 = ChainP.order supa c (ChainP.csupz supb)
          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

init-uchain : (A : HOD)  ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) 
    { 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)   ⟫

ChainP-next : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f)  {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal )
   → {x z : Ordinal } →  ChainP A f mf ay supf x z → ChainP A f mf ay supf x (f z )
ChainP-next A f mf {y} ay supf {x} {z} cp = record { y<s = ChainP.y<s cp ; supfu=u  = ChainP.supfu=u cp  
     ; fcy<sup = ChainP.fcy<sup cp ; csupz = fsuc _ (ChainP.csupz cp) ; order = ChainP.order cp }

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} → {P : Set n} → odef A y ∧ P → y o< & A
     z07 {y} 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 ; A∋maximal = 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) 

     -- Uncountable ascending chain by axiom of choice
     cf : ¬ Maximal A → Ordinal → Ordinal
     cf  nmx x with ODC.∋-p O A (* x)
     ... | no _ = o∅
     ... | yes ax with is-o∅ (& ( Gtx ax ))
     ... | yes nogt = -- no larger element, so it is maximal
         ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ )
     ... | no not =  & (ODC.minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq)))
     is-cf : (nmx : ¬ Maximal A ) → {x : Ordinal} → odef A x → odef A (cf nmx x) ∧ ( * x < * (cf nmx x) )
     is-cf nmx {x} ax with ODC.∋-p O A (* x)
     ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax ))
     ... | yes ax with is-o∅ (& ( Gtx ax ))
     ... | yes nogt = ⊥-elim (no-maximum (z08 nmx) x ⟪ subst (λ k → odef A k) &iso ax , x-is-maximal nmx (subst (λ k → odef A k ) &iso ax) nogt ⟫ )
     ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))

     ---
     --- infintie ascention sequence of f
     ---
     cf-is-<-monotonic : (nmx : ¬ Maximal A ) → (x : Ordinal) →  odef A x → ( * x < * (cf nmx x) ) ∧  odef A (cf nmx x )
     cf-is-<-monotonic nmx x ax = ⟪ proj2 (is-cf nmx ax ) , proj1 (is-cf nmx ax ) ⟫
     cf-is-≤-monotonic : (nmx : ¬ Maximal A ) →  ≤-monotonic-f A ( cf nmx )
     cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫

     sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f )  (zc : ZChain A f mf as0 (& A) ) 
        (total : IsTotalOrderSet (ZChain.chain zc) )  → SUP A (ZChain.chain zc)
     sp0 f mf zc total = supP (ZChain.chain zc) (ZChain.chain⊆A zc) total 
     zc< : {x y z : Ordinal} → {P : Set n} → (x o< y → P) → x o< z → z o< y → P
     zc< {x} {y} {z} {P} prev x<z z<y = prev (ordtrans x<z z<y)

     SZ1 :( A : HOD )    ( 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 A f mf {y} ay zc x = TransFinite { λ x →  ZChain1 A f mf ay zc x } zc1 x where
        chain-mono2 : (x : Ordinal) {a b c : Ordinal} → a o≤ b → b o≤ x →
            odef (UnionCF A f mf ay (ZChain.supf zc) a) c → odef (UnionCF A f mf ay (ZChain.supf zc) b) c
        chain-mono2 x {a} {b} {c} a≤b b≤x ⟪ ua , ch-init fc  ⟫ = 
            ⟪ ua , ch-init fc  ⟫ 
        chain-mono2 x {a} {b} {c} a≤b b≤x ⟪ uaa ,  ch-is-sup ua u<x is-sup fc  ⟫ = 
            ⟪ uaa , ch-is-sup ua (ordtrans<-≤ u<x a≤b ) is-sup fc  ⟫ 
        chain<ZA : {x : Ordinal } → UnionCF A f mf ay (ZChain.supf zc) x ⊆' UnionCF A f mf ay (ZChain.supf zc) (& A)
        chain<ZA {x} ux with proj2 ux
        ... | ch-init fc = ⟪ proj1 ux , ch-init fc ⟫
        ... | ch-is-sup u pu<x is-sup fc = ⟪ proj1 ux , ch-is-sup u u<x is-sup fc ⟫ where
            u<A : (& ( * ( ZChain.supf zc u)))  o< & A
            u<A = c<→o< (subst (λ k → odef A k ) (sym &iso) (A∋fcs _ f mf fc) )
            u<x :  u o< & A
            u<x = subst (λ k → k o< & A ) (trans &iso (ChainP.supfu=u is-sup)) u<A
        is-max-hp : (x : Ordinal) {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a →
            b o< x → (ab : odef A b) →
            HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f → 
            * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
        is-max-hp 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 , 
                 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 (ChainP-next A f mf ay _ is-sup) (fsuc _ fc))  ⟫ 
        zc1 :  (x : Ordinal) →  ((y₁ : Ordinal) → y₁ o< x → ZChain1 A f mf ay zc y₁) → ZChain1 A f mf ay zc x
        zc1 x prev with Oprev-p x
        ... | yes op = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = fcy<sup ; sup=u = sup=u ; order = order } where
           px = Oprev.oprev op
           px<x : px o< x
           px<x = subst (λ k → px o< k ) (Oprev.oprev=x op) <-osuc
           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 
           fcy<sup :  {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u
           fcy<sup {a} {b} a<x fcb with trio< a px
           ... | tri< a<px ¬b ¬c = ZChain1.fcy<sup (prev px px<x) a<px fcb
           ... | tri≈ ¬a a=px ¬c = ?
           ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) a<x ⟫ )
           order : {b sup1 z1 : Ordinal} → b o< x →
                sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b
           order {b} {sup1} {z1} b<x s<b fcz with trio< b px
           ... | tri< b<px ¬b ¬c = ZChain1.order (prev px px<x) b<px s<b fcz
           ... | tri≈ ¬a b=px ¬c = ?
           ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ )
           sup=u : {b : Ordinal} {ab : odef A b} → b o< x →
                IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab → ZChain.supf zc b ≡ b
           sup=u {b} {ab} b<x is-sup with trio< b px
           ... | tri< b<px ¬b ¬c = ZChain1.sup=u (prev px px<x) b<px is-sup
           ... | tri≈ ¬a b=px ¬c = ZChain.sup=u zc ? ?
           ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ )
           is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a →
              b o< x → (ab : odef A b) →
              HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab →
              * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
           is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b 
           is-max {a} {b} ua b<x ab (case2 is-sup) a<b with ODC.p∨¬p O ( HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f )
           ... | case1 has-prev = is-max-hp x {a} {b} ua b<x ab has-prev a<b 
           ... | case2  ¬fy<x  = m01 where
                 m01 : odef (UnionCF A f mf ay (ZChain.supf zc) x) b
                 m01 with trio< b px    --- px  < b < x
                 ... | tri> ¬a ¬b c = ⊥-elim (¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x  ⟫)
                 ... | tri< b<px ¬b ¬c = chain-mono2 x ( o<→≤  (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ))  o≤-refl m04 where
                    m03 :  odef (UnionCF A f mf ay (ZChain.supf zc) px) a
                    m03 with proj2 ua
                    ... | ch-init fc = ⟪ proj1 ua ,  ch-init fc ⟫
                    ... | ch-is-sup u u<x is-sup fc with trio< u px
                    ... | tri< a ¬b ¬c = ⟪ proj1 ua , ch-is-sup u a is-sup fc ⟫
                    ... | tri≈ ¬a u=px ¬c = ?   ---    supf u < a < b , 
                    ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → u o< k ) (sym (Oprev.oprev=x op)) u<x ⟫ )
                    -- = ⟪ proj1 ua , ? ⟫
                    m04 : odef (UnionCF A f mf ay (ZChain.supf zc) px) b
                    m04 = ZChain1.is-max (prev px px<x) m03 b<px ab 
                         (case2 record {x<sup = λ {z} lt → IsSup.x<sup is-sup (chain-mono2 x ( o<→≤  (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ))  o≤-refl lt)  } ) a<b
                 ... | tri≈ ¬a b=px ¬c = ? -- b = px case,  u = px  u<x 
        ... | no lim = record { is-max = is-max ; chain-mono2 = chain-mono2 x ; fcy<sup = fcy<sup ; sup=u = sup=u ; order = order  }  where
           fcy<sup : {u w : Ordinal} → u o< x → FClosure A f y w → w << ZChain.supf zc u
           fcy<sup {u} {w} u<x fc = ZChain1.fcy<sup (prev (osuc u) (ob<x lim u<x)) <-osuc fc
           sup=u :  {b : Ordinal} {ab : odef A b} → b o< x →
                IsSup A (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) ab →
                ZChain.supf zc b ≡ b
           sup=u {b} {ab} b<x is-sup = ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) <-osuc is-sup
           order : {b sup1 z1 : Ordinal} → b o< x → sup1 o< b →
                FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b
           order {b} b<x s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x)) <-osuc s<b fc
           is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) x) a →
              b o< x → (ab : odef A b) →
              HasPrev A (UnionCF A f mf ay (ZChain.supf zc) x) ab f ∨ IsSup A (UnionCF A f mf ay (ZChain.supf zc) x) ab →
              * a < * b → odef (UnionCF A f mf ay (ZChain.supf zc) x) b
           is-max {a} {b} ua b<x ab (case1 has-prev) a<b = is-max-hp x {a} {b} ua b<x ab has-prev a<b 
           is-max {a} {b} ua b<x ab (case2 is-sup) a<b with IsSup.x<sup is-sup (init-uchain A f mf ay )
           ... | case1 b=y = ⊥-elim ( <-irr ( ZChain.initial zc (chain<ZA (chain-mono2 (osuc x) (o<→≤ <-osuc ) o≤-refl ua )) )
                       (subst (λ k → * a < * k ) (sym b=y) a<b ) )
           ... | case2 y<b = chain-mono2 x (o<→≤ (ob<x lim b<x) ) o≤-refl m04 where
                 y<s : y << ZChain.supf zc b
                 y<s = y<s 
                 m07 : {z : Ordinal} → FClosure A f y z → z << ZChain.supf zc b
                 m07 {z} fc = ZChain1.fcy<sup (prev (osuc b) (ob<x lim b<x)) <-osuc fc
                 m08 : {sup1 z1 : Ordinal} → sup1 o< b → FClosure A f (ZChain.supf zc sup1) z1 → z1 << ZChain.supf zc b
                 m08 {sup1} {z1} s<b fc = ZChain1.order (prev (osuc b) (ob<x lim b<x) ) <-osuc s<b fc
                 m05 : b ≡ ZChain.supf zc b
                 m05 = sym (ZChain1.sup=u (prev (osuc b) (ob<x lim b<x)) {_} {ab} <-osuc 
                        record { x<sup = λ lt → IsSup.x<sup is-sup (chain-mono2 x (o<→≤ (ob<x lim b<x)) o≤-refl lt )} )   -- ZChain on x
                 m06 : ChainP A f mf ay (ZChain.supf zc) b b
                 m06 = record { fcy<sup = m07 ; csupz = subst (λ k →  FClosure A f k b ) m05 (init ab) ; order = m08 ; y<s = y<s 
                       ; supfu=u = sym m05 }
                 m04 : odef (UnionCF A f mf ay (ZChain.supf zc) (osuc b)) b
                 m04 = ⟪ ab , ch-is-sup b <-osuc m06 (subst (λ k → FClosure A f k b) m05 (init ab))  ⟫

     ---
     --- the maximum chain  has fix point of any ≤-monotonic function
     ---
     fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f )  (zc : ZChain A f mf as0 (& A) )
            → (total : IsTotalOrderSet (ZChain.chain zc) )
            → f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc  total))
     fixpoint f mf zc total = z14 where
           chain = ZChain.chain zc
           sp1 = sp0 f mf zc total
           z10 :  {a b : Ordinal } → (ca : odef chain a ) → b o< & A → (ab : odef A b ) 
              →  HasPrev A chain ab f ∨  IsSup A chain {b} ab -- (supO  chain  (ZChain.chain⊆A zc) (ZChain.f-total zc) ≡ b )
              → * a < * b  → odef chain b
           z10 = ZChain1.is-max (SZ1 A f mf as0 zc (& A) )
           z11 : & (SUP.sup sp1) o< & A
           z11 = c<→o< ( SUP.A∋maximal sp1)
           z12 : odef chain (& (SUP.sup sp1))
           z12 with o≡? (& s) (& (SUP.sup sp1))
           ... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc )
           ... | no ne = z10 {& s} {& (SUP.sup sp1)} ( ZChain.chain∋init zc ) z11 (SUP.A∋maximal sp1)
                (case2 z19 ) z13 where
               z13 :  * (& s) < * (& (SUP.sup sp1))
               z13 with SUP.x<sup sp1 ( ZChain.chain∋init zc )
               ... | case1 eq = ⊥-elim ( ne (cong (&) eq) )
               ... | case2 lt = subst₂ (λ j k → j < k ) (sym *iso) (sym *iso) lt
               z19 : IsSup A chain {& (SUP.sup sp1)} (SUP.A∋maximal sp1)
               z19 = record {   x<sup = z20 }  where
                   z20 :  {y : Ordinal} → odef chain y → (y ≡ & (SUP.sup sp1)) ∨ (y << & (SUP.sup sp1))
                   z20 {y} zy with SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) zy)
                   ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso ( cong (&) y=p ))
                   ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p )
                   -- λ {y} zy → subst (λ k → (y ≡ & k ) ∨ (y << & k)) ?  (SUP.x<sup sp1 ? ) }
           z14 :  f (& (SUP.sup (sp0 f mf zc total ))) ≡ & (SUP.sup (sp0 f mf zc total ))
           z14 with total (subst (λ k → odef chain k) (sym &iso)  (ZChain.f-next zc z12 )) z12 
           ... | tri< a ¬b ¬c = ⊥-elim z16 where
               z16 : ⊥
               z16 with proj1 (mf (& ( SUP.sup sp1)) ( SUP.A∋maximal sp1 ))
               ... | case1 eq = ⊥-elim (¬b (subst₂ (λ j k → j ≡ k ) refl *iso (sym eq) ))
               ... | case2 lt = ⊥-elim (¬c (subst₂ (λ j k → k < j ) refl *iso lt ))
           ... | tri≈ ¬a b ¬c = subst ( λ k → k ≡ & (SUP.sup sp1) ) &iso ( cong (&) b )
           ... | tri> ¬a ¬b c = ⊥-elim z17 where
               z15 : (* (f ( & ( SUP.sup sp1 ))) ≡ SUP.sup sp1) ∨ (* (f ( & ( SUP.sup sp1 ))) <  SUP.sup sp1)
               z15  = SUP.x<sup sp1 (subst (λ k → odef chain k ) (sym &iso) (ZChain.f-next zc z12 ))
               z17 : ⊥
               z17 with z15
               ... | case1 eq = ¬b eq
               ... | case2 lt = ¬a lt

     -- ZChain contradicts ¬ Maximal
     --
     -- 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)) 
           → IsTotalOrderSet (ZChain.chain zc) → ⊥
     z04 nmx zc total = <-irr0  {* (cf nmx c)} {* c} (subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (SUP.A∋maximal  sp1 ))))
                                               (subst (λ k → odef A (& k)) (sym *iso) (SUP.A∋maximal sp1) )
           (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc total ))) -- x ≡ f x ̄
           (proj1 (cf-is-<-monotonic nmx c (SUP.A∋maximal sp1 ))) where          -- x < f x
          sp1 : SUP A (ZChain.chain zc)
          sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) zc total
          c = & (SUP.sup sp1)

     inititalChain : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → ZChain A f mf ay o∅
     inititalChain f mf {y} ay = record { supf = isupf ; chain⊆A = λ lt → proj1 lt ; chain∋init = cy
      ; initial = isy ; f-next = inext ; f-total = itotal ; sup=u = λ b<0 → ⊥-elim (¬x<0 b<0) ; order = λ b<0 → ⊥-elim (¬x<0 b<0) } where
          isupf : Ordinal → Ordinal
          isupf z = y
          cy : odef (UnionCF A f mf ay isupf o∅) y
          cy = ⟪ ay , ch-init (init ay)  ⟫
          isy : {z : Ordinal } → odef (UnionCF A f mf ay isupf o∅) z → * y ≤ * z
          isy {z} ⟪ az , uz ⟫ with uz 
          ... | ch-init fc = s≤fc y f mf fc 
          ... | ch-is-sup u u<x is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) )
          inext : {a : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a → odef (UnionCF A f mf ay isupf o∅) (f a)
          inext {a} ua with (proj2 ua)
          ... | ch-init fc = ⟪ proj2 (mf _ (proj1 ua))  , ch-init (fsuc _ fc )  ⟫
          ... | ch-is-sup u u<x is-sup fc = ⊥-elim ( <-irr (case1 refl) ( ChainP.fcy<sup is-sup (init ay) ) )
          itotal : IsTotalOrderSet (UnionCF A f mf ay isupf o∅) 
          itotal {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 isupf (proj2 ca) (proj2 cb) 
          imax : {a b : Ordinal} → odef (UnionCF A f mf ay isupf o∅) a →
            b o< o∅ → (ab : odef A b) →
            HasPrev A (UnionCF A f mf ay isupf o∅) ab f ∨ IsSup A (UnionCF A f mf ay isupf o∅) ab →
            * a < * b → odef (UnionCF A f mf ay isupf o∅) b
          imax {a} {b} ua b<x ab (case1 hasp) a<b = subst (λ k → odef  (UnionCF A f mf ay isupf o∅) k ) (sym (HasPrev.x=fy hasp)) ( inext (HasPrev.ay hasp) )
          imax {a} {b} ua b<x ab (case2 sup)  a<b = ⊥-elim ( ¬x<0 b<x )

     --
     -- 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 = zc4 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
          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 

          pchain : HOD
          pchain  = UnionCF A f mf ay (ZChain.supf zc) x
          ptotal : IsTotalOrderSet pchain
          ptotal {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 (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) 
          pchain⊆A : {y : Ordinal} → odef pchain y →  odef A y
          pchain⊆A {y} ny = proj1 ny
          pnext : {a : Ordinal} → odef pchain a → odef pchain (f a)
          pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc)  ⟫
          pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u<x (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc ) ⟫
          pinit :  {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁
          pinit {a} ⟪ aa , ua ⟫  with  ua
          ... | ch-init fc = s≤fc y f mf fc
          ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc)  where
               zc7 : y << (ZChain.supf zc) u 
               zc7 = ChainP.fcy<sup is-sup (init ay)
          pcy : odef pchain y
          pcy = ⟪ ay , ch-init (init ay)    ⟫

          supf0 : Ordinal → Ordinal
          supf0 z with trio< z x
          ... | tri< a ¬b ¬c = ZChain.supf zc z
          ... | tri≈ ¬a b ¬c = ZChain.supf zc px
          ... | tri> ¬a ¬b c = ZChain.supf zc px

          spuf0eq : {z : Ordinal} → z o< x → supf0 z ≡ ZChain.supf zc z 
          spuf0eq {z} z<x with trio< z x
          ... | tri< a ¬b ¬c = refl
          ... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a z<x )
          ... | tri> ¬a ¬b c = ⊥-elim ( ¬a z<x )

          zc9 :  UnionCF A f mf ay (ZChain.supf zc) px ⊆'  UnionCF A f mf ay supf0 x 
          zc9 ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ 
          zc9 {z} ⟪ az , ch-is-sup u u<px is-sup fc ⟫ with trio< z x
          ... | tri< a ¬b ¬c = ⟪ az , ch-is-sup u (ordtrans u<px px<x) ? ? ⟫ 
          ... | tri≈ ¬a b ¬c = ?
          ... | tri> ¬a ¬b c = ?

          -- ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫ 
          --- zc9 {z} ⟪ az , ch-is-sup u u<px is-sup fc ⟫ with trio< z x
          -- ... | tri< a ¬b ¬c = ⟪ az , ch-is-sup u (ordtrans u<px px<x) ? ? ⟫ 
          -- ... | tri≈ ¬a b ¬c = ?
          -- ... | tri> ¬a ¬b c = ?
          -- = ⟪ az , ch-is-sup u (ordtrans u<px px<x) ? ? ⟫ 

          -- if previous chain satisfies maximality, we caan reuse it
          --
          no-extension : ZChain A f mf ay x
          no-extension = record { supf = supf0
               ; initial = ? ; chain∋init = ?  ; sup=u = sup=u  ; chain∋supf = ?
              ;  chain⊆A = ?  ;  f-next = ? ;  f-total = ? ;  order = ? } where
              cs : {z : Ordinal} → odef (UnionCF A f mf ay supf0 x) (supf0 z)
              cs {z} with trio< z x
              ... | tri< a ¬b ¬c = zc9 (ZChain.chain∋supf zc {z} )
              ... | tri≈ ¬a b ¬c = ?
              ... | tri> ¬a ¬b c = ?
              --- = subst (λ k → odef (UnionCF A f mf ay supf0 k) (supf0 z)) ? (ZChain.chain∋supf zc {z} ) 
              sup=u :  {b : Ordinal} {ab : odef A b} → b o< x 
                    → IsSup A (UnionCF A f mf ay supf0 x) ab 
                    → supf0 b ≡ b
              sup=u {b} {ab} b<x is-sup with trio< b px
              ... | tri< a ¬b ¬c = ? where --  ZChain.sup=u zc {b} {ab} a record { x<sup = pc20 } where
                     pc20 : {z : Ordinal} → odef (UnionCF A f mf ay (ZChain.supf zc) (Oprev.oprev op)) z → (z ≡ b) ∨ (z << b)
                     pc20 {z} ⟪ az , ch-init fc  ⟫ = 
                         IsSup.x<sup is-sup ⟪ az ,  ch-init fc  ⟫ 
                     pc20 {z} ⟪ az , ch-is-sup u u<x is-sup-c fc  ⟫ = IsSup.x<sup is-sup 
                         ⟪ az , ch-is-sup u (subst (λ k → u o< k) (Oprev.oprev=x op) (ordtrans u<x <-osuc)) ? ?  ⟫ 
              ... | tri≈ ¬a refl ¬c = ?
              ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op))  b<x  ⟫ )
              order : {b sup1 z1 : Ordinal} → b o< x → sup1 o< b → FClosure A f (supf0 sup1) z1 → z1 << supf0 b
              order {b} {s} {z1} b<x s<b fc with trio< b px 
              ... | tri< a ¬b ¬c = ? -- ZChain.order zc a s<b ?
              ... | tri≈ ¬a refl ¬c = ?
              ... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → b o< k ) (sym (Oprev.oprev=x op))  b<x  ⟫ )

          zc4 : ZChain A f mf ay x 
          zc4 with ODC.∋-p O A (* px)
          ... | no noapx = no-extension -- ¬ A ∋ p, just skip
          ... | yes apx with ODC.p∨¬p O ( HasPrev A (ZChain.chain zc ) apx f )   
               -- we have to check adding x preserve is-max ZChain A y f mf x
          ... | case1 pr = no-extension -- we have previous A ∋ z < x , f z ≡ x, so chain ∋ f z ≡ x because of f-next
          ... | case2 ¬fy<x with ODC.p∨¬p O (IsSup A (ZChain.chain zc ) apx )
          ... | case1 is-sup = -- x is a sup of zc 
                record {  supf = psupf1 ; chain⊆A = ? ; f-next = ? ; f-total = ? ; chain∋supf = ?
                     ; initial = ? ; chain∋init  = ? ; sup=u = ? ; order = ? }  where
             psupf1 : Ordinal → Ordinal
             psupf1 z with trio< z x 
             ... | tri< a ¬b ¬c = ZChain.supf zc z
             ... | tri≈ ¬a b ¬c = x
             ... | tri> ¬a ¬b c = x
             cA : UnionCF A f mf ay psupf1 x ⊆' A
             cA {z} uz with trio< z x
             ... | tri< a ¬b ¬c = ZChain.chain⊆A zc ⟪ proj1 uz , ? ⟫ 
             ... | tri≈ ¬a b ¬c = ?
             ... | tri> ¬a ¬b c = ?

          ... | case2 ¬x=sup =  no-extension -- px is not f y' nor sup of former ZChain from y -- no extention
     ... | no lim = zc5 where

          pzc : (z : Ordinal) → z o< x → ZChain A f mf ay z
          pzc z z<x = prev z z<x

          psupf0 : (z : Ordinal) → Ordinal
          psupf0 z with trio< z x
          ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z
          ... | tri≈ ¬a b ¬c = & A
          ... | tri> ¬a ¬b c = & A

          pchain : HOD
          pchain = UnionCF A f mf ay psupf0 x

          psupf0=pzc : {z : Ordinal} → (z<x : z o< x) → psupf0 z ≡ ZChain.supf (pzc z z<x) z
          psupf0=pzc {z} z<x with trio< z x
          ... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x)
          ... | tri> ¬a ¬b c = ⊥-elim (¬a z<x)
          ... | tri< a ¬b ¬c with o<-irr z<x a 
          ... | refl = refl

          ptotal : IsTotalOrderSet pchain
          ptotal {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 psupf0 ( (proj2 ca)) ( (proj2 cb)) 

          pchain⊆A : {y : Ordinal} → odef pchain y →  odef A y
          pchain⊆A {y} ny = proj1 ny
          pnext : {a : Ordinal} → odef pchain a → odef pchain (f a)
          pnext {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 ( mf a aa ) , ch-init (fsuc _ fc) ⟫
          pnext {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ⟪  proj2 ( mf a aa ) , ch-is-sup u u<x (ChainP-next A f mf ay _ is-sup ) (fsuc _ fc) ⟫
          pinit :  {y₁ : Ordinal} → odef pchain y₁ → * y ≤ * y₁
          pinit {a} ⟪ aa , ua ⟫  with  ua
          ... | ch-init fc = s≤fc y f mf fc
          ... | ch-is-sup u u<x is-sup fc = ≤-ftrans (case2 zc7) (s≤fc _ f mf fc)  where
               zc7 : y << psupf0 ?
               zc7 = ChainP.fcy<sup is-sup (init ay)
          pcy : odef pchain y
          pcy = ⟪ ay , ch-init (init ay)    ⟫

          no-extension : ZChain A f mf ay x
          no-extension  = record { initial = pinit ; chain∋init = pcy  ; supf = psupf0  ; chain∋supf = ?
              ;  chain⊆A = pchain⊆A ;  f-next = pnext ;  f-total = ptotal ; sup=u = ? ; order = ? }

          usup : SUP A pchain
          usup = supP pchain (λ lt → proj1 lt) ptotal
          spu = & (SUP.sup usup)
          psupf : Ordinal → Ordinal
          psupf z with trio< z x 
          ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z
          ... | tri≈ ¬a b ¬c = spu
          ... | tri> ¬a ¬b c = spu

          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 ax f )   
               -- 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 (IsSup A pchain ax )
          ... | case1 is-sup = record { initial = {!!} ; chain∋init = {!!}  ; supf = psupf1  ; sup=u = ? ; order = ? ; chain∋supf = ?
              ;  chain⊆A = {!!} ;  f-next = {!!} ;  f-total = ? } where -- x is a sup of (zc ?) 
             psupf1 : Ordinal → Ordinal
             psupf1 z with trio< z x 
             ... | tri< a ¬b ¬c = ZChain.supf (pzc z a) z
             ... | tri≈ ¬a b ¬c = x
             ... | tri> ¬a ¬b c = x
          ... | case2 ¬x=sup =  no-extension -- x is not f y' nor sup of former ZChain from y -- no extention
         
     SZ : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → ZChain A f mf ay   (& A)
     SZ f mf {y} ay = TransFinite {λ z → ZChain A f mf ay   z  } (λ x → ind f mf ay x   ) (& A)

     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)) ; A∋maximal = 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 zorn04 total ) 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.A∋maximal 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) ) ⟫
         zorn04 : ZChain A (cf nmx) (cf-is-≤-monotonic nmx) as0 (& A)
         zorn04 = SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as ) 
         total : IsTotalOrderSet (ZChain.chain zorn04)
         total {a} {b} = zorn06  where
             zorn06 :  odef (ZChain.chain zorn04) (& a) → odef (ZChain.chain zorn04) (& b) → Tri (a < b) (a ≡ b) (b < a)
             zorn06 = ZChain.f-total (SZ (cf nmx) (cf-is-≤-monotonic nmx) (subst (λ k → odef A k ) &iso as) ) 

-- 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