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

comparison src/zorn.agda @ 1092:87c2da3811c3

index version

author	Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date	Wed, 21 Dec 2022 07:30:18 +0900
parents	63c1167b2343
children	6caa088346f0

comparison

equal deleted inserted replaced

-:63c1167b2343
+:87c2da3811c3
 open import Ordinals
 open import Relation.Binary
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality
 import OD hiding ( _⊆_ )
-module zorn {n : Level } (O : Ordinals {n}) (_<_ : (x y : OD.HOD O ) → Set n ) (PO : IsStrictPartialOrder _≡_ _<_ ) where
+module zorn1 {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
 ≤-monotonic-f A f = (x : Ordinal ) → odef A x → (  x ≤  (f x) ) ∧  odef A (f x )
 <-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set 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 : {s1 : Ordinal } → odef A s → s ≡ s1  → FClosure A f s s1
-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 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 =  subst₂ (λ j k → j ≤ k ) refl x=fx (s≤fc 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 (cong (*) 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 ) (cong (*) x=fx) ( fc07 cx eq )
--- ... | case2 x<fx = ?
-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 (suc i) (suc j) x cx (init x₃ x₄) x₁ x₂ = ⊥-elim ( fc06 x₄ x₂ )
-fc00 (suc i) (suc j) x (init x₃ x₄) (fsuc x₅ cy) x₁ x₂ = ⊥-elim ( fc06 x₄ x₁ )
-fc00 zero zero refl (init _ refl) (init x₁ refl) i=x i=y = refl
-fc00 zero zero refl (init as refl) (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 (fc07 cy i=y) -- ( fc00 zero zero refl (init as refl) cy i=x i=y )
-fc00 zero zero refl (fsuc x cx) (init as refl) i=x i=y with proj1 (mf x (A∋fc s f mf cx ) )
-... | case1 x=fx = subst (λ k → * k ≡ * s ) x=fx (sym (fc07 cx i=x)) -- ( fc00 zero zero refl cx (init as refl) 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 x1 (init x₁ x₂) x = ⊥-elim (fc06 x₂ x)
-fc02 .(f x1) (fsuc x1 cx1) i=x1 with proj1 (mf x1 (A∋fc s f mf cx1 ) )
-... | case1 eq = trans (sym (cong (*) 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 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 ) (cong (*) 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 ) (cong (*) 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
--- 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 )
 _⊆_ : ( A B : HOD ) → Set n
 _⊆_ A B = {x : Ordinal } → odef A x → odef B x
 field
 sup : HOD
 isSUP : IsSUP A B (& sup)
 ax = IsSUP.ax isSUP
 x≤sup = IsSUP.x≤sup isSUP
+record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where
+field
+as : 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
+record MinSUP ( A B : HOD )  : Set n where
+field
+sup : Ordinal
+isMinSUP : IsMinSUP A B sup
+as = IsMinSUP.as isMinSUP
+x≤sup = IsMinSUP.x≤sup isMinSUP
+minsup = IsMinSUP.minsup isMinSUP
+record IChain (A : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where
+field
+y : Ordinal
+x=fy   : x ≡ f y
+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))
 --
 --   Our Proof strategy of the Zorn Lemma
 --
 --         f (f ( ... (supf y))) f (f ( ... (supf z1)))
 --
 --    if sup z1 ≡ sup z2, the chain is stopped at sup z1, then f (sup z1) ≡ sup z1
 --    this means sup z1 is the Maximal, so f is <-monotonic if we have no Maximal.
 --
-fc-stop : ( A : HOD )    ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) { a b : Ordinal }
-→ (aa : odef A a ) →(  {y : Ordinal} → FClosure A f a y → (y ≡ b ) ∨ (y << b )) → a ≡ b → f a ≡ a
-fc-stop A f mf {a} {b} aa x≤sup a=b with x≤sup (fsuc a (init aa refl ))
-... | case1 eq = trans eq (sym a=b)
-... | case2 lt = ⊥-elim (<-irr (case1 (cong (λ k → * (f k) ) (sym a=b))) (ftrans<-≤ lt fc00 ) ) where
-fc00 :   b ≤  (f b)
-fc00 = proj1 (mf _ (subst (λ k → odef A k) a=b aa ))
 ∈∧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 and FClosure A f y
-data UChain  { A : HOD } { f : Ordinal → Ordinal }  {supf : Ordinal → Ordinal} {y : Ordinal } (ay : odef A y )
-(x : Ordinal) : (z : Ordinal) → Set n where
-ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain ay x z
-ch-is-sup  : (u : Ordinal) {z : Ordinal }  (u<x : u o< x) (supu=u : supf u ≡ u) ( fc : FClosure A f (supf u) z ) → UChain ay x z
-UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) {y : Ordinal } (ay : odef A y ) ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD
-UnionCF A f ay supf x
-= record { od = record { def = λ z → odef A z ∧ UChain {A} {f} {supf} ay x z } ;
-odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy }
--- Union of chain lower than x
-data IChain  {A : HOD}  { f : Ordinal → Ordinal } {y : Ordinal } (ay : odef A y )
-{x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) : (z : Ordinal ) → Set n where
-ic-init : {z : Ordinal } (fc : FClosure A f y z) → IChain ay supfz z
-ic-isup : {z : Ordinal} (i : Ordinal) (i<x : i o< x) (s<x : supfz i<x o≤ i ) (fc : FClosure A f (supfz i<x) z) → IChain ay supfz z
-UnionIC : ( A : HOD ) ( f : Ordinal → Ordinal ) { x : Ordinal } {y : Ordinal } (ay : odef A y ) (supfz : {z : Ordinal } → z o< x → Ordinal)  → HOD
-UnionIC A f ay supfz
-= record { od = record { def = λ z → odef A z ∧ IChain {A} {f} ay supfz 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 )
-... | case2 lt = ⊥-elim ( o<> sx<sy lt )
-record IsMinSUP ( A B : HOD ) (sup : Ordinal) : Set n where
-field
-as : 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
-record MinSUP ( A B : HOD )  : Set n where
-field
-sup : Ordinal
-isMinSUP : IsMinSUP A B sup
-as = IsMinSUP.as isMinSUP
-x≤sup = IsMinSUP.x≤sup isMinSUP
-minsup = IsMinSUP.minsup isMinSUP
-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))
-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 ay supf a) c → odef (UnionCF A f 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 ⟪ ua , ch-is-sup u u<x supu=u fc ⟫ = ⟪ ua , ch-is-sup u (ordtrans<-≤ u<x a≤b) supu=u 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
-supf-mono : {a b : Ordinal } → a o≤ b → supf a o≤ supf b
-cfcs  : {a b w : Ordinal } → a o< b → b o≤ z → supf a o< b → FClosure A f (supf a) w → odef (UnionCF A f ay supf b) w
-asupf :  {x : Ordinal } → odef A (supf x)
-zo≤sz : {x : Ordinal } → x o≤ z → x o≤ supf x
-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x)
-chain : HOD
-chain = UnionCF A f ay supf z
-chain⊆A : chain ⊆ A
-chain⊆A = λ lt → proj1 lt
-chain∋init : {x : Ordinal } → odef (UnionCF A f ay supf x) y
-chain∋init {x} = ⟪ ay , ch-init (init ay refl)  ⟫
-mf : ≤-monotonic-f A f
-mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
-mf00 : * x < * (f x)
-mf00 = proj1 ( mf< x ax )
-f-next : {a z : Ordinal} → odef (UnionCF A f ay supf z) a → odef (UnionCF A f ay supf z) (f a)
-f-next {a} ⟪ ua , ch-init fc ⟫ = ⟪ proj2 ( mf _ ua)  , ch-init (fsuc _ fc) ⟫
-f-next {a} ⟪ ua , ch-is-sup u su<x su=u fc ⟫ = ⟪ proj2 ( mf _ ua)  , ch-is-sup u su<x su=u (fsuc _ fc) ⟫
-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 )
-csupf : {b : Ordinal } → supf b o< supf z → supf b o< z → odef (UnionCF A f ay supf z) (supf b) -- supf z is not an element of this chain
-csupf {b} sb<sz sb<z = cfcs (supf-inject sb<sz) o≤-refl sb<z (init asupf refl)
-minsup : {x : Ordinal } → x o≤ z  → MinSUP A (UnionCF A f ay supf x)
-minsup {x} x≤z = record { sup = supf x ; isMinSUP = is-minsup x≤z }
-supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup (minsup x≤z)
-supf-is-minsup _ = refl
--- different from order because y o< supf
-fcy<sup  : {u w : Ordinal } → u o≤ z  → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u )
-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)
-, 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 ) ) ))
-... | case2 lt = case2 (subst₂ (λ j k → j << k ) &iso (sym (supf-is-minsup u≤z )) lt )
-initial : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) x →  y ≤ x
-initial {x} x≤z ⟪ aa , ch-init fc ⟫ = s≤fc y f mf fc
-initial {x} x≤z ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ≤-ftrans (fcy<sup (ordtrans u<x x≤z) (init ay refl)) (s≤fc _ f mf fc)
-sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z
-→ IsSUP A (UnionCF A f ay supf b) b  → supf b ≡ b
-sup=u {b} ab b≤z is-sup = z50 where
-z48 : supf b o≤ b
-z48 = IsMinSUP.minsup (is-minsup b≤z) ab (λ ux → IsSUP.x≤sup is-sup ux )
-z50 : supf b ≡ b
-z50 with trio< (supf b) b
-... | tri< sb<b ¬b ¬c = ⊥-elim ( o≤> z47 sb<b ) where
-z47 : b o≤ supf b
-z47 = zo≤sz b≤z
-... | tri≈ ¬a b ¬c = b
-... | tri> ¬a ¬b b<sb = ⊥-elim ( o≤> z48 b<sb )
-supfeq : {a b : Ordinal } → a o≤ z →  b o≤ z → UnionCF A f ay supf a ≡ UnionCF A f ay supf b → supf a ≡ supf b
-supfeq {a} {b} a≤z b≤z ua=ub with trio< (supf a) (supf b)
-... | tri< sa<sb ¬b ¬c = ⊥-elim ( o≤> (
-IsMinSUP.minsup (is-minsup b≤z) asupf (λ {z} uzb → IsMinSUP.x≤sup (is-minsup a≤z) (subst (λ k → odef k z) (sym ua=ub) uzb)) ) sa<sb )
-... | tri≈ ¬a b ¬c = b
-... | tri> ¬a ¬b sb<sa = ⊥-elim ( o≤> (
-IsMinSUP.minsup (is-minsup a≤z) asupf (λ {z} uza → IsMinSUP.x≤sup (is-minsup b≤z) (subst (λ k → odef k z) ua=ub uza)) ) sb<sa )
-union-max : {a b : Ordinal } → b o≤ supf a → b o≤ z → supf a o< supf b → UnionCF A f ay supf a ≡ UnionCF A f ay supf b
-union-max {a} {b} b≤sa b≤z sa<sb = ==→o≡ record { eq→ = z47 ; eq← = z48 } where
-z47 : {w : Ordinal } → odef (UnionCF A f ay supf a ) w → odef ( UnionCF A f ay supf b ) w
-z47 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
-z47 {w} ⟪ aw , ch-is-sup u u<a su=u fc ⟫ = ⟪ aw , ch-is-sup u u<b su=u fc ⟫ where
-u<b : u o< b
-u<b = ordtrans u<a (supf-inject sa<sb )
-z48 : {w : Ordinal } → odef (UnionCF A f ay supf b ) w → odef ( UnionCF A f ay supf a ) w
-z48 {w} ⟪ aw , ch-init fc ⟫ = ⟪ aw , ch-init fc ⟫
-z48 {w} ⟪ aw , ch-is-sup u u<b su=u fc ⟫ = ⟪ aw , ch-is-sup u u<a su=u fc ⟫ where
-u<a : u o< a
-u<a = supf-inject ( osucprev (begin
-osuc (supf u)  ≡⟨ cong osuc su=u ⟩
-osuc u  ≤⟨ osucc u<b ⟩
-b  ≤⟨ b≤sa ⟩
-supf a ∎ )) where open o≤-Reasoning O
-x≤supfx→¬sa<sa : {a b : Ordinal } → b o≤ z → b o≤ supf a → ¬ (supf a o< supf b )
-x≤supfx→¬sa<sa {a} {b} b≤z b≤sa sa<sb = ⊥-elim ( o<¬≡ z27 sa<sb ) where -- x o≤ supf a ∧ supf a o< supf b → ⊥, because it defines the same UnionCF
-z27 : supf a ≡ supf b
-z27 = supfeq (ordtrans (supf-inject sa<sb) b≤z) b≤z ( union-max  b≤sa b≤z sa<sb)
-order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b
-order {a} {b} {w} b≤z sa<sb fc = IsMinSUP.x≤sup (is-minsup b≤z) (cfcs (supf-inject sa<sb) b≤z sa<b fc) where
-sa<b : supf a o< b
-sa<b with x<y∨y≤x (supf a) b
-... | case1 lt = lt
-... | case2 b≤sa = ⊥-elim (x≤supfx→¬sa<sa b≤z b≤sa sa<sb)
-supf-idem : {b : Ordinal } → b o≤ z → supf b o≤ z  → supf (supf b) ≡ supf b
-supf-idem {b} b≤z sfb≤x = z52 where
-z54 :  {w : Ordinal} → odef (UnionCF A f ay supf (supf b)) w → (w ≡ supf b) ∨ (w << supf b)
-z54 {w} ⟪ aw , ch-init fc ⟫ = fcy<sup b≤z fc
-z54 {w} ⟪ aw , ch-is-sup u u<x su=u fc ⟫ = order b≤z (subst (λ k → k o< supf b) (sym su=u) u<x)  fc where
-u<b : u o< b
-u<b = supf-inject (subst (λ k → k o< supf b ) (sym (su=u)) u<x )
-z52 : supf (supf b) ≡ supf b
-z52 = sup=u asupf sfb≤x  record { ax = asupf  ; x≤sup = z54  }
-supf-mono< : {a b : Ordinal } → b o≤ z → supf a o< supf b → supf a << supf b
-supf-mono< {a} {b} b≤z sa<sb  with order {a} {b} b≤z sa<sb (init asupf refl)
-... | case2 lt = lt
-... | case1 eq = ⊥-elim ( o<¬≡ eq sa<sb )
-f-total : IsTotalOrderSet chain
-f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ =
-subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso fc-total where
-fc-total : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-fc-total with trio< ua ub
-... | tri< a₁ ¬b ¬c with ≤-ftrans  (order (o<→≤ sub<x) (subst₂ (λ j k → j o< k) (sym sua=ua) (sym sub=ub) a₁) fca ) (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 = cong (*) eq1
-... | case2 a<b =  tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)
-fc-total | tri≈ _ refl _ = fcn-cmp _ f mf fca fcb
-fc-total | tri> ¬a ¬b c with ≤-ftrans  (order (o<→≤ sua<x) (subst₂ (λ j k → j o< k) (sym sub=ub) (sym sua=ua) c) fcb ) (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 = cong (*) (sym eq1)
-... | case2 b<a =  tri> (λ lt → <-irr (case2 b<a ) lt)  (λ eq → <-irr (case1 eq) b<a )  b<a
-f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-is-sup ub sub<x sub=ub fcb ⟫ = ft00 where
-ft01 : (& a) ≤ (& b) → Tri ( a <  b) ( a ≡  b) ( b <  a )
-ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
-a=b : a ≡ b
-a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) eq)
-ft01 (case2 lt) = tri< a<b (λ eq → <-irr (case1 (sym eq)) a<b ) (λ lt → <-irr (case2 a<b ) lt)  where
-a<b : a < b
-a<b = subst₂ (λ j k → j < k ) *iso *iso lt
-ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
-ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sub<x) fca) (s≤fc {A} _ f mf fcb))
-f-total {a} {b} ⟪ uaa , ch-is-sup ua sua<x sua=ua fca ⟫ ⟪ uab , ch-init fcb ⟫ = ft00 where
-ft01 : (& b) ≤ (& a) → Tri ( a <  b) ( a ≡  b) ( b <  a )
-ft01 (case1 eq) = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym a=b)) lt)) a=b  (λ lt → ⊥-elim (<-irr (case1 a=b) lt))  where
-a=b : a ≡ b
-a=b = subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) (sym eq))
-ft01 (case2 lt) = tri> (λ lt → <-irr (case2 b<a ) lt) (λ eq → <-irr (case1 eq) b<a ) b<a where
-b<a : b < a
-b<a = subst₂ (λ j k → j < k ) *iso *iso lt
-ft00 :   Tri ( a <  b) ( a ≡  b) ( b <  a )
-ft00 = ft01 (≤-ftrans (fcy<sup (o<→≤ sua<x) fcb) (s≤fc {A} _ f mf fca))
-f-total {a} {b} ⟪ uaa , ch-init fca ⟫ ⟪ uab , ch-init fcb ⟫ =
-subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso  (fcn-cmp y f mf fca fcb )
-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 ay supf z) a ) → b o< z  → (ab : odef A b)
-→ HasPrev A (UnionCF A f ay supf z) f b ∨  IsSUP A (UnionCF A f ay supf b) b
-→ * a < * b  → odef ((UnionCF A f ay supf z)) 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
---
--- supf in TransFinite indution may differ each other, but it is the same because of the minimul sup
---
-supf-unique :  ( A : HOD )    ( f : Ordinal → Ordinal )  (mf< : <-monotonic-f A f)
-{y xa xb : Ordinal} → (ay : odef A y) →  (xa o≤ xb ) → (za : ZChain A f mf< ay xa ) (zb : ZChain A f mf< ay xb )
-→ {z : Ordinal } → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z
-supf-unique A f mf< {y} {xa} {xb} ay xa≤xb za zb {z} z≤xa = TransFinite0 {λ z → z o≤ xa → ZChain.supf za z ≡ ZChain.supf zb z } ind z z≤xa  where
-supfa = ZChain.supf za
-supfb = ZChain.supf zb
-ind : (x : Ordinal) → ((w : Ordinal) → w o< x → w o≤ xa → supfa w ≡ supfb w) → x o≤ xa → supfa x ≡ supfb x
-ind x prev x≤xa = sxa=sxb where
-ma = ZChain.minsup za x≤xa
-mb = ZChain.minsup zb (OrdTrans x≤xa xa≤xb )
-spa = MinSUP.sup ma
-spb = MinSUP.sup mb
-sax=spa : supfa x ≡ spa
-sax=spa = ZChain.supf-is-minsup za x≤xa
-sbx=spb : supfb x ≡ spb
-sbx=spb = ZChain.supf-is-minsup zb (OrdTrans x≤xa xa≤xb )
-sxa=sxb : supfa x ≡ supfb x
-sxa=sxb with trio< (supfa x) (supfb x)
-... | tri≈ ¬a b ¬c = b
-... | tri< a ¬b ¬c = ⊥-elim ( o≤> (
-begin
-supfb x  ≡⟨ sbx=spb ⟩
-spb  ≤⟨ MinSUP.minsup mb (MinSUP.as ma) (λ {z} uzb → MinSUP.x≤sup ma (z53 uzb)) ⟩
-spa ≡⟨ sym sax=spa ⟩
-supfa x ∎ ) a ) where
-open o≤-Reasoning O
-z53 : {z : Ordinal } →  odef (UnionCF A f ay (ZChain.supf zb) x) z →  odef (UnionCF A f ay (ZChain.supf za) x) z
-z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫
-z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ = ⟪ as , ch-is-sup u u<x (trans ua=ub su=u) z55 ⟫ where
-ua=ub : supfa u ≡ supfb u
-ua=ub = prev u u<x (ordtrans u<x x≤xa )
-z55 : FClosure A f (ZChain.supf za u) z
-z55 = subst (λ k → FClosure A f k z ) (sym ua=ub) fc
-... | tri> ¬a ¬b c = ⊥-elim ( o≤> (
-begin
-supfa x  ≡⟨ sax=spa ⟩
-spa  ≤⟨ MinSUP.minsup ma (MinSUP.as mb) (λ uza → MinSUP.x≤sup mb (z53 uza)) ⟩
-spb  ≡⟨ sym sbx=spb ⟩
-supfb x ∎ ) c ) where
-open o≤-Reasoning O
-z53 : {z : Ordinal } →  odef (UnionCF A f ay (ZChain.supf za) x) z →  odef (UnionCF A f ay (ZChain.supf zb) x) z
-z53 ⟪ as , ch-init fc ⟫ = ⟪ as , ch-init fc ⟫
-z53 {z} ⟪ as , ch-is-sup u u<x su=u fc ⟫ =  ⟪ as , ch-is-sup u u<x (trans ub=ua su=u) z55  ⟫ where
-ub=ua : supfb u ≡ supfa u
-ub=ua = sym ( prev u u<x (ordtrans u<x x≤xa ))
-z55 : FClosure A f (ZChain.supf zb u) z
-z55 = subst (λ k → FClosure A f k z ) (sym ub=ua) fc
 Zorn-lemma : { A : HOD }
 → o∅ o< & A
 → ( ( B : HOD) → (B⊆A : B ⊆ A) → IsTotalOrderSet B → SUP A B   ) -- SUP condition
 → Maximal A
 ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (trans &iso eq))
 ... | case2 lt = case2 lt
 m02 : MinSUP A B
 m02 = dont-or (m00 (& A)) m03
--- Uncountable ascending chain by axiom of choice
+--   -- Uncountable ascending chain by axiom of choice
-cf : ¬ Maximal A → Ordinal → Ordinal
+--   cf : ¬ Maximal A → Ordinal → Ordinal
-cf  nmx x with ODC.∋-p O A (* x)
+--   cf  nmx x with ODC.∋-p O A (* x)
-... | no _ = o∅
+--   ... | no _ = o∅
-... | yes ax with is-o∅ (& ( Gtx ax ))
+--   ... | yes ax with is-o∅ (& ( Gtx ax ))
-... | yes nogt = -- no larger element, so it is maximal
+--   ... | 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 ⟫ )
+--       ⊥-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)))
+--   ... | 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 : ¬ 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)
+--   is-cf nmx {x} ax with ODC.∋-p O A (* x)
-... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax ))
+--   ... | no not = ⊥-elim ( not (subst (λ k → odef A k ) (sym &iso) ax ))
-... | yes ax with is-o∅ (& ( Gtx 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 ⟫ )
+--   ... | 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))
+--   ... | no not = ODC.x∋minimal O (Gtx ax) (λ eq → not (=od∅→≡o∅ eq))
+--
----
+--   ---
---- infintie ascention sequence of f
+--   --- 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 : ¬ 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 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 : ¬ 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  ) ⟫
+--   cf-is-≤-monotonic nmx x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic nmx x ax  ))  , proj2 ( cf-is-<-monotonic nmx x ax  ) ⟫
---
--- maximality of chain
---
---     supf is fixed for z ≡ & A , we can prove order and is-max
---     we have supf-unique now, it is provable in the first Tranfinte induction
-SZ1 : ( f : Ordinal → Ordinal )  (mf : ≤-monotonic-f A f) (mf< : <-monotonic-f A f)
-{init : Ordinal} (ay : odef A init) (zc : ZChain A f mf< ay (& A)) (x : Ordinal) → x o≤ & A → ZChain1 A f mf< ay zc x
-SZ1 f mf mf< {y} ay zc x x≤A = zc1 x x≤A  where
-chain-mono1 :  {a b c : Ordinal} → a o≤ b
-→ odef (UnionCF A f ay (ZChain.supf zc) a) c → odef (UnionCF A f 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 ay (ZChain.supf zc) x) a → (ab : odef A b)
-→ HasPrev A (UnionCF A f ay (ZChain.supf zc) x) f b
-→ * a < * b → odef (UnionCF A f 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 su=u fc ⟫ = ⟪ ab , subst (λ k → UChain ay x k )
-(sym (HasPrev.x=fy has-prev)) ( ch-is-sup u u<x su=u (fsuc _ fc))  ⟫
-supf = ZChain.supf zc
-zc1 :  (x : Ordinal ) → x o≤ & A →   ZChain1 A f mf< ay zc x
-zc1 x x≤A with Oprev-p x
-... | yes op = record { is-max = is-max } where
-px = Oprev.oprev op
-is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a →
-b o< x → (ab : odef A b) →
-HasPrev A (UnionCF A f ay supf x) f b  ∨ IsSUP A (UnionCF A f ay supf b) b →
-* a < * b → odef (UnionCF A f 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 b<x ab P a<b | case2 is-sup with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x))
-... | case2 sb<sx = m10 where
-b<A : b o< & A
-b<A = z09 ab
-m05 : ZChain.supf zc b ≡ b
-m05 =  ZChain.sup=u zc ab (o<→≤ (z09 ab) )  record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz  }
-m10 : odef (UnionCF A f ay supf x) b
-m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05)
-... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where
-m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x )
-m17 = ZChain.minsup zc x≤A
-m18 : supf x ≡ MinSUP.sup m17
-m18 = ZChain.supf-is-minsup zc x≤A
-m10 : f (supf b) ≡ supf b
-m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where
-m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x)
-m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where
-m04 : ¬ HasPrev A (UnionCF A f 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 { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz  }
-m14 : ZChain.supf zc b o< x
-m14 = subst (λ k → k o< x ) (sym m05)  b<x
-m13 :  odef (UnionCF A f ay supf x) z
-m13 = ZChain.cfcs zc b<x x≤A m14 fc
-... | no lim = record { is-max = is-max }  where
-is-max :  {a : Ordinal} {b : Ordinal} → odef (UnionCF A f ay supf x) a →
-b o< x → (ab : odef A b) →
-HasPrev A (UnionCF A f ay supf x) f b  ∨ IsSUP A (UnionCF A f ay supf b) b →
-* a < * b → odef (UnionCF A f 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 b<x ab P a<b | case2 is-sup with IsSUP.x≤sup (proj2 is-sup) (ZChain.chain∋init zc  )
-... | 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 with osuc-≡< (ZChain.supf-mono zc (o<→≤ b<x))
-... | case2 sb<sx = m10 where
-m09 : b o< & A
-m09 = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab))
-m05 : ZChain.supf zc b ≡ b
-m05 = ZChain.sup=u zc ab (o<→≤  m09) record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt }
-m10 : odef (UnionCF A f ay supf x) b
-m10 = ZChain.cfcs zc b<x x≤A (subst (λ k → k o< x) (sym m05) b<x) (init (ZChain.asupf zc) m05)
-... | case1 sb=sx = ⊥-elim (<-irr (case1 (cong (*) m10)) (proj1 (mf< (supf b) (ZChain.asupf zc)))) where
-m17 : MinSUP A (UnionCF A f ay supf x) -- supf z o< supf ( supf x )
-m17 = ZChain.minsup zc x≤A
-m18 : supf x ≡ MinSUP.sup m17
-m18 = ZChain.supf-is-minsup zc x≤A
-m10 : f (supf b) ≡ supf b
-m10 = fc-stop A f mf (ZChain.asupf zc) m11 sb=sx where
-m11 : {z : Ordinal} → FClosure A f (supf b) z → (z ≡ ZChain.supf zc x) ∨ (z << ZChain.supf zc x)
-m11 {z} fc = subst (λ k → (z ≡ k) ∨ (z << k)) (sym m18) ( MinSUP.x≤sup m17 m13 ) where
-m05 =  ZChain.sup=u zc ab (o<→≤ (z09 ab) ) record { ax = ab ; x≤sup = λ {z} uz → IsSUP.x≤sup (proj2 is-sup) uz }
-m14 : ZChain.supf zc b o< x
-m14 = subst (λ k → k o< x ) (sym m05)  b<x
-m13 :  odef (UnionCF A f ay supf x) z
-m13 = ZChain.cfcs zc b<x x≤A m14 fc
-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)
---
--- 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 sup1 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 ay supf0 px
-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  ⟫ )
-mf : ≤-monotonic-f A f
-mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
-mf00 : * x < * (f x)
-mf00 = proj1 ( mf< x ax )
---
--- find the next value of supf
---
-pchainpx : HOD
-pchainpx = record { od = record { def = λ z →  (odef A z  ∧ UChain  ay px z )
-∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x)) } ; odmax = & A ; <odmax = zc00 } where
-zc00 : {z : Ordinal } → (odef A z ∧ UChain ay px z ) ∨ (FClosure A f (supf0 px) z ∧ (supf0 px o< x) )→ z o< & A
-zc00 {z} (case1 lt) = z07 lt
-zc00 {z} (case2 fc) = z09 ( A∋fc (supf0 px) f mf (proj1 fc) )
-zc02 : { a b : Ordinal } → odef A a ∧ UChain ay px a → FClosure A f (supf0 px) b ∧ ( supf0 px o< x) → a ≤ b
-zc02 {a} {b} ca fb = zc05 (proj1 fb) where
-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 ) eq (zc05 fb)
-... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt)
-zc05 (init b1 refl) = MinSUP.x≤sup (ZChain.minsup zc o≤-refl) ca
-ptotal : IsTotalOrderSet pchainpx
-ptotal (case1 a) (case1 b) =  ZChain.f-total zc a 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
-lt1 : a0 < b0
-lt1 = subst₂ (λ j k → j < k ) *iso *iso lt
-ptotal {b0} {a0} (case2 b) (case1 a) with zc02 a b
-... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where
-eq1 : a0 ≡ b0
-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 (proj1 a) (proj1 b))
-pcha : pchainpx ⊆ A
-pcha (case1 lt) = proj1 lt
-pcha (case2 fc) = A∋fc _ f mf (proj1 fc)
-sup1 : MinSUP A pchainpx
-sup1 = minsupP pchainpx pcha ptotal
---
---     supf0 px o≤ sp1
---
-zc41 : MinSUP A pchainpx → ZChain A f mf< ay x
-zc41 sup1 =  record { supf = supf1 ; asupf = asupf1 ; zo≤sz = zo≤sz ;  is-minsup = is-minsup ;  cfcs = cfcs ; supf-mono = supf1-mono }  where
-sp1 = MinSUP.sup sup1
-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 = refl
-... | tri> ¬a ¬b c = ⊥-elim ( o≤> z≤px c )
-sf1=sp1 : {z : Ordinal } → px o< z → supf1 z ≡ sp1
-sf1=sp1 {z} px<z with trio< z px
-... | tri< a ¬b ¬c = ⊥-elim ( o<> px<z a )
-... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) px<z )
-... | tri> ¬a ¬b c = refl
-sf=eq :  { z : Ordinal } → z o< x → supf0 z ≡ supf1 z
-sf=eq {z} z<x = sym (sf1=sf0 (subst (λ k → z o< k ) (sym (Oprev.oprev=x op)) z<x ))
-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.as 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 ay supf0 a)
-zc21 = ZChain.minsup zc (o<→≤ a<px)
-zc24 : {x₁ : Ordinal} → odef (UnionCF A f 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.as sup1) zc24 )
-... | tri≈ ¬a b ¬c = zc18 where
-zc21 : MinSUP A (UnionCF A f 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 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.as 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
--- this is a kind of maximality, so we cannot prove this without <-monotonicity
---
-cfcs : {a b w : Ordinal }
-→ a o< b → b o≤ x → supf1 a o< b  → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w
-cfcs {a} {b} {w} a<b b≤x sa<b fc with x<y∨y≤x  (supf0 a) px
-... | case2 px≤sa = z50 where
-a<x : a o< x
-a<x = ordtrans<-≤ a<b b≤x
-a≤px : a o≤ px
-a≤px = subst (λ k → a o< k) (sym (Oprev.oprev=x op)) (ordtrans<-≤ a<b b≤x)
---  supf0 a ≡ px we cannot use previous cfcs, it is in the chain because
---       supf0 a ≡ supf0 (supf0 a) ≡ supf0 px o< x
-z50 : odef (UnionCF A f ay supf1 b) w
-z50 with osuc-≡< px≤sa
-... | case1 px=sa = ⟪ A∋fc {A} _ f mf fc , cp  ⟫ where
-sa≤px : supf0 a o≤ px
-sa≤px = subst₂ (λ j k → j o< k) px=sa (sym (Oprev.oprev=x op)) px<x
-spx=sa : supf0 px ≡ supf0 a
-spx=sa = begin
-supf0 px ≡⟨ cong supf0 px=sa  ⟩
-supf0 (supf0 a) ≡⟨ ZChain.supf-idem zc a≤px sa≤px  ⟩
-supf0 a ∎  where open ≡-Reasoning
-z51 : supf0 px o< b
-z51 = subst (λ k → k o< b ) (sym ( begin supf0 px ≡⟨ spx=sa ⟩
-supf0 a ≡⟨ sym (sf1=sf0 a≤px) ⟩
-supf1 a ∎ )) sa<b where open ≡-Reasoning
-z52 : supf1 a ≡ supf1 (supf0 px)
-z52 = begin supf1 a ≡⟨ sf1=sf0 a≤px ⟩
-supf0 a ≡⟨ sym (ZChain.supf-idem zc a≤px sa≤px ) ⟩
-supf0 (supf0 a) ≡⟨ sym (sf1=sf0 sa≤px)  ⟩
-supf1 (supf0 a) ≡⟨ cong supf1 (sym spx=sa) ⟩
-supf1 (supf0 px) ∎ where open ≡-Reasoning
-z53 : supf1 (supf0 px) ≡ supf0 px
-z53 = begin
-supf1 (supf0 px)  ≡⟨ cong supf1 spx=sa ⟩
-supf1 (supf0 a)  ≡⟨ sf1=sf0 sa≤px ⟩
-supf0 (supf0 a)  ≡⟨ sym ( cong supf0 px=sa ) ⟩
-supf0 px  ∎  where open ≡-Reasoning
-cp : UChain ay b w
-cp = ch-is-sup (supf0 px) z51 z53 (subst (λ k → FClosure A f k w) z52 fc)
-... | case2 px<sa = ⊥-elim ( ¬p<x<op ⟪ px<sa , subst₂ (λ j k → j o< k ) (sf1=sf0 a≤px) (sym (Oprev.oprev=x op)) z53 ⟫  ) where
-z53  : supf1 a o< x
-z53  = ordtrans<-≤ sa<b b≤x
-... | case1 sa<px with trio< a px
-... | tri< a<px ¬b ¬c = z50 where
-z50 : odef (UnionCF A f ay supf1 b) w
-z50 with osuc-≡< b≤x
-... | case2 lt with ZChain.cfcs zc a<b (subst (λ k → b o< k) (sym (Oprev.oprev=x op)) lt) sa<b fc
-... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-... | ⟪ az , ch-is-sup u u<b su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc u≤px )  ⟫ where
-u≤px : u o≤ px
-u≤px = subst (λ k → u o< k) (sym (Oprev.oprev=x op))  (ordtrans<-≤ u<b b≤x )
-u<x : u o< x
-u<x = ordtrans<-≤ u<b b≤x
-z50 | case1 eq with ZChain.cfcs zc a<px o≤-refl sa<px fc
-... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-... | ⟪ az , ch-is-sup u u<px su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans (sym (sf=eq u<x)) su=u) (fcpu fc (o<→≤ u<px)) ⟫  where -- u o< px → u o< b ?
-u<b : u o< b
-u<b = subst (λ k → u o< k ) (trans (Oprev.oprev=x op) (sym eq) ) (ordtrans u<px <-osuc )
-u<x : u o< x
-u<x = subst (λ k → u o< k ) (Oprev.oprev=x op) ( ordtrans u<px <-osuc )
-... | tri≈ ¬a a=px ¬c = csupf1 where
--- a ≡ px , b ≡ x, sp o≤ x
-px<b : px o< b
-px<b = subst₂ (λ j k → j o< k) a=px refl a<b
-b=x : b ≡ x
-b=x with trio< b x
-... | tri< a ¬b ¬c = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) a ⟫ ) --  px o< b o< x
-... | tri≈ ¬a b ¬c = b
-... | tri> ¬a ¬b c = ⊥-elim ( o≤> b≤x c ) --   x o< b
-z51 : FClosure A f (supf1 px) w
-z51 = subst (λ k → FClosure A f k w) (sym (trans (cong supf1 (sym a=px)) (sf1=sf0 (o≤-refl0 a=px) ))) fc
-z53 : odef A w
-z53 = A∋fc {A} _ f mf fc
-csupf1 : odef (UnionCF A f ay supf1 b) w
-csupf1 with x<y∨y≤x  px (supf0 px)
-... | case2 spx≤px = ⟪ z53 , ch-is-sup (supf0 px) z54 z52 fc1 ⟫  where
-z54 : supf0 px o< b
-z54 = subst (λ k → supf0 px o< k ) (trans (Oprev.oprev=x op) (sym b=x) ) spx≤px
-z52 : supf1 (supf0 px) ≡ supf0 px
-z52 = trans (sf1=sf0 spx≤px ) ( ZChain.supf-idem zc o≤-refl spx≤px  )
-fc1 : FClosure A f (supf1 (supf0 px)) w
-fc1 = subst (λ k → FClosure A f k w ) (trans (cong supf0 a=px) (sym z52) ) fc
-... | case1 px<spx = ⊥-elim (¬p<x<op ⟪ px<spx , z54  ⟫ ) where  -- supf1 px o≤ spuf1 x → supf1 px ≡ x o< x
-z54 : supf0 px o≤ px
-z54 = subst₂ (λ j k → supf0 j o< k ) a=px (trans b=x (sym (Oprev.oprev=x op))) sa<b
-... | tri> ¬a ¬b c = ⊥-elim ( ¬p<x<op ⟪ c , subst (λ k → a o< k ) (sym (Oprev.oprev=x op)) ( ordtrans<-≤ a<b b≤x) ⟫ ) --  px o< a o< b o≤ x
-zc11 : {z : Ordinal} → odef (UnionCF A f 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 u<x su=u fc ⟫ = zc21 fc where
-zc21 : {z1 : Ordinal } → FClosure A f (supf1 u) z1 → odef pchainpx z1
-zc21 {z1} (fsuc z2 fc ) with zc21 fc
-... | case1 ⟪ ua1 , ch-init fc₁ ⟫ = case1 ⟪ proj2 ( mf _ ua1)  , ch-init (fsuc _ fc₁)  ⟫
-... | case1 ⟪ ua1 ,  ch-is-sup u u<x su=u fc₁   ⟫ = case1 ⟪ proj2 ( mf _ ua1)  ,  ch-is-sup u u<x su=u (fsuc _ fc₁) ⟫
-... | case2 fc = case2 ⟪ fsuc _ (proj1 fc) , proj2 fc ⟫
-zc21 (init asp refl ) with trio< (supf0 u) (supf0 px)
-... | tri< a ¬b ¬c = case1 ⟪ asp , ch-is-sup u u<px (trans (sym (sf1=sf0 (o<→≤ u<px))) su=u )(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
-... | tri≈ ¬a b ¬c = case2 ⟪ (init (subst (λ k → odef A k) b (ZChain.asupf zc) )
-(sym (trans (sf1=sf0 (zc-b<x _ u<x))  b ))) , spx<x ⟫ where
-spx<x : supf0 px o< x
-spx<x = osucprev ( begin
-osuc (supf0 px) ≡⟨ cong osuc (sym b) ⟩
-osuc (supf0 u) ≡⟨ cong osuc  (sym (sf1=sf0 (zc-b<x _ u<x) ))  ⟩
-osuc (supf1 u) ≡⟨ cong osuc  su=u ⟩
-osuc u ≤⟨ osucc u<x ⟩
-x ∎ ) where open o≤-Reasoning O
-... | 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  ⟫ )
-is-minsup :  {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z)
-is-minsup {z} z≤x with osuc-≡< z≤x
-... | case1 z=x = record { as = zc22 ; x≤sup = z23 ; minsup = z24  }  where
-px<z : px o< z
-px<z = subst (λ k → px o< k) (sym z=x) px<x
-zc22 : odef A (supf1 z)
-zc22 = subst (λ k → odef A k ) (sym (sf1=sp1 px<z ))  ( MinSUP.as sup1 )
-z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z
-z23 {w} uz  = subst (λ k → w ≤ k ) (sym (sf1=sp1 px<z)) ( MinSUP.x≤sup sup1 (
-zc11 (subst (λ k → odef (UnionCF A f ay supf1 k) w) z=x uz )))
-z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s )
-→ supf1 z o≤ s
-z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sp1 px<z)) ( MinSUP.minsup sup1 as z25 ) where
-z25 : {w : Ordinal } → odef pchainpx w → w ≤ s
-z25 {w} (case2 fc) = sup ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) z28 z27 fc1 ⟫ where
--- z=x , supf0 px o< x
-z28 : supf0 px o< z --    supf0 px ≡ supf1 px o≤ supf1 x ≡ sp1 o≤ x ≡ z
-z28 = subst (λ k → supf0 px o< k) (sym z=x) (proj2 fc)
-z29 : supf0 px o≤ px
-z29 = zc-b<x _ (proj2 fc)
-z27 : supf1 (supf0 px) ≡ supf0 px
-z27 = trans (sf1=sf0 z29) ( ZChain.supf-idem zc o≤-refl z29 )
-fc1 : FClosure A f (supf1 (supf0 px)) w
-fc1 = subst (λ k → FClosure A f k w) (sym z27) (proj1 fc)
-z25 {w} (case1 ⟪ ua , ch-init fc ⟫) = sup ⟪ ua , ch-init fc ⟫
-z25 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc  ⟫) = sup ⟪ ua , ch-is-sup u u<z
-(trans (sf1=sf0 u≤px)  su=u)  (fcpu fc u≤px)  ⟫ where
-u≤px : u o< osuc px
-u≤px = ordtrans u<x <-osuc
-u<z : u o< z
-u<z = ordtrans u<x (subst (λ k → px o< k ) (sym z=x) px<x )
-... | case2 z<x = record { as = zc22 ; x≤sup = z23 ; minsup = z24  } where
-z≤px = zc-b<x _ z<x
-m =  ZChain.is-minsup zc z≤px
-zc22 : odef A (supf1 z)
-zc22 = subst (λ k → odef A k ) (sym (sf1=sf0 z≤px))  ( IsMinSUP.as m )
-z23 : {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ supf1 z
-z23 {w} ⟪ ua , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px)) ( ZChain.fcy<sup zc z≤px fc )
-z23 {w} ⟪ ua ,  ch-is-sup u u<x su=u fc  ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf0 z≤px))
-(IsMinSUP.x≤sup m ⟪ ua ,  ch-is-sup u u<x (trans (sym (sf1=sf0 u≤px )) su=u)  (fcup fc u≤px )  ⟫ ) where
-u≤px : u o≤ px
-u≤px = ordtrans u<x z≤px
-z24 : {s : Ordinal } → odef A s → ( {w : Ordinal } → odef ( UnionCF A f ay supf1 z ) w → w ≤ s )
-→ supf1 z o≤ s
-z24 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf0 z≤px)) ( IsMinSUP.minsup m as z25 ) where
-z25 : {w : Ordinal } → odef ( UnionCF A f ay supf0 z ) w → w ≤ s
-z25 {w} ⟪ ua , ch-init fc ⟫ = sup ⟪ ua , ch-init fc ⟫
-z25 {w} ⟪ ua , ch-is-sup u u<x su=u fc  ⟫ = sup ⟪ ua , ch-is-sup u u<x
-(trans (sf1=sf0 u≤px)  su=u)  (fcpu fc u≤px)  ⟫ where
-u≤px : u o≤ px
-u≤px = ordtrans u<x z≤px
-zo≤sz : {z : Ordinal} →  z o≤ x → z o≤ supf1 z
-zo≤sz {z} z≤x with osuc-≡< z≤x
-... | case2 z<x = subst (λ k → z o≤ k) (sym (sf1=sf0 (zc-b<x _ z<x ))) (ZChain.zo≤sz zc (zc-b<x _ z<x ))
-... | case1 refl with osuc-≡< (supf1-mono (o<→≤ (px<x))) --   px o≤ supf1 px o≤ supf1 x ≡ sp1 → x o≤ sp1
-... | case2 lt = begin
-x ≡⟨ sym (Oprev.oprev=x op) ⟩
-osuc px ≤⟨ osucc (ZChain.zo≤sz zc o≤-refl)  ⟩
-osuc (supf0 px) ≡⟨ sym (cong osuc (sf1=sf0 o≤-refl )) ⟩
-osuc (supf1 px) ≤⟨ osucc lt ⟩
-supf1 x ∎ where open o≤-Reasoning O
-... | case1 spx=sx with osuc-≡< ( ZChain.zo≤sz zc o≤-refl )
-... | case2 lt = begin
-x ≡⟨ sym (Oprev.oprev=x op) ⟩
-osuc px ≤⟨ osucc lt ⟩
-supf0 px ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩
-supf1 px ≤⟨ supf1-mono (o<→≤ px<x)  ⟩
-supf1 x ∎ where open o≤-Reasoning O
-... | case1 px=spx =  ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where
-zc37 : supf0 px ≡ px
-zc37 = sym px=spx
-zc39 : supf0 px ≡ sp1
-zc39 = begin
-supf0 px ≡⟨ sym (sf1=sf0 o≤-refl)  ⟩
-supf1 px ≡⟨ spx=sx ⟩
-supf1 x ≡⟨ sf1=sp1 px<x ⟩
-sp1 ∎ where open ≡-Reasoning
-zc40 :  f (supf0 px) ≤ supf0 px
-zc40 = subst (λ k → f (supf0 px) ≤ k ) (sym zc39)
-( MinSUP.x≤sup sup1 (case2 ⟪ fsuc _ (init (ZChain.asupf zc) refl) , subst (λ k → k o< x) (sym zc37) px<x ⟫  ))
-... | no lim with trio< x o∅
-... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )
-... | tri≈ ¬a x=0 ¬c = record { supf = λ _ → MinSUP.sup (ysup f mf ay) ; asupf = MinSUP.as (ysup f mf ay) ; supf-mono = λ _ → o≤-refl
-; zo≤sz = zo≤sz ; is-minsup = is-minsup ; cfcs = λ a<b b≤0 → ⊥-elim ( ¬x<0 (subst (λ k → _ o< k ) x=0 (ordtrans<-≤ a<b b≤0)))    } where
-mf : ≤-monotonic-f A f
-mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
-mf00 : * x < * (f x)
-mf00 = proj1 ( mf< x ax )
-ym = MinSUP.sup (ysup f mf ay)
-zo≤sz : {z : Ordinal} → z o≤ x → z o≤ MinSUP.sup (ysup f mf ay)
-zo≤sz {z} z≤x with osuc-≡< z≤x
-... | case1 refl = subst (λ k → k o≤ _) (sym x=0) o∅≤z
-... | case2 lt = ⊥-elim ( ¬x<0  (subst (λ k → z o< k ) x=0 lt ) )
-is-minsup : {z : Ordinal} → z o≤ x →
-IsMinSUP A (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) z) (MinSUP.sup (ysup f mf ay))
-is-minsup {z} z≤x with osuc-≡< z≤x
-... | case1 refl = record { as = MinSUP.as  (ysup f mf ay) ; x≤sup = λ {w} uw → is00 uw ; minsup = λ {s} as sup → is01 as sup } where
-is00 : {w : Ordinal } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x ) w → w ≤ MinSUP.sup (ysup f mf ay)
-is00 {w} ⟪ aw , ch-init fc ⟫ = MinSUP.x≤sup (ysup f mf ay) fc
-is00 {w} ⟪ aw , ch-is-sup u u<z su=u fc ⟫ = ⊥-elim (¬x<0 (subst (λ k → u o< k ) x=0 u<z ))
-is01 : { s : Ordinal } → odef A s →  ( {w : Ordinal  } → odef (UnionCF A f ay (λ _ → MinSUP.sup (ysup f mf ay)) x )  w → w ≤ s )
-→ ym o≤ s
-is01 {s} as sup = MinSUP.minsup (ysup f mf ay) as is02 where
-is02 : {w : Ordinal } →  odef (uchain f mf ay) w → (w ≡ s) ∨ (w << s)
-is02 fc = sup ⟪ A∋fc _ f mf fc , ch-init fc ⟫
-... | case2 lt = ⊥-elim ( ¬x<0  (subst (λ k → z o< k ) x=0 lt ) )
-... | tri> ¬a ¬b 0<x = zc400 usup ssup where
-mf : ≤-monotonic-f A f
-mf x ax = ⟪ case2 mf00 , proj2 (mf< x ax ) ⟫ where
-mf00 : * x < * (f x)
-mf00 = proj1 ( mf< x ax )
-pzc : {z : Ordinal} → z o< x → ZChain A f mf< ay z
-pzc {z} z<x = prev z z<x
-ysp =  MinSUP.sup (ysup f mf ay)
-supfz : {z : Ordinal } → z o< x → Ordinal
-supfz {z} z<x = ZChain.supf (pzc  (ob<x lim z<x)) z
-pchainU : HOD
-pchainU = UnionIC A f ay supfz
-zeq : {xa xb z : Ordinal }
-→ (xa<x : xa o< x) → (xb<x : xb o< x) → xa o≤ xb → z o≤ xa
-→ ZChain.supf (pzc  xa<x) z ≡  ZChain.supf (pzc  xb<x) z
-zeq {xa} {xb} {z} xa<x xb<x xa≤xb z≤xa =  supf-unique A f mf< ay xa≤xb
-(pzc xa<x)  (pzc xb<x)  z≤xa
-iceq : {ix iy : Ordinal } → ix ≡ iy → {i<x : ix o< x } {i<y : iy o< x } → supfz i<x ≡ supfz i<y
-iceq refl = cong supfz  o<-irr
-IChain-i : {z : Ordinal } → IChain ay supfz z → Ordinal
-IChain-i (ic-init fc) = o∅
-IChain-i (ic-isup ia ia<x sa<x fca) = ia
-pic<x : {z : Ordinal } → (ic : IChain ay supfz z ) → osuc (IChain-i ic) o< x
-pic<x {z} (ic-init fc) = ob<x lim 0<x   -- 0<x ∧ lim x → osuc o∅ o< x
-pic<x {z} (ic-isup ia ia<x sa<x fca) = ob<x lim ia<x
-pchainU⊆chain : {z : Ordinal } → (pz : odef pchainU z) → odef (ZChain.chain (pzc (pic<x (proj2 pz)))) z
-pchainU⊆chain {z} ⟪ aw , ic-init fc ⟫ = ⟪ aw , ch-init fc ⟫
-pchainU⊆chain {z} ⟪ aw , (ic-isup ia ia<x sa<x fca) ⟫ = ZChain.cfcs (pzc (ob<x lim ia<x) ) <-osuc o≤-refl uz03 fca where
-uz02 : FClosure A f (ZChain.supf (pzc (ob<x lim ia<x)) ia ) z
-uz02 = fca
-uz03 : ZChain.supf (pzc (ob<x lim ia<x)) ia o≤ ia
-uz03 = sa<x
-chain⊆pchainU : {z w : Ordinal } → (z<x : z o< x) → odef (UnionCF A f ay (ZChain.supf (pzc (ob<x lim z<x))) z) w → odef pchainU w
-chain⊆pchainU {z} {w} z<x ⟪ aw , ch-init fc ⟫ = ⟪ aw , ic-init fc ⟫
-chain⊆pchainU {z} {w} z<x ⟪ aw , ch-is-sup u u<oz su=u fc ⟫
-= ⟪ aw , ic-isup u u<x (o≤-refl0 su≡u) (subst (λ  k → FClosure A f k w ) su=su fc) ⟫ where
-u<x : u o< x
-u<x = ordtrans u<oz z<x
-su=su : ZChain.supf (pzc (ob<x lim z<x)) u ≡ supfz u<x
-su=su = sym ( zeq _ _  (o<→≤ (osucc u<oz)) (o<→≤ <-osuc) )
-su≡u :  supfz u<x ≡ u
-su≡u = begin
-ZChain.supf (pzc (ob<x lim u<x )) u ≡⟨ sym su=su ⟩
-ZChain.supf (pzc (ob<x lim z<x)) u  ≡⟨ su=u ⟩
-u ∎ where open ≡-Reasoning
-IC⊆ : {a b : Ordinal } (ia : IChain ay supfz a ) (ib : IChain ay supfz b )
-→ IChain-i ia o< IChain-i ib → odef (ZChain.chain (pzc (pic<x ib))) a
-IC⊆ {a} {b} (ic-init fc ) ib ia<ib = ⟪ A∋fc _ f mf fc , ch-init fc ⟫
-IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-init fcb ) ia<ib = ⊥-elim ( ¬x<0 ia<ib  )
-IC⊆ {a} {b} (ic-isup i i<x sa<x fc ) (ic-isup j j<x sb<x fcb ) ia<ib
-= ZChain.cfcs (pzc (ob<x lim j<x) ) (o<→≤ ia<ib) o≤-refl (OrdTrans (ZChain.supf-mono (pzc (ob<x lim j<x)) (o<→≤ ia<ib)) sb<x)
-(subst (λ k → FClosure A f k a) (zeq _ _ (osucc (o<→≤ ia<ib)) (o<→≤ <-osuc)) fc )
-ptotalU : IsTotalOrderSet pchainU
-ptotalU {a} {b} ia ib with trio< (IChain-i (proj2 ia)) (IChain-i (proj2 ib))
-... | tri< ia<ib ¬b ¬c = ZChain.f-total (pzc (pic<x (proj2 ib))) (IC⊆ (proj2 ia) (proj2 ib) ia<ib) (pchainU⊆chain ib)
-... | tri≈ ¬a ia=ib ¬c = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso ( pcmp (proj2 ia) (proj2 ib) ia=ib ) where
-pcmp : (ia : IChain ay supfz (& a)) → (ib : IChain ay supfz (& b)) → IChain-i ia ≡ IChain-i ib
-→ Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-pcmp (ic-init fca) (ic-init fcb) eq = fcn-cmp _ f mf fca fcb
-pcmp (ic-init fca) (ic-isup i i<x s<x fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fca
-... | case1 eq1 = ct22 where
-ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-ct22 with subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb )
-... | case1 eq2 =  tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
-ct00 : * (& a) ≡ * (& b)
-ct00 = cong (*) (trans eq1 eq2)
-... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
-fc11 : * (& a) < * (& b)
-fc11 = subst (λ k →  k < * (& b) ) (cong (*) (sym eq1)) lt
-... | case2 lt = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where
-fc11 : * (& a) < * (& b)
-fc11 = ftrans<-≤ lt (subst (λ k → k ≤ & b) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fcb ) )
-pcmp (ic-isup i i<x s<x fca) (ic-init fcb) eq with ZChain.fcy<sup (pzc i<x) o≤-refl fcb
-... | case1 eq1 =  ct22 where
-ct22 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) )
-ct22 with subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca )
-... | case1 eq2 =  tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00  (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where
-ct00 : * (& a) ≡ * (& b)
-ct00 = cong (*) (sym (trans eq1 eq2))
-... | case2 lt = tri> (λ lt → <-irr (case2 fc11) lt) (λ eq → <-irr (case1 eq) fc11) fc11  where
-fc11 : * (& b) < * (& a)
-fc11 = subst (λ k →  k < * (& a) ) (cong (*) (sym eq1)) lt
-... | case2 lt = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12  where
-fc12 : * (& b) < * (& a)
-fc12 = ftrans<-≤ lt (subst (λ k → k ≤ & a) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl )) (s≤fc _ f mf fca ) )
-pcmp (ic-isup i i<x s<x fca) (ic-isup i i<y s<y fcb) refl = fcn-cmp _ f mf fca (subst (λ k → FClosure A f k (& b)) pc01 fcb ) where
-pc01 : supfz i<y ≡ supfz i<x
-pc01 = cong supfz  o<-irr
-... | tri> ¬a ¬b ib<ia = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainU⊆chain ia) (IC⊆ (proj2 ib) (proj2 ia) ib<ia)
-usup : MinSUP A pchainU
-usup = minsupP pchainU (λ ic → proj1 ic ) ptotalU
-spu0 = MinSUP.sup usup
-pchainS : HOD
-pchainS = record { od = record { def = λ z →  (odef A z  ∧ IChain  ay supfz z )
-∨ (FClosure A f spu0 z ∧ (spu0 o< x)) } ; odmax = & A ; <odmax = zc00 } where
-zc00 : {z : Ordinal } → (odef A z ∧ IChain ay supfz z ) ∨ (FClosure A f spu0 z ∧ (spu0 o< x) )→ z o< & A
-zc00 {z} (case1 lt) = z07 lt
-zc00 {z} (case2 fc) = z09 ( A∋fc spu0 f mf (proj1 fc) )
-zc02 : { a b : Ordinal } → odef A a ∧ IChain ay supfz a → FClosure A f spu0 b ∧ ( spu0 o< x) → a ≤ b
-zc02 {a} {b} ca fb = zc05 (proj1 fb) where
-zc05 : {b : Ordinal } → FClosure A f spu0 b → a ≤ b
-zc05 (fsuc b1 fb ) with proj1 ( mf b1 (A∋fc spu0 f mf fb ))
-... | case1 eq = subst (λ k → a ≤ k ) eq (zc05 fb)
-... | case2 lt = ≤-ftrans (zc05 fb) (case2 lt)
-zc05 (init b1 refl) = MinSUP.x≤sup usup ca
-ptotalS : IsTotalOrderSet pchainS
-ptotalS (case1 a) (case1 b) =  ptotalU a b
-ptotalS {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
-lt1 : a0 < b0
-lt1 = subst₂ (λ j k → j < k ) *iso *iso lt
-ptotalS {b0} {a0} (case2 b) (case1 a) with zc02 a b
-... | case1 eq = tri≈ (<-irr (case1 eq1)) (sym eq1) (<-irr (case1 (sym eq1))) where
-eq1 : a0 ≡ b0
-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
-ptotalS (case2 a) (case2 b) = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (fcn-cmp spu0 f mf (proj1 a) (proj1 b))
-S⊆A : pchainS ⊆ A
-S⊆A (case1 lt) = proj1 lt
-S⊆A (case2 fc) = A∋fc _ f mf (proj1 fc)
-ssup : MinSUP A pchainS
-ssup = minsupP pchainS S⊆A ptotalS
-zc400 : MinSUP A pchainU → MinSUP A pchainS → ZChain A f mf< ay x
-zc400 usup ssup = record { supf = supf1 ; asupf = asupf ; zo≤sz = zo≤sz   ; is-minsup = is-minsup ; cfcs = cfcs ; supf-mono = supf-mono  }  where
-spu = MinSUP.sup usup
-sps = MinSUP.sup ssup
-supf1 : Ordinal → Ordinal
-supf1 z with trio< z x
-... | tri< a ¬b ¬c = ZChain.supf (pzc  (ob<x lim a)) z   -- each sup o< x
-... | tri≈ ¬a b ¬c = spu                                 -- sup of all sup o< x
-... | tri> ¬a ¬b c = sps                                 -- sup of spu which o< x
---  if x o< spu, spu is not included in UnionCF x
--- the chain
-pchain : HOD
-pchain = UnionCF A f ay supf1 x
--- pchain ⊆ pchainU ⊆ pchianS
-sf1=sf : {z : Ordinal } → (a : z o< x ) → supf1 z ≡ ZChain.supf (pzc  (ob<x lim a)) z
-sf1=sf {z} z<x with trio< z x
-... | tri< a ¬b ¬c = cong ( λ k → ZChain.supf (pzc (ob<x lim k)) z) o<-irr
-... | tri≈ ¬a b ¬c = ⊥-elim (¬a z<x)
-... | tri> ¬a ¬b c = ⊥-elim (¬a z<x)
-sf1=spu : {z : Ordinal } → x ≡ z → supf1 z ≡ spu
-sf1=spu {z} eq with trio< z x
-... | tri< a ¬b ¬c = ⊥-elim (¬b (sym eq))
-... | tri≈ ¬a b ¬c = refl
-... | tri> ¬a ¬b c = ⊥-elim (¬b (sym eq))
-sf1=sps : {z : Ordinal } → (a : x o< z ) → supf1 z ≡ sps
-sf1=sps {z} x<z with trio< z x
-... | tri< a ¬b ¬c = ⊥-elim (o<> x<z a)
-... | tri≈ ¬a b ¬c = ⊥-elim (¬c x<z )
-... | tri> ¬a ¬b c = refl
-asupf : {z : Ordinal } → odef A (supf1 z)
-asupf {z} with trio< z x
-... | tri< a ¬b ¬c = ZChain.asupf (pzc  (ob<x lim a))
-... | tri≈ ¬a b ¬c = MinSUP.as usup
-... | tri> ¬a ¬b c = MinSUP.as ssup
-supf-mono : {z y : Ordinal } → z o≤ y → supf1 z o≤ supf1 y
-supf-mono {z} {y} z≤y with trio< y x
-... | tri< y<x ¬b ¬c = zc01 where
-open o≤-Reasoning O
-zc01 : supf1 z o≤ ZChain.supf (pzc  (ob<x lim y<x)) y
-zc01 = begin
-supf1 z ≡⟨ sf1=sf (ordtrans≤-< z≤y y<x)  ⟩
-ZChain.supf (pzc  (ob<x lim (ordtrans≤-< z≤y y<x))) z ≡⟨ zeq _ _ (osucc z≤y) (o<→≤ <-osuc)  ⟩
-ZChain.supf (pzc  (ob<x lim y<x)) z ≤⟨ ZChain.supf-mono (pzc  (ob<x lim y<x)) z≤y  ⟩
-ZChain.supf (pzc  (ob<x lim y<x)) y ∎
-... | tri≈ ¬a b ¬c = zc01 where  -- supf1 z o≤ spu
-open o≤-Reasoning O
-zc01 : supf1 z o≤ spu
-zc01 with osuc-≡< (subst (λ k → z o≤ k) b z≤y)
-... | case1 z=x = o≤-refl0 (sf1=spu (sym z=x))
-... | case2 z<x = subst (λ k → k o≤ spu ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) )
-(MinSUP.as usup) (λ uw → MinSUP.x≤sup usup (chain⊆pchainU z<x uw)) )
-... | tri> ¬a ¬b c = zc01 where  -- supf1 z o≤ sps
-zc01 : supf1 z o≤ sps
-zc01 with trio< z x
-... | tri< z<x ¬b ¬c = IsMinSUP.minsup (ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc) )
-(MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 (chain⊆pchainU z<x uw)) )
-... | tri≈ ¬a z=x ¬c = MinSUP.minsup usup (MinSUP.as ssup) (λ uw → MinSUP.x≤sup ssup (case1 uw) )
-... | tri> ¬a ¬b c = o≤-refl -- (sf1=sps c)
-is-minsup :  {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z)
-is-minsup {z} z≤x with osuc-≡< z≤x
-... | case1 z=x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where
-zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z
-zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x))) ( MinSUP.x≤sup usup ⟪ az , ic-init fc ⟫ )
-zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=spu (sym z=x)))
-( MinSUP.x≤sup usup  ⟪ az , ic-isup u u<x (o≤-refl0 zm05) (subst (λ k → FClosure A f k w) (sym zm06) fc)  ⟫  ) where
-u<x : u o< x
-u<x = subst (λ k → u o< k) z=x u<b
-zm06 : supfz (subst (λ k → u o< k) z=x u<b) ≡ supf1 u
-zm06 = trans (zeq _ _  o≤-refl (o<→≤ <-osuc) ) (sym (sf1=sf u<x ))
-zm05 : supfz (subst (λ k → u o< k) z=x u<b) ≡ u
-zm05 = trans zm06 su=u
-zm01 : { s : Ordinal } → odef A s →  ( {x : Ordinal  } → odef (UnionCF A f ay supf1 z) x → x ≤ s )  → supf1 z o≤ s
-zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=spu (sym z=x))) ( MinSUP.minsup usup as zm02 ) where
-zm02 : {w : Ordinal } →  odef pchainU w → w ≤ s
-zm02 {w} uw with pchainU⊆chain uw
-... | ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫
-... | ⟪ az , ch-is-sup u1 u<b su=u fc ⟫ = sup  ⟪ az , ch-is-sup u1 (ordtrans u<b zm05) (trans zm03 su=u) zm04 ⟫  where
-zm05 : osuc (IChain-i (proj2 uw)) o< z
-zm05 = subst (λ k → osuc  (IChain-i (proj2 uw)) o< k) (sym z=x) ( pic<x (proj2 uw) )
-u<x : u1 o< x
-u<x = subst (λ k → u1 o< k) z=x ( ordtrans u<b zm05 )
-zm03 : supf1 u1 ≡ ZChain.supf (prev (osuc (IChain-i (proj2 uw))) (pic<x (proj2 uw))) u1
-zm03 = trans (sf1=sf u<x) (zeq _ _ (osucc u<b) (o<→≤ <-osuc) )
-zm04 : FClosure A f (supf1 u1) w
-zm04 = subst (λ k → FClosure A f k w) (sym zm03) fc
-... | case2 z<x = record { as = asupf ; x≤sup = zm00 ; minsup = zm01 } where
-supf0 = ZChain.supf (pzc (ob<x lim z<x))
-msup : IsMinSUP A (UnionCF A f ay supf0 z) (supf0 z)
-msup = ZChain.is-minsup (pzc (ob<x lim z<x)) (o<→≤ <-osuc)
-s1=0 : {u : Ordinal } → u o< z → supf1 u ≡ supf0 u
-s1=0 {u} u<z = trans (sf1=sf (ordtrans u<z z<x)) (zeq _ _ (o<→≤ (osucc u<z))  (o<→≤ <-osuc) )
-zm00 : {w : Ordinal } → odef (UnionCF A f ay supf1 z) w → w ≤ supf1 z
-zm00 {w} ⟪ az , ch-init fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x)) ( IsMinSUP.x≤sup msup  ⟪ az , ch-init fc ⟫ )
-zm00 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = subst (λ k → w ≤ k ) (sym (sf1=sf z<x))
-( IsMinSUP.x≤sup msup  ⟪ az , ch-is-sup u u<b (trans (sym (s1=0 u<b)) su=u)  (subst (λ k → FClosure A f k w) (s1=0 u<b) fc)  ⟫  )
-zm01 : { s : Ordinal } → odef A s →  ( {x : Ordinal  } → odef (UnionCF A f ay supf1 z) x → x ≤ s )  → supf1 z o≤ s
-zm01 {s} as sup = subst (λ k → k o≤ s ) (sym (sf1=sf z<x)) ( IsMinSUP.minsup msup as zm02 ) where
-zm02 : {w : Ordinal } →  odef (UnionCF A f ay supf0 z) w → w ≤ s
-zm02 {w} ⟪ az , ch-init fc ⟫ = sup ⟪ az , ch-init fc ⟫
-zm02 {w} ⟪ az , ch-is-sup u u<b su=u fc ⟫ = sup
-⟪ az , ch-is-sup u u<b (trans (s1=0 u<b) su=u) (subst (λ k → FClosure A f k w) (sym (s1=0 u<b)) fc) ⟫
-cfcs :  {a b w : Ordinal } → a o< b → b o≤ x →  supf1 a o< b → FClosure A f (supf1 a) w → odef (UnionCF A f ay supf1 b) w
-cfcs {a} {b} {w} a<b b≤x sa<b fc with osuc-≡< b≤x
-... | case1 b=x with trio< a x
-... | tri< a<x ¬b ¬c = zc40 where
-sa = ZChain.supf (pzc  (ob<x lim a<x)) a
-m =  omax a sa     -- x is limit ordinal, so we have sa o< m o< x
-m<x : m o< x
-m<x with trio< a sa | inspect (omax a) sa
-... | tri< a<sa ¬b ¬c | record { eq = eq } = ob<x lim (ordtrans<-≤ sa<b b≤x )
-... | tri≈ ¬a a=sa ¬c | record { eq = eq } = subst (λ k → k o< x) eq zc41 where
-zc41 : omax a sa o< x
-zc41 = osucprev ( begin
-osuc ( omax a sa ) ≡⟨ cong (λ k → osuc (omax a k)) (sym a=sa) ⟩
-osuc ( omax a a ) ≡⟨ cong osuc (omxx _) ⟩
-osuc ( osuc  a ) ≤⟨ o<→≤ (ob<x lim (ob<x lim a<x))  ⟩
-x ∎ ) where open o≤-Reasoning O
-... | tri> ¬a ¬b c | record { eq = eq } = ob<x lim a<x
-sam = ZChain.supf (pzc (ob<x lim m<x)) a
-zc42 : osuc a o≤ osuc m
-zc42 = osucc (o<→≤ ( omax-x _ _ ) )
-sam<m : sam o< m
-sam<m = subst (λ k → k o< m ) (supf-unique A f mf< ay zc42 (pzc (ob<x lim a<x)) (pzc (ob<x lim m<x)) (o<→≤ <-osuc)) ( omax-y _ _ )
-fcm : FClosure A f (ZChain.supf (pzc (ob<x lim m<x)) a) w
-fcm = subst (λ k → FClosure A f k w ) (zeq (ob<x lim a<x) (ob<x lim m<x) zc42 (o<→≤ <-osuc) ) fc
-zcm : odef (UnionCF A f ay (ZChain.supf (pzc  (ob<x lim m<x))) (osuc (omax a sa))) w
-zcm = ZChain.cfcs (pzc  (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm
-zc40 : odef (UnionCF A f ay supf1 b) w
-zc40 with ZChain.cfcs (pzc  (ob<x lim m<x)) (o<→≤ (omax-x _ _)) o≤-refl (o<→≤ sam<m) fcm
-... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<b (trans zc45 su=u) zc44 ⟫ where
-u<b : u o< b
-u<b = osucprev ( begin
-osuc u ≤⟨ osucc u<x ⟩
-osuc m ≤⟨ osucc m<x ⟩
-x ≡⟨ sym b=x ⟩
-b ∎ ) where open o≤-Reasoning O
-zc45 : supf1 u ≡  ZChain.supf (pzc  (ob<x lim m<x)) u
-zc45 = begin
-supf1 u ≡⟨ sf1=sf (subst (λ k → u o< k) b=x u<b )  ⟩
-ZChain.supf (pzc  (ob<x lim (subst (λ k → u o< k) b=x u<b ))) u  ≡⟨ zeq _ _ (osucc u<x) (o<→≤ <-osuc)  ⟩
-ZChain.supf (pzc  (ob<x lim m<x)) u ∎  where open ≡-Reasoning
-zc44 : FClosure A f (supf1 u) w
-zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc
-... | tri≈ ¬a b ¬c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x))
-... | tri> ¬a ¬b c = ⊥-elim ( ¬a (ordtrans<-≤ a<b b≤x))
-cfcs {a} {b} {w} a<b b≤x sa<b fc | case2 b<x = zc40 where
-supfb =  ZChain.supf (pzc (ob<x lim b<x))
-sb=sa : {a : Ordinal } → a o< b → supf1 a ≡ ZChain.supf (pzc (ob<x lim b<x)) a
-sb=sa {a} a<b = trans (sf1=sf (ordtrans<-≤ a<b b≤x)) (zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ <-osuc) )
-fcb : FClosure A f (supfb a) w
-fcb = subst (λ k → FClosure A f k w) (sb=sa a<b) fc
---  supfb a o< b assures it is in Union b
-zcb : odef (UnionCF A f ay supfb b) w
-zcb = ZChain.cfcs (pzc (ob<x lim b<x)) a<b (o<→≤ <-osuc) (subst (λ k → k o< b) (sb=sa a<b) sa<b) fcb
-zc40 : odef (UnionCF A f ay supf1 b) w
-zc40 with zcb
-... | ⟪ az , ch-init fc ⟫ = ⟪ az , ch-init fc ⟫
-... | ⟪ az , ch-is-sup u u<x su=u fc ⟫ = ⟪ az , ch-is-sup u u<x (trans zc45 su=u) zc44  ⟫ where
-zc45 : supf1 u ≡  ZChain.supf (pzc  (ob<x lim b<x)) u
-zc45 = begin
-supf1 u ≡⟨ sf1=sf (ordtrans u<x b<x)  ⟩
-ZChain.supf (pzc  (ob<x lim (ordtrans u<x b<x) )) u  ≡⟨ zeq _ _ (o<→≤ (osucc u<x)) (o<→≤ <-osuc)  ⟩
-ZChain.supf (pzc  (ob<x lim b<x )) u ∎  where open ≡-Reasoning
-zc44 : FClosure A f (supf1 u) w
-zc44 = subst (λ k → FClosure A f k w ) (sym zc45) fc
-zo≤sz : {z : Ordinal} →  z o≤ x → z o≤ supf1 z
-zo≤sz {z} z≤x with osuc-≡< z≤x
-... | case2 z<x = subst (λ k → z o≤ k) (sym (trans (sf1=sf z<x) (sym (zeq _ _ (o<→≤ <-osuc) o≤-refl)))) ( ZChain.zo≤sz (pzc z<x) o≤-refl )
-... | case1 refl with x<y∨y≤x (supf1 spu) x
-... | case2 x≤ssp = z40 where
-z40 : z o≤ supf1 z
-z40 with  x<y∨y≤x z spu
-... | case1 z<spu = o<→≤ ( subst (λ k → z o< k ) (sym (sf1=spu refl)) z<spu )
-... | case2 spu≤z =  begin   -- x ≡ supf1 spu ≡ spu ≡ supf1 x
-x ≤⟨ x≤ssp ⟩
-supf1 spu ≤⟨ supf-mono spu≤z ⟩
-supf1 x ∎   where open o≤-Reasoning O
-... | case1 ssp<x = subst (λ k → x o≤ k) (sym (sf1=spu refl)) z47 where
-z47 : x o≤ spu
-z47 with x<y∨y≤x spu x
-... | case2 lt = lt
-... | case1 spu<x = ⊥-elim ( <<-irr (MinSUP.x≤sup usup z48) (proj1 ( mf< spu (MinSUP.as usup))))  where
-z70 : odef (UnionCF A f ay supf1 z) (supf1 spu)
-z70 = cfcs spu<x o≤-refl ssp<x (init asupf refl )
-z73 : IsSUP A (UnionCF A f ay (ZChain.supf (pzc (ob<x lim spu<x))) spu) spu
-z73 = record { ax = MinSUP.as usup ; x≤sup = λ uw → MinSUP.x≤sup usup (chain⊆pchainU spu<x uw ) }
-z49 : supfz spu<x ≡ spu
-z49 = begin
-supfz spu<x ≡⟨ ZChain.sup=u (pzc (ob<x lim spu<x)) (MinSUP.as usup) (o<→≤ <-osuc) z73 ⟩
-spu ∎ where open ≡-Reasoning
-z50 : supfz spu<x o≤ spu
-z50 = o≤-refl0 z49
-z48 : odef pchainU (f spu)
-z48 = ⟪  proj2 (mf _ (MinSUP.as usup) ) , ic-isup _ (subst (λ k → k o< x) refl spu<x) z50
-(fsuc _ (init (ZChain.asupf (pzc (ob<x lim spu<x))) z49)) ⟫
 ---
 --- 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
+record ZChain ( A : HOD ) {y : Ordinal} (ay : odef A y) (x : Ordinal) : Set (Level.suc n) where
-SZ f mf< {y} ay x = TransFinite {λ z → ZChain A f mf< ay z  } (λ x → ind f mf< ay x   ) x
+field
+chain : HOD
-msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {x y : Ordinal} (ay : odef A y)
+chain⊆A : chain ⊆ A
-→ (zc : ZChain A f mf< ay x )
+f-total  : IsTotalOrderSet chain
-→ MinSUP A (UnionCF A f ay (ZChain.supf zc) x)
+cf : Ordinal → Ordinal
-msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc)
+is-cf : {x : Ordinal} → odef A x → odef A (cf x) ∧ ( * x < * (cf x) )
+f-next  : {x : Ordinal } → odef chain x → odef chain (cf x)
+fixpoint : (sp1 : MinSUP A chain ) → odef chain (MinSUP.sup sp1)
+cf-is-<-monotonic : <-monotonic-f A cf
+cf-is-<-monotonic x ax = ⟪ proj2 (is-cf ax ) , proj1 (is-cf ax ) ⟫
+cf-is-≤-monotonic : ≤-monotonic-f A cf
+cf-is-≤-monotonic x ax = ⟪ case2 (proj1 ( cf-is-<-monotonic x ax  ))  , proj2 ( cf-is-<-monotonic x ax  ) ⟫
+SZ : ¬ Maximal A → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A ay x
+SZ nmx  {y} ay x =  TransFinite {λ z → ZChain A ay z  } (λ x → ind x ) x where
+ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ZChain A ay z) → ZChain A ay x
+ind x prev = ?  -- with Oprev-p x
+--  record {
+--       chain = record { od = record { def = λ x → odef A x ∧ IChain A f x } ; odmax = & A ; <odmax = λ lt → z09 (proj1 lt) } ;
+--       chain⊆A = λ cx → proj1 cx ;
+--       f-total = λ ia ib → subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (f-total (proj2 ia) (proj2 ib))  ;
+--       f-next = λ ix → ⟪ ? , f-next (proj2 ix) ⟫ ;
+--       fixpoint = λ sp1 → ⟪ ? , ? ⟫
+--    } where
+--       f-total : {a b : Ordinal } → IChain A f a → IChain A f b → Tri (a << b) (* a ≡ * b) (b << a)
+--       f-total = ?
+--       f-next : {a : Ordinal } → IChain A f a → IChain A f (f a)
+--       f-next record { y = y ; x=fy = x=fy } = record { y = f y ; x=fy = cong f x=fy }
+msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y)
+→ (zc : ZChain A ay (& A) )
+→ MinSUP A (ZChain.chain zc)
+msp0 f mf< {x} ay zc = minsupP (ZChain.chain zc)  (ZChain.chain⊆A zc) (ZChain.f-total zc)
 -- f eventualy stop
 --    we can prove contradict here, it is here for a historical reason
 --
-fixpoint :  ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f )  (zc : ZChain A f mf< as0 (& A) )
+fixpoint : (zc : ZChain A  as0 (& A))
 → (sp1 : MinSUP A (ZChain.chain zc))
-→ f (MinSUP.sup sp1)  ≡ MinSUP.sup sp1
+→ ZChain.cf zc (MinSUP.sup sp1)  ≡ MinSUP.sup sp1
-fixpoint f mf mf< zc sp1 = z14 where
+fixpoint zc sp1 = z14 where
 chain = ZChain.chain zc
-supf = ZChain.supf zc
 sp : Ordinal
 sp = MinSUP.sup sp1
 asp : odef A sp
 asp = MinSUP.as sp1
-ay = as0
+f = ZChain.cf zc
-z10 :  {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b )
+mf : ≤-monotonic-f A f
-→  HasPrev A chain f b  ∨  IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b
+mf = ZChain.cf-is-≤-monotonic zc
-→ * a < * b  → odef chain b
-z10 = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl )
-z22 : sp o< & A
-z22 = z09 asp
 z12 : odef chain sp
-z12 with o≡? (& s) sp
+z12 = ZChain.fixpoint zc sp1
-... | yes eq = subst (λ k → odef chain k) eq ( ZChain.chain∋init zc )
-... | no ne = ZChain1.is-max (SZ1 f mf mf< as0 zc (& A) o≤-refl) {& s} {sp} ( ZChain.chain∋init zc ) (z09 asp) 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 ay (ZChain.supf zc) sp) sp
-z19 = record { ax = asp ;   x≤sup = z20 }  where
-z20 : {y : Ordinal} → odef (UnionCF A f ay (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.as sp1 ))
 --
 -- 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
 --
-¬Maximal→¬cf-mono :  (nmx : ¬ Maximal A ) → (zc : ZChain A (cf nmx) (cf-is-<-monotonic nmx) as0 (& A)) → ⊥
+¬Maximal→¬cf-mono :  (nmx : ¬ Maximal A ) → (zc : ZChain A as0 (& A)) → ⊥
-¬Maximal→¬cf-mono nmx zc = <-irr0  {* (cf nmx c)} {* c}
+¬Maximal→¬cf-mono nmx zc = <-irr0  {* (ZChain.cf zc c)} {* c}
-(subst (λ k → odef A k ) (sym &iso) (proj1 (is-cf nmx (MinSUP.as  msp1 ))))
+(subst (λ k → odef A k ) (sym &iso) (proj1 (ZChain.is-cf zc (MinSUP.as  msp1 ))))
 (subst (λ k → odef A k) (sym &iso) (MinSUP.as msp1) )
-(case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1  ))) -- x ≡ f x ̄
+(case1 ( cong (*)( fixpoint zc msp1  ))) -- x ≡ f x ̄
-(proj1 (cf-is-<-monotonic nmx c (MinSUP.as msp1 ))) where          -- x < f x
+(proj1 (ZChain.cf-is-<-monotonic zc c (MinSUP.as 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
+msp1 = msp0 (ZChain.cf zc) (ZChain.cf-is-<-monotonic zc) as0 zc
 c : Ordinal
 c = MinSUP.sup msp1
 zorn00 : Maximal A
 zorn00 with is-o∅ ( & HasMaximal )
 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 ( ¬Maximal→¬cf-mono nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as0 (& A) )) where
+... | yes ¬Maximal = ⊥-elim ( ¬Maximal→¬cf-mono nmx (SZ 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) ) ⟫

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

comparison src/zorn.agda @ 1092:87c2da3811c3