Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 930:0e0608b1953b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 23 Oct 2022 23:29:30 +0900 |
| parents | a6d97e6e5309 |
| children | 307ad8807963 |
| files | src/zorn.agda src/zorn1.agda |
| diffstat | 2 files changed, 790 insertions(+), 13 deletions(-) [+] |
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--- a/src/zorn.agda Sat Oct 22 18:26:16 2022 +0900 +++ b/src/zorn.agda Sun Oct 23 23:29:30 2022 +0900 @@ -1427,35 +1427,46 @@ msup = ZChain.minsup zc (o<→≤ d<A) sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) + -- z26 : {x : Ordinal } → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) x + -- → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) c) x ∨ odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x + -- z26 = ? is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd) is-sup = record { x<sup = z22 } where - -- z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) z23 lt = MinSUP.x<sup spd lt z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) - z22 {y} zy = ? - -- with MinSUP.x<sup spd ? -- (subst (λ k → odef _ k ) (sym &iso) zy) - -- ... | case1 y=p = case1 (subst (λ k → k ≡ _ ) &iso y=p ) - -- ... | case2 y<p = case2 (subst (λ k → * y < k ) (sym *iso) y<p ) + z22 {a} ⟪ aa , ch-init fc ⟫ = ? + z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ? + -- u<x : ZChain.supf zc u o< ZChain.supf zc d + -- supf u o< spuf c → order not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx) not-hasprev hp = ? where y : Ordinal y = HasPrev.y hp z24 : y << d z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) )) + -- z26 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ d ) ∨ (x << d ) + -- z26 lt with MinSUP.x<sup spd (subst (λ k → odef _ k ) ? lt) + -- ... | case1 eq = ? + -- ... | case2 lt = ? + -- z25 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ y ) ∨ (x << y ) + -- z25 {x} (init au eq ) = ? -- sup c = x, cf y ≡ d, sup c =< d + -- z25 (fsuc x lt) = ? -- cf (sup c) sd=d : supf d ≡ d sd=d = ZChain.sup=u zc (MinSUP.asm spd) (o<→≤ d<A) ⟪ is-sup , not-hasprev ⟫ - sc<d : supf c << d - sc<d = ? where - z21 = proj1 ( cf-is-<-monotonic nmx ? ? ) - sco<d : supf c o< supf d - sco<d = ? + sc<sd : supf c << supf d + sc<sd = ? + -- z21 = proj1 ( cf-is-<-monotonic nmx ? ? ) + -- sco<d : supf c o< supf d + -- sco<d with osuc-≡< ( ZChain.supf-<= zc (case2 sc<sd ) ) + -- ... | case1 eq = ⊥-elim ( <-irr eq sc<sd ) + -- ... | case2 lt = lt ss<sa : supf c o< supf (& A) - ss<sa with osuc-≡< ( ZChain.supf-mono zc ? ) - ... | case2 lt = lt - ... | case1 eq = ? -- where + ss<sa = ? -- with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ SUP.as sp0 , lift true ⟫) )) + -- ... | case2 lt = lt + -- ... | case1 eq = ? -- where 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
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/zorn1.agda Sun Oct 23 23:29:30 2022 +0900 @@ -0,0 +1,766 @@ +{-# 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 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 +-- → 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 } } + +_≤_ : (x y : HOD) → Set (Level.suc n) +x ≤ y = ( x ≡ y ) ∨ ( x < y ) + +≤-ftrans : {x y z : HOD} → x ≤ y → y ≤ z → x ≤ z +≤-ftrans {x} {y} {z} (case1 refl ) (case1 refl ) = case1 refl +≤-ftrans {x} {y} {z} (case1 refl ) (case2 y<z) = case2 y<z +≤-ftrans {x} {_} {z} (case2 x<y ) (case1 refl ) = case2 x<y +≤-ftrans {x} {y} {z} (case2 x<y) (case2 y<z) = case2 ( IsStrictPartialOrder.trans PO x<y y<z ) + +<-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 ) + +<=to≤ : {x y : Ordinal } → x <= y → * x ≤ * y +<=to≤ (case1 eq) = case1 (cong (*) eq) +<=to≤ (case2 lt) = case2 lt + +≤to<= : {x y : Ordinal } → * x ≤ * y → x <= y +≤to<= (case1 eq) = case1 ( subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) eq) ) +≤to<= (case2 lt) = case2 lt + +<-irr : {a b : HOD} → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ +<-irr {a} {b} (case1 a=b) b<a = IsStrictPartialOrder.irrefl PO (sym a=b) b<a +<-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 (Level.suc n) +≤-monotonic-f A f = (x : Ordinal ) → odef A x → ( * x ≤ * (f x) ) ∧ odef A (f x ) + +data FClosure (A : HOD) (f : Ordinal → Ordinal ) (s : Ordinal) : Ordinal → Set n where + init : {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 = subst (λ k → * s ≤ * k ) (*≡*→≡ x=fx) ( s≤fc {A} s f mf fcy ) +... | case2 x<fx with s≤fc {A} s f mf fcy +... | case1 s≡x = case2 ( subst₂ (λ j k → j < k ) (sym s≡x) refl x<fx ) +... | case2 s<x = case2 ( IsStrictPartialOrder.trans PO s<x x<fx ) + +fcn : {A : HOD} (s : Ordinal) { x : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) → FClosure A f s x → ℕ +fcn s mf (init as refl) = zero +fcn {A} s {x} {f} mf (fsuc y p) with proj1 (mf y (A∋fc s f mf p)) +... | case1 eq = fcn s mf p +... | case2 y<fy = suc (fcn s mf p ) + +fcn-inject : {A : HOD} (s : Ordinal) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) + → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx ≡ fcn s mf cy → * x ≡ * y +fcn-inject {A} s {x} {y} {f} mf cx cy eq = fc00 (fcn s mf cx) (fcn s mf cy) eq cx cy refl refl where + fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) + fc06 {x} {y} refl {j} not = fc08 not where + fc08 : {j : ℕ} → ¬ suc j ≡ 0 + fc08 () + fc07 : {x : Ordinal } (cx : FClosure A f s x ) → 0 ≡ fcn s mf cx → * s ≡ * x + fc07 {x} (init as refl) eq = refl + fc07 {.(f x)} (fsuc x cx) eq with proj1 (mf x (A∋fc s f mf cx ) ) + ... | case1 x=fx = subst (λ k → * s ≡ k ) x=fx ( fc07 cx eq ) + -- ... | 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 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 ) eq + ... | case2 lt = subst₂ (λ j k → * (f j) ≡ * (f k )) &iso &iso ( cong (λ k → * ( f (& k ))) fc05) where + fc05 : * x ≡ * y1 + fc05 = fc00 i j (cong pred i=j) cx cy1 (cong pred i=x) (cong pred j=y1) + ... | case2 x₁ | case2 x₂ = subst₂ (λ j k → * (f j) ≡ * (f k) ) &iso &iso (cong (λ k → * (f (& k))) (fc00 i j (cong pred i=j) cx cy (cong pred i=x) (cong pred j=y))) + + +fcn-< : {A : HOD} (s : Ordinal ) { x y : Ordinal} {f : Ordinal → Ordinal} → (mf : ≤-monotonic-f A f) + → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → fcn s mf cx Data.Nat.< fcn s mf cy → * x < * y +fcn-< {A} s {x} {y} {f} mf cx cy x<y = fc01 (fcn s mf cy) cx cy refl x<y where + fc06 : {y : Ordinal } { as : odef A s } (eq : s ≡ y ) { j : ℕ } → ¬ suc j ≡ fcn {A} s {y} {f} mf (init as eq ) + fc06 {x} {y} refl {j} not = fc08 not where + fc08 : {j : ℕ} → ¬ suc j ≡ 0 + fc08 () + fc01 : (i : ℕ ) → {y : Ordinal } → (cx : FClosure A f s x ) (cy : FClosure A f s y ) → (i ≡ fcn s mf cy ) → fcn s mf cx Data.Nat.< i → * x < * y + fc01 (suc i) cx (init x₁ x₂) x (s≤s x₃) = ⊥-elim (fc06 x₂ x) + fc01 (suc i) {y} cx (fsuc y1 cy) i=y (s≤s x<i) with proj1 (mf y1 (A∋fc s f mf cy ) ) + ... | case1 y=fy = subst (λ k → * x < k ) y=fy ( fc01 (suc i) {y1} cx cy i=y (s≤s x<i) ) + ... | case2 y<fy with <-cmp (fcn s mf cx ) i + ... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> x<i c ) + ... | tri≈ ¬a b ¬c = subst (λ k → k < * (f y1) ) (fcn-inject s mf cy cx (sym (trans b (cong pred i=y) ))) y<fy + ... | tri< a ¬b ¬c = IsStrictPartialOrder.trans PO fc02 y<fy where + fc03 : suc i ≡ suc (fcn s mf cy) → i ≡ fcn s mf cy + fc03 eq = cong pred eq + fc02 : * x < * y1 + fc02 = fc01 i cx cy (fc03 i=y ) a + + +fcn-cmp : {A : HOD} (s : Ordinal) { x y : Ordinal } (f : Ordinal → Ordinal) (mf : ≤-monotonic-f A f) + → (cx : FClosure A f s x) → (cy : FClosure A f s y ) → Tri (* x < * y) (* x ≡ * y) (* y < * x ) +fcn-cmp {A} s {x} {y} f mf cx cy with <-cmp ( fcn s mf cx ) (fcn s mf cy ) +... | tri< a ¬b ¬c = tri< fc11 (λ eq → <-irr (case1 (sym eq)) fc11) (λ lt → <-irr (case2 fc11) lt) where + fc11 : * x < * y + fc11 = fcn-< {A} s {x} {y} {f} mf cx cy a +... | tri≈ ¬a b ¬c = tri≈ (λ lt → <-irr (case1 (sym fc10)) lt) fc10 (λ lt → <-irr (case1 fc10) lt) where + fc10 : * x ≡ * y + fc10 = fcn-inject {A} s {x} {y} {f} mf cx cy b +... | tri> ¬a ¬b c = tri> (λ lt → <-irr (case2 fc12) lt) (λ eq → <-irr (case1 eq) fc12) fc12 where + fc12 : * y < * x + fc12 = fcn-< {A} s {y} {x} {f} mf cy cx c + + + +-- open import Relation.Binary.Properties.Poset as Poset + +IsTotalOrderSet : ( A : HOD ) → Set (Level.suc n) +IsTotalOrderSet A = {a b : HOD} → odef A (& a) → odef A (& b) → Tri (a < b) (a ≡ b) (b < a ) + +⊆-IsTotalOrderSet : { A B : HOD } → B ⊆ A → IsTotalOrderSet A → IsTotalOrderSet B +⊆-IsTotalOrderSet {A} {B} B⊆A T ax ay = T (incl B⊆A ax) (incl B⊆A ay) + +_⊆'_ : ( A B : HOD ) → Set n +_⊆'_ A B = {x : Ordinal } → odef A x → odef B x + +-- +-- inductive maxmum tree from x +-- tree structure +-- + +record HasPrev (A B : HOD) (x : Ordinal ) ( f : Ordinal → Ordinal ) : Set n where + field + ax : odef A x + y : Ordinal + ay : odef B y + x=fy : x ≡ f y + +record IsSup (A B : HOD) {x : Ordinal } (xa : odef A x) : Set n where + field + x<sup : {y : Ordinal} → odef B y → (y ≡ x ) ∨ (y << x ) + +record SUP ( A B : HOD ) : Set (Level.suc n) where + field + sup : HOD + as : A ∋ sup + x<sup : {x : HOD} → B ∋ x → (x ≡ sup ) ∨ (x < sup ) -- B is Total, use positive + +-- +-- sup and its fclosure is in a chain HOD +-- chain HOD is sorted by sup as Ordinal and <-ordered +-- whole chain is a union of separated Chain +-- minimum index is sup of y not ϕ +-- + +record ChainP (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal) (u : Ordinal) : Set n where + field + fcy<sup : {z : Ordinal } → FClosure A f y z → (z ≡ supf u) ∨ ( z << supf u ) + order : {s z1 : Ordinal} → (lt : supf s o< supf u ) → FClosure A f (supf s ) z1 → (z1 ≡ supf u ) ∨ ( z1 << supf u ) + supu=u : supf u ≡ u + +data UChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) + (supf : Ordinal → Ordinal) (x : Ordinal) : (z : Ordinal) → Set n where + ch-init : {z : Ordinal } (fc : FClosure A f y z) → UChain A f mf ay supf x z + ch-is-sup : (u : Ordinal) {z : Ordinal } (u<x : supf u o< supf x) ( is-sup : ChainP A f mf ay supf u ) + ( fc : FClosure A f (supf u) z ) → UChain A f mf ay supf x z + +-- +-- f (f ( ... (sup y))) f (f ( ... (sup z1))) +-- / | / | +-- / | / | +-- sup y < sup z1 < sup z2 +-- o< o< +-- data UChain is total + +chain-total : (A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) + {s s1 a b : Ordinal } ( ca : UChain A f mf ay supf s a ) ( cb : UChain A f mf ay supf s1 b ) → Tri (* a < * b) (* a ≡ * b) (* b < * a ) +chain-total A f mf {y} ay supf {xa} {xb} {a} {b} ca cb = ct-ind xa xb ca cb where + ct-ind : (xa xb : Ordinal) → {a b : Ordinal} → UChain A f mf ay supf xa a → UChain A f mf ay supf xb b → Tri (* a < * b) (* a ≡ * b) (* b < * a) + ct-ind xa xb {a} {b} (ch-init fca) (ch-init fcb) = fcn-cmp y f mf fca fcb + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) with ChainP.fcy<sup supb fca + ... | case1 eq with s≤fc (supf ub) f mf fcb + ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where + ct00 : * a ≡ * b + ct00 = trans (cong (*) eq) eq1 + ... | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where + ct01 : * a < * b + ct01 = subst (λ k → * k < * b ) (sym eq) lt + ct-ind xa xb {a} {b} (ch-init fca) (ch-is-sup ub u≤x supb fcb) | case2 lt = tri< ct01 (λ eq → <-irr (case1 (sym eq)) ct01) (λ lt → <-irr (case2 ct01) lt) where + ct00 : * a < * (supf ub) + ct00 = lt + ct01 : * a < * b + ct01 with s≤fc (supf ub) f mf fcb + ... | case1 eq = subst (λ k → * a < k ) eq ct00 + ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt + ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) with ChainP.fcy<sup supa fcb + ... | case1 eq with s≤fc (supf ua) f mf fca + ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where + ct00 : * a ≡ * b + ct00 = sym (trans (cong (*) eq) eq1 ) + ... | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where + ct01 : * b < * a + ct01 = subst (λ k → * k < * a ) (sym eq) lt + ct-ind xa xb {a} {b} (ch-is-sup ua u≤x supa fca) (ch-init fcb) | case2 lt = tri> (λ lt → <-irr (case2 ct01) lt) (λ eq → <-irr (case1 eq) ct01) ct01 where + ct00 : * b < * (supf ua) + ct00 = lt + ct01 : * b < * a + ct01 with s≤fc (supf ua) f mf fca + ... | case1 eq = subst (λ k → * b < k ) eq ct00 + ... | case2 lt = IsStrictPartialOrder.trans POO ct00 lt + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) with trio< ua ub + ... | tri< a₁ ¬b ¬c with ChainP.order supb (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supa )) (sym (ChainP.supu=u supb )) a₁) fca + ... | case1 eq with s≤fc (supf ub) f mf fcb + ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where + ct00 : * a ≡ * b + ct00 = trans (cong (*) eq) eq1 + ... | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where + ct02 : * a < * b + ct02 = subst (λ k → * k < * b ) (sym eq) lt + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri< a₁ ¬b ¬c | case2 lt = tri< ct02 (λ eq → <-irr (case1 (sym eq)) ct02) (λ lt → <-irr (case2 ct02) lt) where + ct03 : * a < * (supf ub) + ct03 = lt + ct02 : * a < * b + ct02 with s≤fc (supf ub) f mf fcb + ... | case1 eq = subst (λ k → * a < k ) eq ct03 + ... | case2 lt = IsStrictPartialOrder.trans POO ct03 lt + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri≈ ¬a eq ¬c + = fcn-cmp (supf ua) f mf fca (subst (λ k → FClosure A f k b ) (cong supf (sym eq)) fcb ) + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c with ChainP.order supa (subst₂ (λ j k → j o< k ) (sym (ChainP.supu=u supb )) (sym (ChainP.supu=u supa )) c) fcb + ... | case1 eq with s≤fc (supf ua) f mf fca + ... | case1 eq1 = tri≈ (λ lt → ⊥-elim (<-irr (case1 (sym ct00)) lt)) ct00 (λ lt → ⊥-elim (<-irr (case1 ct00) lt)) where + ct00 : * a ≡ * b + ct00 = sym (trans (cong (*) eq) eq1) + ... | case2 lt = tri> (λ lt → <-irr (case2 ct02) lt) (λ eq → <-irr (case1 eq) ct02) ct02 where + ct02 : * b < * a + ct02 = subst (λ k → * k < * a ) (sym eq) lt + ct-ind xa xb {a} {b} (ch-is-sup ua ua<x supa fca) (ch-is-sup ub ub<x supb fcb) | tri> ¬a ¬b c | case2 lt = tri> (λ lt → <-irr (case2 ct04) lt) (λ eq → <-irr (case1 (eq)) ct04) ct04 where + ct05 : * b < * (supf ua) + ct05 = lt + ct04 : * b < * a + ct04 with s≤fc (supf ua) f mf fca + ... | case1 eq = subst (λ k → * b < k ) eq ct05 + ... | case2 lt = IsStrictPartialOrder.trans POO ct05 lt + +∈∧P→o< : {A : HOD } {y : Ordinal} → {P : Set n} → odef A y ∧ P → y o< & A +∈∧P→o< {A } {y} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) + +-- Union of supf z which o< x +-- +UnionCF : ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } (ay : odef A y ) + ( supf : Ordinal → Ordinal ) ( x : Ordinal ) → HOD +UnionCF A f mf ay supf x + = record { od = record { def = λ z → odef A z ∧ UChain A f mf ay supf x z } ; odmax = & A ; <odmax = λ {y} sy → ∈∧P→o< sy } + +supf-inject0 : {x y : Ordinal } {supf : Ordinal → Ordinal } → (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) + → supf x o< supf y → x o< y +supf-inject0 {x} {y} {supf} supf-mono sx<sy with trio< x y +... | tri< a ¬b ¬c = a +... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) +... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) +... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) +... | case2 lt = ⊥-elim ( o<> sx<sy lt ) + +record MinSUP ( A B : HOD ) : Set n where + field + sup : Ordinal + asm : odef A sup + x<sup : {x : Ordinal } → odef B x → (x ≡ sup ) ∨ (x << sup ) + minsup : { sup1 : Ordinal } → odef A sup1 + → ( {x : Ordinal } → odef B x → (x ≡ sup1 ) ∨ (x << sup1 )) → sup o≤ sup1 + +z09 : {b : Ordinal } { A : HOD } → odef A b → b o< & A +z09 {b} {A} ab = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) ab)) + +M→S : { A : HOD } { f : Ordinal → Ordinal } {mf : ≤-monotonic-f A f} {y : Ordinal} {ay : odef A y} { x : Ordinal } + → (supf : Ordinal → Ordinal ) + → MinSUP A (UnionCF A f mf ay supf x) + → SUP A (UnionCF A f mf ay supf x) +M→S {A} {f} {mf} {y} {ay} {x} supf ms = record { sup = * (MinSUP.sup ms) + ; as = subst (λ k → odef A k) (sym &iso) (MinSUP.asm ms) ; x<sup = ms00 } where + msup = MinSUP.sup ms + ms00 : {z : HOD} → UnionCF A f mf ay supf x ∋ z → (z ≡ * msup) ∨ (z < * msup) + ms00 {z} uz with MinSUP.x<sup ms uz + ... | case1 eq = case1 (subst (λ k → k ≡ _) *iso ( cong (*) eq)) + ... | case2 lt = case2 (subst₂ (λ j k → j < k ) *iso refl lt ) + + +chain-mono : {A : HOD} ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) (supf : Ordinal → Ordinal ) + (supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y ) {a b c : Ordinal} → a o≤ b + → odef (UnionCF A f mf ay supf a) c → odef (UnionCF A f mf ay supf b) c +chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ ua , ch-init fc ⟫ = + ⟪ ua , ch-init fc ⟫ +chain-mono f mf ay supf supf-mono {a} {b} {c} a≤b ⟪ uaa , ch-is-sup ua ua<x is-sup fc ⟫ = + ⟪ uaa , ch-is-sup ua (ordtrans<-≤ ua<x (supf-mono a≤b ) ) is-sup fc ⟫ + +record ZChain ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) + {y : Ordinal} (ay : odef A y) ( z : Ordinal ) : Set (Level.suc n) where + field + supf : Ordinal → Ordinal + sup=u : {b : Ordinal} → (ab : odef A b) → b o≤ z + → IsSup A (UnionCF A f mf ay supf b) ab ∧ (¬ HasPrev A (UnionCF A f mf ay supf b) b f) → supf b ≡ b + + asupf : {x : Ordinal } → odef A (supf x) + supf-mono : {x y : Ordinal } → x o≤ y → supf x o≤ supf y + supf-< : {x y : Ordinal } → supf x o< supf y → supf x << supf y + supfmax : {x : Ordinal } → z o< x → supf x ≡ supf z + + minsup : {x : Ordinal } → x o≤ z → MinSUP A (UnionCF A f mf ay supf x) + supf-is-minsup : {x : Ordinal } → (x≤z : x o≤ z) → supf x ≡ MinSUP.sup ( minsup x≤z ) + csupf : {b : Ordinal } → supf b o< z → odef (UnionCF A f mf ay supf z) (supf b) -- supf z is not an element of this chain + + chain : HOD + chain = UnionCF A f mf ay supf z + chain⊆A : chain ⊆' A + chain⊆A = λ lt → proj1 lt + sup : {x : Ordinal } → x o≤ z → SUP A (UnionCF A f mf ay supf x) + sup {x} x≤z = M→S supf (minsup x≤z) + -- supf-sup<minsup : {x : Ordinal } → (x≤z : x o≤ z) → & (SUP.sup (M→S supf (minsup x≤z) )) o≤ supf x ... supf-mono + + chain∋init : odef chain y + chain∋init = ⟪ ay , ch-init (init ay refl) ⟫ + f-next : {a z : Ordinal} → odef (UnionCF A f mf ay supf z) a → odef (UnionCF A f mf ay supf z) (f a) + f-next {a} ⟪ aa , ch-init fc ⟫ = ⟪ proj2 (mf a aa) , ch-init (fsuc _ fc) ⟫ + f-next {a} ⟪ aa , ch-is-sup u u≤x is-sup fc ⟫ = ⟪ proj2 (mf a aa) , ch-is-sup u u≤x is-sup (fsuc _ fc ) ⟫ + initial : {z : Ordinal } → odef chain z → * y ≤ * z + initial {a} ⟪ aa , ua ⟫ with ua + ... | ch-init fc = s≤fc y f mf fc + ... | ch-is-sup u u≤x is-sup fc = ≤-ftrans (<=to≤ zc7) (s≤fc _ f mf fc) where + zc7 : y <= supf u + zc7 = ChainP.fcy<sup is-sup (init ay refl) + f-total : IsTotalOrderSet chain + f-total {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where + uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) + uz01 = chain-total A f mf ay supf ( (proj2 ca)) ( (proj2 cb)) + + supf-<= : {x y : Ordinal } → supf x <= supf y → supf x o≤ supf y + supf-<= {x} {y} (case1 sx=sy) = o≤-refl0 sx=sy + supf-<= {x} {y} (case2 sx<sy) with trio< (supf x) (supf y) + ... | tri< a ¬b ¬c = o<→≤ a + ... | tri≈ ¬a b ¬c = o≤-refl0 b + ... | tri> ¬a ¬b c = ⊥-elim (<-irr (case2 sx<sy ) (supf-< c) ) + + supf-inject : {x y : Ordinal } → supf x o< supf y → x o< y + supf-inject {x} {y} sx<sy with trio< x y + ... | tri< a ¬b ¬c = a + ... | tri≈ ¬a refl ¬c = ⊥-elim ( o<¬≡ (cong supf refl) sx<sy ) + ... | tri> ¬a ¬b y<x with osuc-≡< (supf-mono (o<→≤ y<x) ) + ... | case1 eq = ⊥-elim ( o<¬≡ (sym eq) sx<sy ) + ... | case2 lt = ⊥-elim ( o<> sx<sy lt ) + + fcy<sup : {u w : Ordinal } → u o≤ z → FClosure A f y w → (w ≡ supf u ) ∨ ( w << supf u ) -- different from order because y o< supf + fcy<sup {u} {w} u≤z fc with 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 ) + + -- ordering is not proved here but in ZChain1 + +record ZChain1 ( A : HOD ) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) + {y : Ordinal} (ay : odef A y) (zc : ZChain A f mf ay (& A)) ( z : Ordinal ) : Set (Level.suc n) where + supf = ZChain.supf zc + field + is-max : {a b : Ordinal } → (ca : odef (UnionCF A f mf ay supf z) a ) → supf b o< supf z → (ab : odef A b) + → HasPrev A (UnionCF A f mf ay supf z) b f ∨ IsSup A (UnionCF A f mf ay supf z) ab + → * a < * b → odef ((UnionCF A f mf 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 + +init-uchain : (A : HOD) ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal } → (ay : odef A y ) + { supf : Ordinal → Ordinal } { x : Ordinal } → odef (UnionCF A f mf ay supf x) y +init-uchain A f mf ay = ⟪ ay , ch-init (init ay refl) ⟫ + +Zorn-lemma : { A : HOD } + → o∅ o< & A + → ( ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → SUP A B ) -- SUP condition + → Maximal A +Zorn-lemma {A} 0<A supP = zorn00 where + <-irr0 : {a b : HOD} → A ∋ a → A ∋ b → (a ≡ b ) ∨ (a < b ) → b < a → ⊥ + <-irr0 {a} {b} A∋a A∋b = <-irr + z07 : {y : Ordinal} {A : HOD } → {P : Set n} → odef A y ∧ P → y o< & A + z07 {y} {A} p = subst (λ k → k o< & A) &iso ( c<→o< (subst (λ k → odef A k ) (sym &iso ) (proj1 p ))) + s : HOD + s = ODC.minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) + as : A ∋ * ( & s ) + as = subst (λ k → odef A (& k) ) (sym *iso) ( ODC.x∋minimal O A (λ eq → ¬x<0 ( subst (λ k → o∅ o< k ) (=od∅→≡o∅ eq) 0<A )) ) + as0 : odef A (& s ) + as0 = subst (λ k → odef A k ) &iso as + s<A : & s o< & A + s<A = c<→o< (subst (λ k → odef A (& k) ) *iso as ) + HasMaximal : HOD + HasMaximal = record { od = record { def = λ x → odef A x ∧ ( (m : Ordinal) → odef A m → ¬ (* x < * m)) } ; odmax = & A ; <odmax = z07 } + no-maximum : HasMaximal =h= od∅ → (x : Ordinal) → odef A x ∧ ((m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m) )) → ⊥ + no-maximum nomx x P = ¬x<0 (eq→ nomx {x} ⟪ proj1 P , (λ m ma p → proj2 ( proj2 P m ma ) p ) ⟫ ) + Gtx : { x : HOD} → A ∋ x → HOD + Gtx {x} ax = record { od = record { def = λ y → odef A y ∧ (x < (* y)) } ; odmax = & A ; <odmax = z07 } + z08 : ¬ Maximal A → HasMaximal =h= od∅ + z08 nmx = record { eq→ = λ {x} lt → ⊥-elim ( nmx record {maximal = * x ; as = subst (λ k → odef A k) (sym &iso) (proj1 lt) + ; ¬maximal<x = λ {y} ay → subst (λ k → ¬ (* x < k)) *iso (proj2 lt (& y) ay) } ) ; eq← = λ {y} lt → ⊥-elim ( ¬x<0 lt )} + x-is-maximal : ¬ Maximal A → {x : Ordinal} → (ax : odef A x) → & (Gtx (subst (λ k → odef A k ) (sym &iso) ax)) ≡ o∅ → (m : Ordinal) → odef A m → odef A x ∧ (¬ (* x < * m)) + x-is-maximal nmx {x} ax nogt m am = ⟪ subst (λ k → odef A k) &iso (subst (λ k → odef A k ) (sym &iso) ax) , ¬x<m ⟫ where + ¬x<m : ¬ (* x < * m) + ¬x<m x<m = ∅< {Gtx (subst (λ k → odef A k ) (sym &iso) ax)} {* m} ⟪ subst (λ k → odef A k) (sym &iso) am , subst (λ k → * x < k ) (cong (*) (sym &iso)) x<m ⟫ (≡o∅→=od∅ nogt) + + minsupP : ( B : HOD) → (B⊆A : B ⊆' A) → IsTotalOrderSet B → MinSUP A B + minsupP B B⊆A total = m02 where + xsup : (sup : Ordinal ) → Set n + xsup sup = {w : Ordinal } → odef B w → (w ≡ sup ) ∨ (w << sup ) + ∀-imply-or : {A : Ordinal → Set n } {B : Set n } + → ((x : Ordinal ) → A x ∨ B) → ((x : Ordinal ) → A x) ∨ B + ∀-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 ; asm = 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.as S)) ⟪ SUP.as 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 {* w} (subst (λ k → odef B k) (sym &iso) bw ) + ... | case1 eq = case1 ( subst₂ (λ j k → j ≡ k ) &iso refl (cong (&) eq) ) + ... | case2 lt = case2 ( subst (λ k → _ < k ) (sym *iso) lt ) + m02 : MinSUP A B + m02 = dont-or (m00 (& A)) m03 + + -- Uncountable ascending chain by axiom of choice + cf : ¬ Maximal A → Ordinal → Ordinal + cf nmx x with ODC.∋-p O A (* x) + ... | 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 ) ⟫ + + -- + -- Second TransFinite Pass for maximality + -- + + SZ1 : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) + {init : Ordinal} (ay : odef A init) (zc : ZChain A f mf ay (& A)) (x : Ordinal) → ZChain1 A f mf ay zc x + SZ1 f mf {y} ay zc x = ? + + 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 ; as = SUP.as sup ; x<sup = λ lt → SUP.x<sup sup (B⊆C lt) } + + record xSUP (B : HOD) (x : Ordinal) : Set n where + field + ax : odef A x + is-sup : IsSup A B ax + + -- + -- 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 = ? + + --- + --- 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 + + data ZChainP ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + ( supf : Ordinal → Ordinal ) (z : Ordinal) : Set n where + zchain : (uz : Ordinal ) → odef (UnionCF A f mf ay supf uz) z → ZChainP f mf ay supf z + + auzc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → odef A x + auzc f mf {y} ay supf {x} (zchain uz ucf) = proj1 ucf + + zp-uz : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + (supf : Ordinal → Ordinal ) → {x : Ordinal } → ZChainP f mf ay supf x → Ordinal + zp-uz f mf ay supf (zchain uz _) = uz + + uzc⊆zc : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + (supf : Ordinal → Ordinal ) → {x : Ordinal } → (zp : ZChainP f mf ay supf x ) → UChain A f mf ay supf (zp-uz f mf ay supf zp) x + uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-init fc ⟫) = ch-init fc + uzc⊆zc f mf {y} ay supf {x} (zchain uz ⟪ ua , ch-is-sup u u<x is-sup fc ⟫) with ChainP.supu=u is-sup + ... | eq = ch-is-sup u u<x is-sup fc + + UnionZF : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + (supf : Ordinal → Ordinal ) → HOD + UnionZF f mf {y} ay supf = record { od = record { def = λ x → ZChainP f mf ay supf x } + ; odmax = & A ; <odmax = λ lt → ∈∧P→o< ⟪ auzc f mf ay supf lt , lift true ⟫ } + + uzctotal : ( f : Ordinal → Ordinal ) (mf : ≤-monotonic-f A f) {y : Ordinal} (ay : odef A y) + → ( supf : Ordinal → Ordinal ) + → IsTotalOrderSet (UnionZF f mf ay supf ) + uzctotal f mf ay supf {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso (uz01 ca cb) where + uz01 : {ua ub : Ordinal } → ZChainP f mf ay supf ua → ZChainP f mf ay supf ub + → Tri (* ua < * ub) (* ua ≡ * ub) (* ub < * ua ) + uz01 {ua} {ub} (zchain uza uca) (zchain uzb ucb) = chain-total A f mf ay supf (proj2 uca) (proj2 ucb) + + usp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {y : Ordinal} (ay : odef A y) + → ( supf : Ordinal → Ordinal ) + → SUP A (UnionZF f mf ay supf ) + usp0 f mf ay supf = supP (UnionZF f mf ay supf ) (λ lt → auzc f mf ay supf lt ) (uzctotal f mf ay supf ) + + sp0 : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) {x y : Ordinal} (ay : odef A y) + → (zc : ZChain A f mf ay x ) + → SUP A (UnionCF A f mf ay (ZChain.supf zc) x) + sp0 f mf {x} ay zc = supP (UnionCF A f mf ay (ZChain.supf zc) x) (ZChain.chain⊆A zc) ztotal where + ztotal : IsTotalOrderSet (ZChain.chain zc) + ztotal {a} {b} ca cb = subst₂ (λ j k → Tri (j < k) (j ≡ k) (k < j)) *iso *iso uz01 where + uz01 : Tri (* (& a) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a) ) + uz01 = chain-total A f mf ay (ZChain.supf zc) ( (proj2 ca)) ( (proj2 cb)) + + fixpoint : ( f : Ordinal → Ordinal ) → (mf : ≤-monotonic-f A f ) (zc : ZChain A f mf as0 (& A) ) + → ZChain.supf zc (& (SUP.sup (sp0 f mf as0 zc))) o< ZChain.supf zc (& A) + → f (& (SUP.sup (sp0 f mf as0 zc ))) ≡ & (SUP.sup (sp0 f mf as0 zc )) + fixpoint f mf zc ss<sa = ? + + + -- 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 (SUP.as sp1 )))) + (subst (λ k → odef A (& k)) (sym *iso) (SUP.as sp1) ) + (case1 ( cong (*)( fixpoint (cf nmx) (cf-is-≤-monotonic nmx ) zc ss<sa ))) -- x ≡ f x ̄ + (proj1 (cf-is-<-monotonic nmx c (SUP.as sp1 ))) where -- x < f x + supf = ZChain.supf zc + sp1 : SUP A (ZChain.chain zc) + sp1 = sp0 (cf nmx) (cf-is-≤-monotonic nmx) as0 zc + c = & (SUP.sup sp1) + z20 : c << cf nmx c + z20 = proj1 (cf-is-<-monotonic nmx c (SUP.as sp1) ) + asc : odef A (supf c) + asc = ZChain.asupf zc + spd : MinSUP A (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) + spd = ysup (cf nmx) (cf-is-≤-monotonic nmx) asc + d = MinSUP.sup spd + d<A : d o< & A + d<A = ∈∧P→o< ⟪ MinSUP.asm spd , lift true ⟫ + msup : MinSUP A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 supf d) + msup = ZChain.minsup zc (o<→≤ d<A) + sd=ms : supf d ≡ MinSUP.sup ( ZChain.minsup zc (o<→≤ d<A) ) + sd=ms = ZChain.supf-is-minsup zc (o<→≤ d<A) + -- z26 : {x : Ordinal } → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) x + -- → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) c) x ∨ odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x + -- z26 = ? + is-sup : IsSup A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) (MinSUP.asm spd) + is-sup = record { x<sup = z22 } where + z23 : {z : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc) z → (z ≡ MinSUP.sup spd) ∨ (z << MinSUP.sup spd) + z23 lt = MinSUP.x<sup spd lt + z22 : {y : Ordinal} → odef (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) y → + (y ≡ MinSUP.sup spd) ∨ (y << MinSUP.sup spd) + z22 {a} ⟪ aa , ch-init fc ⟫ = ? + z22 {a} ⟪ aa , ch-is-sup u u<x is-sup fc ⟫ = ? + -- u<x : ZChain.supf zc u o< ZChain.supf zc d + -- supf u o< spuf c → order + not-hasprev : ¬ HasPrev A (UnionCF A (cf nmx) (cf-is-≤-monotonic nmx) as0 (ZChain.supf zc) d) d (cf nmx) + not-hasprev hp = ? where + y : Ordinal + y = HasPrev.y hp + z24 : y << d + z24 = subst (λ k → y << k) (sym (HasPrev.x=fy hp)) ( proj1 (cf-is-<-monotonic nmx y (proj1 (HasPrev.ay hp) ) )) + -- z26 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ d ) ∨ (x << d ) + -- z26 lt with MinSUP.x<sup spd (subst (λ k → odef _ k ) ? lt) + -- ... | case1 eq = ? + -- ... | case2 lt = ? + -- z25 : {x : Ordinal } → odef (uchain (cf nmx) (cf-is-≤-monotonic nmx) asc ) x → (x ≡ y ) ∨ (x << y ) + -- z25 {x} (init au eq ) = ? -- sup c = x, cf y ≡ d, sup c =< d + -- z25 (fsuc x lt) = ? -- cf (sup c) + sd=d : supf d ≡ d + sd=d = ZChain.sup=u zc (MinSUP.asm spd) (o<→≤ d<A) ⟪ is-sup , not-hasprev ⟫ + sc<sd : supf c << supf d + sc<sd = ? + -- z21 = proj1 ( cf-is-<-monotonic nmx ? ? ) + -- sco<d : supf c o< supf d + -- sco<d with osuc-≡< ( ZChain.supf-<= zc (case2 sc<sd ) ) + -- ... | case1 eq = ⊥-elim ( <-irr eq sc<sd ) + -- ... | case2 lt = lt + + ss<sa : supf c o< supf (& A) + ss<sa = ? -- with osuc-≡< ( ZChain.supf-mono zc (o<→≤ ( ∈∧P→o< ⟪ SUP.as sp0 , lift true ⟫) )) + -- ... | case2 lt = lt + -- ... | case1 eq = ? -- where + 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