Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 09 Dec 2022 11:10:41 +0900 |
| parents | 323e6e6622a2 |
| children | 0b6cee971cba |
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{-# 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 x << y = * x < * y _≤_ : (x y : Ordinal ) → Set n -- we can't use * x ≡ * y, it is Set (Level.suc n). Level (suc n) troubles Chain x ≤ y = (x ≡ y ) ∨ ( * x < * y ) POO : IsStrictPartialOrder _≡_ _<<_ POO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = IsStrictPartialOrder.trans PO ; irrefl = λ x=y x<y → IsStrictPartialOrder.irrefl PO (cong (*) x=y) x<y ; <-resp-≈ = record { fst = λ {x} {y} {y1} y=y1 xy1 → subst (λ k → x << k ) y=y1 xy1 ; snd = λ {x} {x1} {y} x=x1 x1y → subst (λ k → k << x ) x=x1 x1y } } ≤-ftrans : {x y z : Ordinal } → 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 ) ftrans≤-< : {x y z : Ordinal } → x ≤ y → y << z → x << z ftrans≤-< {x} {y} {z} (case1 eq) y<z = subst (λ k → k < * z) (sym (cong (*) eq)) y<z ftrans≤-< {x} {y} {z} (case2 lt) y<z = IsStrictPartialOrder.trans PO lt y<z ftrans<-≤ : {x y z : Ordinal } → x << y → y ≤ z → x << z ftrans<-≤ {x} {y} {z} x<y (case1 eq) = subst (λ k → * x < k ) ((cong (*) eq)) x<y ftrans<-≤ {x} {y} {z} x<y (case2 lt) = IsStrictPartialOrder.trans PO x<y lt <-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ <-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a <-irr {a} {b} (case2 a<b) b<a = IsStrictPartialOrder.irrefl PO refl (IsStrictPartialOrder.trans PO b<a a<b) <<-irr : {a b : Ordinal } → a ≤ b → b << a → ⊥ <<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (cong (*) (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 _⊆_ -- <-TransFinite : {A x : HOD} → {P : HOD → Set n} → x ∈ A -- → ({x : HOD} → A ∋ x → ({y : HOD} → A ∋ y → y < x → P y ) → P x) → P x -- <-TransFinite = ? -- -- Closure of ≤-monotonic function f has total order -- ≤-monotonic-f : (A : HOD) → ( Ordinal → Ordinal ) → Set n ≤-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 ) ⊆-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 masum tree from x -- tree structure -- record HasPrev (A B : HOD) ( f : Ordinal → Ordinal ) (x : Ordinal ) : Set n where field ax : odef A x y : Ordinal ay : odef B y x=fy : x ≡ f y record IsSUP (A B : HOD) (x : Ordinal ) : Set n where field ax : odef A x x≤sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) -- B is Total, use positive record SUP ( A B : HOD ) : Set (Level.suc n) where field sup : HOD isSUP : IsSUP A B (& sup) ax = IsSUP.ax isSUP x≤sup = IsSUP.x≤sup isSUP -- -- -- f (f ( ... (supf y))) f (f ( ... (supf z1))) -- / | / | -- / | / | -- supf y < supf z1 < supf z2 -- o< o< -- -- 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 } 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)) M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } → (supf : Ordinal → Ordinal ) → MinSUP A (UnionCF A f ay supf x) → SUP A (UnionCF A f ay supf x) M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) ; isSUP = record { ax = subst (λ k → odef A k) (sym &iso) (MinSUP.as ms) ; x≤sup = ms00 } } where ms00 : {z : Ordinal} → odef (UnionCF A f ay supf x) z → (z ≡ & (* (MinSUP.sup ms))) ∨ (z << & (* (MinSUP.sup ms))) ms00 {z} uz with MinSUP.x≤sup ms uz ... | case1 eq = case1 (subst (λ k → z ≡ k) (sym &iso) eq) ... | case2 lt = case2 (subst (λ k → * z < k ) (sym *iso) lt ) chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b → odef (UnionCF A f 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 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 order : {a b w : Ordinal } → b o≤ z → supf a o< supf b → FClosure A f (supf a) w → w ≤ supf b asupf : {x : Ordinal } → odef A (supf x) supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → IsMinSUP A (UnionCF A f ay supf x) (supf x) sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z → IsSUP A (UnionCF A f ay supf b) b ∧ (¬ HasPrev A (UnionCF A f ay supf b) f b ) → supf b ≡ b chain : HOD chain = UnionCF A f ay supf z chain⊆A : chain ⊆' A chain⊆A = λ lt → proj1 lt chain∋init : {x : Ordinal } → x o≤ z → odef (UnionCF A f ay supf x) y chain∋init {x} x≤z = ⟪ 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 ) supf<A : {x : Ordinal } → supf x o< & A supf<A = z09 asupf 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) 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 ) IsMinSUP→NotHasPrev : {x sp : Ordinal } → odef A sp → ({y : Ordinal} → odef (UnionCF A f ay supf x) y → (y ≡ sp ) ∨ (y << sp )) → ( {a : Ordinal } → odef A a → a << f a ) → ¬ ( HasPrev A (UnionCF A f ay supf x) f sp ) IsMinSUP→NotHasPrev {x} {sp} asp is-sup <-mono-f hp = ⊥-elim (<<-irr fsp≤sp sp<fsp ) where sp<fsp : sp << f sp sp<fsp = <-mono-f asp pr = HasPrev.y hp im00 : f (f pr) ≤ sp im00 = is-sup ( f-next (f-next (HasPrev.ay hp))) fsp≤sp : f sp ≤ sp fsp≤sp = subst (λ k → f k ≤ sp ) (sym (HasPrev.x=fy hp)) im00 supf-¬hp : {x : Ordinal } → x o≤ z → ( {a : Ordinal } → odef A a → a << f a ) → ¬ ( HasPrev A (UnionCF A f ay supf x) f (supf x) ) supf-¬hp {x} x≤z <-mono hp = IsMinSUP→NotHasPrev asupf (λ {w} uw → (subst (λ k → w ≤ k) (sym (supf-is-minsup x≤z)) ( MinSUP.x≤sup (minsup x≤z) uw) )) <-mono hp 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 } , IsMinSUP→NotHasPrev asupf z54 ( λ ax → proj1 (mf< _ ax)) ⟫ x≤supfx : {x : Ordinal } → x o≤ z → supf x o≤ z → x o≤ supf x x≤supfx {x} x≤z sx≤z with x<y∨y≤x (supf x) x ... | case2 le = le ... | case1 spx<px = ⊥-elim ( <<-irr z45 (proj1 ( mf< (supf x) asupf ))) where z46 : odef (UnionCF A f ay supf x) (f (supf x)) z46 = ⟪ proj2 ( mf (supf x) asupf ) , ch-is-sup (supf x) spx<px z47 (fsuc _ (init asupf z47 )) ⟫ where z47 : supf (supf x) ≡ supf x z47 = supf-idem x≤z sx≤z z45 : f (supf x) ≤ supf x z45 = IsMinSUP.x≤sup (is-minsup x≤z ) z46 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 ) -- cp : (mf< : <-monotonic-f A f) {b : Ordinal } → b o≤ z → supf b o≤ z → ChainP A f supf (supf b) -- the condition of cfcs is satisfied, this is obvious 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 record IChain (A : HOD) ( f : Ordinal → Ordinal ) {x : Ordinal } (supfz : {z : Ordinal } → z o< x → Ordinal) (z : Ordinal ) : Set n where field i : Ordinal i<x : i o< x fc : FClosure A f (supfz i<x) z -- -- 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 Zorn-lemma {A} 0<A supP = zorn00 where <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ <-irr0 {a} {b} A∋a A∋b = <-irr z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) s : HOD s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) as : A ∋ * ( & s ) as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) as0 : odef A (& s ) as0 = subst (λ k → odef A k ) &iso as s<A : & s o< & A s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) HasMaximal : HOD HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) Gtx : { x : HOD} → A ∋ x → HOD Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } z08 : ¬ Maximal A → HasMaximal =h= od∅ z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where ¬x<m : ¬ (* x < * m) ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) -- -- we have minsup using LEM, this is similar to the proof of the axiom of choice -- minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B minsupP B B⊆A total = m02 where xsup : (sup : Ordinal ) → Set n xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) ∀-imply-or : {A : Ordinal → Set n } {B : Set n } → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B ∀-imply-or {A} {B} ∀AB with ODC.p∨¬p O ((x : Ordinal ) → A x) -- LEM ∀-imply-or {A} {B} ∀AB | case1 t = case1 t ∀-imply-or {A} {B} ∀AB | case2 x = case2 (lemma (λ not → x not )) where lemma : ¬ ((x : Ordinal ) → A x) → B lemma not with ODC.p∨¬p O B lemma not | case1 b = b lemma not | case2 ¬b = ⊥-elim (not (λ x → dont-orb (∀AB x) ¬b )) m00 : (x : Ordinal ) → ( ( z : Ordinal) → z o< x → ¬ (odef A z ∧ xsup z) ) ∨ MinSUP A B m00 x = TransFinite0 ind x where ind : (x : Ordinal) → ((z : Ordinal) → z o< x → ( ( w : Ordinal) → w o< z → ¬ (odef A w ∧ xsup w )) ∨ MinSUP A B) → ( ( w : Ordinal) → w o< x → ¬ (odef A w ∧ xsup w) ) ∨ MinSUP A B ind x prev = ∀-imply-or m01 where m01 : (z : Ordinal) → (z o< x → ¬ (odef A z ∧ xsup z)) ∨ MinSUP A B m01 z with trio< z x ... | tri≈ ¬a b ¬c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) ... | tri> ¬a ¬b c = case1 ( λ lt → ⊥-elim ( ¬a lt ) ) ... | tri< a ¬b ¬c with prev z a ... | case2 mins = case2 mins ... | case1 not with ODC.p∨¬p O (odef A z ∧ xsup z) ... | case1 mins = case2 record { sup = z ; isMinSUP = record { as = proj1 mins ; x≤sup = proj2 mins ; minsup = m04 } } where m04 : {sup1 : Ordinal} → odef A sup1 → ({w : Ordinal} → odef B w → (w ≡ sup1) ∨ (w << sup1)) → z o≤ sup1 m04 {s} as lt with trio< z s ... | tri< a ¬b ¬c = o<→≤ a ... | tri≈ ¬a b ¬c = o≤-refl0 b ... | tri> ¬a ¬b s<z = ⊥-elim ( not s s<z ⟪ as , lt ⟫ ) ... | case2 notz = case1 (λ _ → notz ) m03 : ¬ ((z : Ordinal) → z o< & A → ¬ odef A z ∧ xsup z) m03 not = ⊥-elim ( not s1 (z09 (SUP.ax S)) ⟪ SUP.ax S , m05 ⟫ ) where S : SUP A B S = supP B B⊆A total s1 = & (SUP.sup S) m05 : {w : Ordinal } → odef B w → (w ≡ s1 ) ∨ (w << s1 ) m05 {w} bw with SUP.x≤sup S bw ... | 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 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 ) ⟫ -- -- 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 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 } , m04 ⟫ 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 } , m04 ⟫ 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 (ordtrans b<x x≤A) ) ... | 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)) 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<→≤ m09) ⟪ record { ax = ab ; x≤sup = λ lt → IsSUP.x≤sup (proj2 is-sup) lt } , m04 ⟫ -- ZChain on x 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 } , m04 ⟫ 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) SUP⊆ : { B C : HOD } → B ⊆' C → SUP A C → SUP A B SUP⊆ {B} {C} B⊆C sup = record { sup = SUP.sup sup ; isSUP = record { ax = SUP.ax sup ; x≤sup = λ lt → SUP.x≤sup sup (B⊆C lt) } } zc43 : (x sp1 : Ordinal ) → ( x o< sp1 ) ∨ ( sp1 o≤ x ) zc43 x sp1 with trio< x sp1 ... | tri< a ¬b ¬c = case1 a ... | tri≈ ¬a b ¬c = case2 (o≤-refl0 (sym b)) ... | tri> ¬a ¬b c = case2 (o<→≤ c) -- -- create all ZChains under o< x -- ind : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ((z : Ordinal) → z o< x → ZChain A f mf< ay z) → ZChain A f mf< ay x ind f mf< {y} ay x prev with Oprev-p x ... | yes op = zc41 where -- -- we have previous ordinal to use induction -- px = Oprev.oprev op zc : ZChain A f mf< ay (Oprev.oprev op) zc = prev px (subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc ) px<x : px o< x px<x = subst (λ k → px o< k) (Oprev.oprev=x op) <-osuc opx=x : osuc px ≡ x opx=x = Oprev.oprev=x op zc-b<x : (b : Ordinal ) → b o< x → b o< osuc px zc-b<x b lt = subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) lt supf0 = ZChain.supf zc pchain : HOD pchain = UnionCF A f ay supf0 px supf-mono : {a b : Ordinal } → a o≤ b → supf0 a o≤ supf0 b supf-mono = ZChain.supf-mono zc zc04 : {b : Ordinal} → b o≤ x → (b o≤ px ) ∨ (b ≡ x ) zc04 {b} b≤x with trio< b px ... | tri< a ¬b ¬c = case1 (o<→≤ a) ... | tri≈ ¬a b ¬c = case1 (o≤-refl0 b) ... | tri> ¬a ¬b px<b with osuc-≡< b≤x ... | case1 eq = case2 eq ... | case2 b<x = ⊥-elim ( ¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) 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 sp1 = MinSUP.sup sup1 -- -- supf0 px o≤ sp1 -- sfpx≤sp1 : supf0 px o< x → supf0 px ≤ sp1 sfpx≤sp1 spx<x = MinSUP.x≤sup sup1 (case2 ⟪ init (ZChain.asupf zc {px}) refl , spx<x ⟫ ) m13 : supf0 px o< x → supf0 px o≤ sp1 m13 spx<x = IsMinSUP.minsup (ZChain.is-minsup zc o≤-refl ) (MinSUP.as sup1) m14 where m14 : {z : Ordinal} → odef (UnionCF A f ay (ZChain.supf zc) px) z → (z ≡ sp1) ∨ (z << sp1) m14 {z} uz = MinSUP.x≤sup sup1 (case1 uz) zc41 : ZChain A f mf< ay x zc41 = record { supf = supf1 ; sup=u = sup=u ; asupf = asupf1 ; supf-mono = supf1-mono ; order = order ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs } where 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 sf≤ : { z : Ordinal } → x o≤ z → supf0 x o≤ supf1 z sf≤ {z} x≤z with trio< z px ... | tri< a ¬b ¬c = ⊥-elim ( o<> (osucc a) (subst (λ k → k o≤ z) (sym (Oprev.oprev=x op)) x≤z ) ) ... | tri≈ ¬a b ¬c = ⊥-elim ( o≤> x≤z (subst (λ k → k o< x ) (sym b) px<x )) ... | tri> ¬a ¬b c = subst₂ (λ j k → j o≤ k ) (trans (sf1=sf0 o≤-refl ) (sym (ZChain.supfmax zc px<x))) (sf1=sp1 c) (supf1-mono (o<→≤ c )) -- px o<z → supf x ≡ supf0 px ≡ supf1 px o≤ supf1 z 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 zc43 (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 zc43 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 sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsSUP A (UnionCF A f ay supf1 b) b ∧ (¬ HasPrev A (UnionCF A f ay supf1 b) f b ) → supf1 b ≡ b sup=u {b} ab b≤x is-sup with trio< b px ... | tri< a ¬b ¬c = ZChain.sup=u zc ab (o<→≤ a) ⟪ record { ax = ? ; x≤sup = ? } , ? ⟫ ... | tri≈ ¬a b=px ¬c = ZChain.sup=u zc ab (o≤-refl0 b=px) ⟪ record { ax = ? ; x≤sup = ? }, ? ⟫ ... | tri> ¬a ¬b px<b = trans zc36 x=b where x=b : x ≡ b x=b with osuc-≡< b≤x ... | case1 eq = sym (eq) ... | case2 b<x = ⊥-elim (¬p<x<op ⟪ px<b , subst (λ k → b o< k ) (sym (Oprev.oprev=x op)) b<x ⟫ ) zc31 : sp1 o≤ b zc31 = MinSUP.minsup sup1 ab zc32 where zc32 : {w : Ordinal } → odef pchainpx w → (w ≡ b) ∨ (w << b) zc32 {w} (case1 ⟪ ua , ch-init fc ⟫ ) = IsSUP.x≤sup (proj1 is-sup) ⟪ ua , ch-init fc ⟫ zc32 {w} (case1 ⟪ ua , ch-is-sup u u<x su=u fc ⟫ ) = IsSUP.x≤sup (proj1 is-sup) ⟪ ua , ch-is-sup u z01 z02 z03 ⟫ where z01 : u o< b z01 = ordtrans u<x (subst (λ k → px o< k ) x=b px<x ) z02 : supf1 u ≡ u z02 = trans (sf1=sf0 (o<→≤ u<x)) su=u z03 : FClosure A f (supf1 u) w z03 = fcpu fc (o<→≤ u<x) zc32 {w} (case2 fc) = IsSUP.x≤sup (proj1 is-sup) ⟪ A∋fc _ f mf (proj1 fc) , ch-is-sup (supf0 px) sa<x su=u fc1 ⟫ where su=u : supf1 (supf0 px) ≡ supf0 px su=u = trans (sf1=sf0 (zc-b<x _ (proj2 fc))) ( ZChain.supf-idem zc o≤-refl (zc-b<x _ (proj2 fc)) ) fc1 : FClosure A f (supf1 (supf0 px)) w fc1 = subst (λ k → FClosure A f k w ) (sym su=u) (proj1 fc) sa<x : supf0 px o< b sa<x = subst (λ k → supf0 px o< k ) x=b ( proj2 fc) zc36 : sp1 ≡ x zc36 with osuc-≡< zc31 ... | case1 eq = trans eq (sym x=b) ... | case2 sp1<b = ⊥-elim ( <<-irr zc40 (proj1 ( mf< (supf0 px) (ZChain.asupf zc))) ) where -- sp1 o< x → ⊥ -- supf0 px o≤ sp1 o< x → supf0 px o≤ px -- px o≤ supf0 px → px ≡ spuf0 px o≤ spuf1 x o< x -- px ≡ supf1 x sp1<x : sp1 o< x sp1<x = subst (λ k → sp1 o< k ) (sym x=b) sp1<b zc38 : supf0 px o≤ px zc38 = begin supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩ supf1 x ≡⟨ sf1=sp1 px<x ⟩ sp1 ≤⟨ zc-b<x _ sp1<x ⟩ px ∎ where open o≤-Reasoning O zc37 : supf0 px ≡ px zc37 with trio< (supf0 px) px ... | tri< a ¬b ¬c = ⊥-elim ( o≤> (ZChain.x≤supfx zc o≤-refl zc38) a ) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim ( o≤> zc38 c ) zc44 : sp1 o≤ supf0 px zc44 = begin sp1 ≤⟨ zc-b<x _ sp1<x ⟩ px ≡⟨ sym zc37 ⟩ supf0 px ∎ where open o≤-Reasoning O zc45 : supf0 px o≤ sp1 zc45 = begin supf0 px ≡⟨ sym (sf1=sf0 o≤-refl) ⟩ supf1 px ≤⟨ supf1-mono (o<→≤ px<x) ⟩ supf1 x ≡⟨ sf1=sp1 px<x ⟩ sp1 ∎ where open o≤-Reasoning O zc39 : supf0 px ≡ sp1 zc39 with trio< (supf0 px) sp1 ... | tri< a ¬b ¬c = ⊥-elim ( o≤> zc44 a ) --- sp1 o< x ≡ osuc px ≡ osuc (supf0 px) ... | tri≈ ¬a b ¬c = b ... | tri> ¬a ¬b c = ⊥-elim ( o≤> zc45 c ) --- supf0 px o≤ supf1 x ≡ sp1 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 ⟫ )) order : {a b : Ordinal} {w : Ordinal} → b o≤ x → supf1 a o< supf1 b → FClosure A f (supf1 a) w → w ≤ supf1 b order {a} {b} {w} b≤x sa<sb fc = z20 where a<b : a o< b a<b = supf-inject0 supf1-mono sa<sb a≤px : a o≤ px a≤px with trio< a px ... | tri< a ¬b ¬c = o<→≤ a ... | tri≈ ¬a b ¬c = o≤-refl0 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 spx<x : supf0 px o< x spx<x = ? spx<=sb : supf0 px ≤ sp1 spx<=sb = MinSUP.x≤sup sup1 (case2 ⟪ init (ZChain.asupf zc) refl , ? ⟫ ) sa<<sb : supf1 a << supf1 b sa<<sb with osuc-≡< b≤x ... | case2 b<x = subst₂ (λ j k → j << k ) (sym (sf1=sf0 ?)) (sym (sf1=sf0 ?)) ( ZChain.supf-mono< zc (zc-b<x _ b<x) (subst₂ (λ j k → j o< k) (sf1=sf0 ?) (sf1=sf0 ?) sa<sb )) ... | case1 b=x with osuc-≡< ( supf1-mono a≤px ) -- supf1 a ≤ supf1 px << sp1 ... | case2 sa<spx = subst₂ (λ j k → j << k ) (sym (sf1=sf0 ?)) (sym (sf1=sp1 ?)) ( ftrans<-≤ ( ZChain.supf-mono< zc o≤-refl (subst₂ (λ j k → j o< k) (sf1=sf0 ?) (sf1=sf0 ?) sa<spx)) spx<=sb ) ... | case1 sa=spx with spx<=sb ... | case2 lt = subst₂ (λ j k → j << k ) ? ? lt ... | case1 eq = ? -- supf1 a o≤ x -- x o< supf1 a o< supf1 b -> UnionCF px ⊆ UnionCF a → supf0 px ≡ supf0 a → ⊥ -- a o≤ supf1 a sa<x : supf1 a o< x -- supf1 a o< supf1 x ≡ sp1 ( supf of fc (supf0 px) ∧ (supf0 px o< x) sa<x with x<y∨y≤x (supf1 a) x ... | case1 lt = lt ... | case2 x≤sa = ⊥-elim ( <<-irr z27 sa<<sb ) where z27 : supf1 b ≤ supf1 a z27 = subst (λ k → supf1 b ≤ k ) ? (IsMinSUP.x≤sup (is-minsup ? ) ? ) ssa=sa : supf1 a ≡ supf1 (supf1 a) -- supf0 a o≤ px ssa=sa = sym ( sup=u ? ? ? ) sa<b : supf1 a o< b -- supf1 (supf1 a) ≡ supf1 a o< supf1 b → inject supf1 a o< b sa<b = supf-inject0 supf1-mono (subst (λ k → k o< supf1 b ) ? sa<sb ) z20 : w ≤ supf1 b z20 with trio< b px ... | tri< b<px ¬b ¬c = ZChain.order zc (o<→≤ b<px) (subst (λ k → k o< supf0 b) (sf1=sf0 (o<→≤ (ordtrans a<b b<px))) sa<sb) (fcup fc (o<→≤ (ordtrans a<b b<px))) ... | tri≈ ¬a b=px ¬c = IsMinSUP.x≤sup (ZChain.is-minsup zc (o≤-refl0 b=px)) z26 where -- sa<b : supf1 a o< b -- px -- sa<b = ? z26 : odef ( UnionCF A f ay supf0 b ) w z26 with cfcs a<b b≤x sa<b fc ... | ⟪ ua , ch-init fc ⟫ = ⟪ ua , ch-init fc ⟫ ... | ⟪ ua , ch-is-sup u u<x su=u fc ⟫ = ⟪ ua , ch-is-sup u ? ? ? ⟫ ... | tri> ¬a ¬b px<b = MinSUP.x≤sup sup1 (zc11 ( chain-mono f mf ay supf1 supf1-mono b≤x (cfcs a<b b≤x sa<b fc))) -- sa<b : supf1 a o< b -- b ≡ x -- sa<b = ? ... | no lim with trio< x o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a b ¬c = record { supf = ? ; sup=u = ? ; asupf = ? ; supf-mono = ? ; order = ? ; supfmax = ? ; is-minsup = ? ; cfcs = ? } ... | tri> ¬a ¬b 0<x = record { supf = supf1 ; sup=u = ? ; asupf = ? ; supf-mono = supf-mono ; order = ? ; supfmax = ? ; is-minsup = ? ; cfcs = cfcs } where -- mf : ≤-monotonic-f A f -- mf x ax = ? -- ⟪ case2 ( proj1 (mf< x ax)) , proj2 (mf< x ax ) ⟫ will result memory exhaust 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 pchainx : HOD pchainx = record { od = record { def = λ z → IChain A f supfz z } ; odmax = & A ; <odmax = zc00 } where zc00 : {z : Ordinal } → IChain A f supfz z → z o< & A zc00 {z} ic = z09 ( A∋fc (supfz (IChain.i<x ic)) f mf (IChain.fc ic) ) aic : {z : Ordinal } → IChain A f supfz z → odef A z aic {z} ic = A∋fc _ f mf (IChain.fc ic ) 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 ifc≤ : {za zb : Ordinal } ( ia : IChain A f supfz za ) ( ib : IChain A f supfz zb ) → (ia≤ib : IChain.i ia o≤ IChain.i ib ) → FClosure A f (ZChain.supf (pzc (ob<x lim (IChain.i<x ib))) (IChain.i ia)) za ifc≤ {za} {zb} ia ib ia≤ib = subst (λ k → FClosure A f k _ ) (zeq _ _ (osucc ia≤ib) (o<→≤ <-osuc) ) (IChain.fc ia) ptotalx : IsTotalOrderSet pchainx ptotalx {a} {b} ia ib = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) uz01 with trio< (IChain.i ia) (IChain.i ib) ... | tri< ia<ib ¬b ¬c with ≤-ftrans (ZChain.order (pzc (ob<x lim (IChain.i<x ib))) ? ? (ifc≤ ia ib (o<→≤ ia<ib))) (s≤fc _ f mf (IChain.fc ib)) ... | 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 lt = tri< lt (λ eq → <-irr (case1 (sym eq)) lt) (λ lt1 → <-irr (case2 lt) lt1) uz01 | tri≈ ¬a ia=ib ¬c = fcn-cmp _ f mf (IChain.fc ia) (subst (λ k → FClosure A f k (& b)) (sym (iceq ia=ib)) (IChain.fc ib)) uz01 | tri> ¬a ¬b ib<ia with ≤-ftrans (ZChain.order (pzc (ob<x lim (IChain.i<x ia))) ? ? (ifc≤ ib ia (o<→≤ ib<ia))) (s≤fc _ f mf (IChain.fc ia)) ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where ct00 : * (& a) ≡ * (& b) ct00 = sym (cong (*) eq1) ... | case2 lt = tri> (λ lt1 → <-irr (case2 lt) lt1) (λ eq → <-irr (case1 eq) lt) lt usup : MinSUP A pchainx usup = minsupP pchainx (λ ic → A∋fc _ f mf (IChain.fc ic ) ) ptotalx spu = MinSUP.sup usup supf1 : Ordinal → Ordinal supf1 z with trio< z x ... | tri< a ¬b ¬c = ZChain.supf (pzc (ob<x lim a)) z ... | tri≈ ¬a b ¬c = spu ... | tri> ¬a ¬b c = spu pchain : HOD pchain = UnionCF A f ay supf1 x -- pchain ⊆ pchainx 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 supf1 ( (proj2 ca)) ( (proj2 cb)) 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 } → (a : x o≤ z ) → supf1 z ≡ spu sf1=spu {z} x≤z with trio< z x ... | tri< a ¬b ¬c = ⊥-elim (o≤> x≤z a) ... | tri≈ ¬a b ¬c = refl ... | tri> ¬a ¬b c = refl zc11 : {z : Ordinal } → odef pchain z → odef pchainx z zc11 {z} lt = ? sfpx≤spu : {z : Ordinal } → supf1 z ≤ spu sfpx≤spu {z} with trio< z x ... | tri< a ¬b ¬c = MinSUP.x≤sup usup ? -- (init (ZChain.asupf (pzc (ob<x lim a)) ) refl ) ... | tri≈ ¬a b ¬c = case1 refl ... | tri> ¬a ¬b c = case1 refl supf-mono : {x y : Ordinal } → x o≤ y → supf1 x o≤ supf1 y supf-mono {x} {y} x≤y with trio< y x ... | tri< a ¬b ¬c = ? ... | tri≈ ¬a b ¬c = ? ... | tri> ¬a ¬b c = ? is-minsup : {z : Ordinal} → z o≤ x → IsMinSUP A (UnionCF A f ay supf1 z) (supf1 z) is-minsup = ? 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 , cp ⟫ = ⟪ az , ? ⟫ ... | 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 , cp ⟫ = ⟪ az , ? ⟫ sup=u : {b : Ordinal} (ab : odef A b) → b o≤ x → IsMinSUP A (UnionCF A f ay supf1 b) b ∧ (¬ HasPrev A (UnionCF A f ay supf1 b) f b ) → supf1 b ≡ b sup=u {b} ab b≤x is-sup with osuc-≡< b≤x ... | case1 b=x = ? where zc31 : spu o≤ b zc31 = MinSUP.minsup usup ab zc32 where zc32 : {w : Ordinal } → odef pchainx w → (w ≡ b) ∨ (w << b) zc32 = ? ... | case2 b<x = trans (sf1=sf ?) (ZChain.sup=u (pzc (ob<x lim b<x)) ab ? ? ) --- --- the maximum chain has fix point of any ≤-monotonic function --- SZ : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) → {y : Ordinal} (ay : odef A y) → (x : Ordinal) → ZChain A f mf< ay x SZ f mf< {y} ay x = TransFinite {λ z → ZChain A f mf< ay z } (λ x → ind f mf< ay x ) x msp0 : ( f : Ordinal → Ordinal ) → (mf< : <-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) → (zc : ZChain A f mf< ay x ) → MinSUP A (UnionCF A f ay (ZChain.supf zc) x) msp0 f mf< {x} ay zc = minsupP (UnionCF A f ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) (ZChain.f-total zc) fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (mf< : <-monotonic-f A f ) (zc : ZChain A f mf< as0 (& A) ) → (sp1 : MinSUP A (ZChain.chain zc)) → f (MinSUP.sup sp1) ≡ MinSUP.sup sp1 fixpoint f mf mf< 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 z10 : {a b : Ordinal } → (ca : odef chain a ) → b o< (& A) → (ab : odef A b ) → HasPrev A chain f b ∨ IsSUP A (UnionCF A f ay (ZChain.supf zc) b) b → * 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 ... | 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 = ? ; 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 )) ... | case1 eq = ⊥-elim (¬b (sym (cong (*) eq ) )) ... | case2 lt = ⊥-elim (¬c lt ) ... | tri≈ ¬a b ¬c = subst₂ (λ j k → j ≡ k ) &iso &iso ( cong (&) b ) ... | tri> ¬a ¬b c = ⊥-elim z17 where z15 : (f sp ≡ MinSUP.sup sp1) ∨ (* (f sp) < * (MinSUP.sup sp1) ) z15 = MinSUP.x≤sup sp1 (ZChain.f-next zc z12 ) z17 : ⊥ z17 with z15 ... | case1 eq = ¬b (cong (*) 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)) → ⊥ z04 nmx zc = <-irr0 {* (cf nmx 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) (MinSUP.as msp1) ) (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) (cf-is-<-monotonic nmx ) zc msp1 ))) -- x ≡ f x ̄ (proj1 (cf-is-<-monotonic nmx 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 c : Ordinal c = MinSUP.sup msp1 zorn00 : Maximal A zorn00 with is-o∅ ( & HasMaximal ) -- we have no Level (suc n) LEM ... | no not = record { maximal = ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ; as = zorn01 ; ¬maximal<x = zorn02 } where -- yes we have the maximal zorn03 : odef HasMaximal ( & ( ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) ) ) zorn03 = ODC.x∋minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) -- Axiom of choice zorn01 : A ∋ ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) zorn01 = proj1 zorn03 zorn02 : {x : HOD} → A ∋ x → ¬ (ODC.minimal O HasMaximal (λ eq → not (=od∅→≡o∅ eq)) < x) zorn02 {x} ax m<x = proj2 zorn03 (& x) ax (subst₂ (λ j k → j < k) (sym *iso) (sym *iso) m<x ) ... | yes ¬Maximal = ⊥-elim ( z04 nmx (SZ (cf nmx) (cf-is-<-monotonic nmx) as0 (& A) )) where -- if we have no maximal, make ZChain, which contradict SUP condition nmx : ¬ Maximal A nmx mx = ∅< {HasMaximal} zc5 ( ≡o∅→=od∅ ¬Maximal ) where zc5 : odef A (& (Maximal.maximal mx)) ∧ (( y : Ordinal ) → odef A y → ¬ (* (& (Maximal.maximal mx)) < * y)) zc5 = ⟪ Maximal.as mx , (λ y ay mx<y → Maximal.¬maximal<x mx (subst (λ k → odef A k ) (sym &iso) ay) (subst (λ k → k < * y) *iso mx<y) ) ⟫ -- usage (see filter.agda ) -- -- _⊆'_ : ( A B : HOD ) → Set n -- _⊆'_ A B = (x : Ordinal ) → odef A x → odef B x -- MaximumSubset : {L P : HOD} -- → o∅ o< & L → o∅ o< & P → P ⊆ L -- → IsPartialOrderSet P _⊆'_ -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) -- → Maximal P (_⊆'_) -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP