Mercurial > hg > Members > kono > Proof > ZF-in-agda
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 13 Jan 2023 13:03:45 +0900 |
| parents | d618788852e5 |
| children | 4517d0721b59 |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module filter {n : Level } (O : Ordinals {n}) where open import zf open import logic import OD open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality import BAlgebra open BAlgebra O 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 ODC O open _∧_ open _∨_ open Bool -- L is a boolean algebra, but we don't assume this explicitly -- -- NEG : {p : HOD} → L ∋ p → L ∋ (P ＼ p) -- CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q ) -- -- Kunen p.76 and p.53, we use ⊆ record Filter { L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field filter : HOD f⊆L : filter ⊆ L filter1 : { p q : HOD } → L ∋ q → filter ∋ p → p ⊆ q → filter ∋ q filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → L ∋ (p ∩ q) → filter ∋ (p ∩ q) open Filter record prime-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) prime : {p q : HOD } → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) record ultra-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where field proper : ¬ (filter F ∋ od∅) ultra : {p : HOD } → L ∋ p → L ∋ ( P ＼ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) ∈-filter : {L P p : HOD} → {LP : L ⊆ Power P} → (F : Filter {L} {P} LP ) → filter F ∋ p → L ∋ p ∈-filter {L} {p} {LP} F lt = ( f⊆L F) lt ⊆-filter : {L P p q : HOD } → {LP : L ⊆ Power P } → (F : Filter {L} {P} LP) → L ∋ q → q ⊆ P ⊆-filter {L} {P} {p} {q} {LP} F lt = power→⊆ P q ( LP lt ) ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L ∪-lemma1 {L} {p} {q} lt p∋x = lt (case1 p∋x) ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L ∪-lemma2 {L} {p} {q} lt p∋x = lt (case2 p∋x) q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q q∩q⊆q lt = proj1 lt open HOD ----- -- -- ultra filter is prime -- filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ (P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) → (F : Filter {L} {P} LP) → ultra-filter F → prime-filter F filter-lemma1 {P} {L} LP NG IN F u = record { proper = ultra-filter.proper u ; prime = lemma3 } where lemma3 : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) lemma3 {p} {q} Lp Lq _ lt with ultra-filter.ultra u Lp (NG Lp) ... | case1 p∈P = case1 p∈P ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P ＼ p)} Lq lemma7 lemma8) where lemma5 : ((p ∪ q ) ∩ (P ＼ p)) =h= (q ∩ (P ＼ p)) lemma5 = record { eq→ = λ {x} lt → ⟪ lemma4 x lt , proj2 lt ⟫ ; eq← = λ {x} lt → ⟪ case2 (proj1 lt) , proj2 lt ⟫ } where lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (P ＼ p)) x → odef q x lemma4 x lt with proj1 lt lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) lemma4 x lt | case2 qx = qx lemma9 : L ∋ ((p ∪ q ) ∩ (P ＼ p)) lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (IN Lq (NG Lp)) lemma6 : filter F ∋ ((p ∪ q ) ∩ (P ＼ p)) lemma6 = filter2 F lt ¬p∈P lemma9 lemma7 : filter F ∋ (q ∩ (P ＼ p)) lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 lemma8 : (q ∩ (P ＼ p)) ⊆ q lemma8 lt = proj1 lt ----- -- -- if Filter {L} {P} contains L, prime filter is ultra -- filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → (F : Filter {L} {P} LP) → filter F ∋ P → prime-filter F → ultra-filter F filter-lemma2 {P} {L} LP Lm F f∋P prime = record { proper = prime-filter.proper prime ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (Lm L∋p) (lemma10 L∋p ((f⊆L F) f∋P) ) (lemma p (p⊆L L∋p )) } where open _==_ p⊆L : {p : HOD} → L ∋ p → p ⊆ P p⊆L {p} lt = power→⊆ P p ( LP lt ) p+1-p=1 : {p : HOD} → p ⊆ P → P =h= (p ∪ (P ＼ p)) eq→ (p+1-p=1 {p} p⊆P) {x} lt with ODC.decp O (odef p x) eq→ (p+1-p=1 {p} p⊆P) {x} lt | yes p∋x = case1 p∋x eq→ (p+1-p=1 {p} p⊆P) {x} lt | no ¬p = case2 ⟪ lt , ¬p ⟫ eq← (p+1-p=1 {p} p⊆P) {x} ( case1 p∋x ) = subst (λ k → odef P k ) &iso (p⊆P ( subst (λ k → odef p k) (sym &iso) p∋x )) eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P ＼ p)) lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P lemma10 : {p : HOD} → L ∋ p → L ∋ P → L ∋ (p ∪ (P ＼ p)) lemma10 {p} L∋p lt = subst (λ k → L ∋ k ) (==→o≡ (p+1-p=1 {p} (p⊆L L∋p))) lt record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field dense : HOD d⊆P : dense ⊆ L dense-f : {p : HOD} → L ∋ p → HOD dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p record Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where field ideal : HOD i⊆L : ideal ⊆ L ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) open Ideal proper-ideal : {L P : HOD} → (LP : L ⊆ Power P) → (P : Ideal {L} {P} LP ) → {p : HOD} → Set n proper-ideal {L} {P} LP I = ideal I ∋ od∅ prime-ideal : {L P : HOD} → (LP : L ⊆ Power P) → Ideal {L} {P} LP → ∀ {p q : HOD } → Set n prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) open import Relation.Binary.Definitions record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where field genf : Filter {L} {P} LP generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) (F : Filter {L} {P} LP) : Set (suc n) where field mf : Filter {L} {P} LP F⊆mf : filter F ⊆ filter mf proper : ¬ (filter mf ∋ od∅) is-maximum : ( f : Filter {L} {P} LP ) → ¬ (filter f ∋ od∅) → ¬ ( filter mf ⊂ filter f ) record Fp {L P : HOD} (LP : L ⊆ Power P) (F : Filter {L} {P} LP) (mx : MaximumFilter {L} {P} LP F ) (p x : Ordinal ) : Set n where field y : Ordinal mfy : odef (filter (MaximumFilter.mf mx)) y y-p⊂x : ( * y ＼ * p ) ⊆ * x max→ultra : {L P : HOD} (LP : L ⊆ Power P) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F0 : Filter {L} {P} LP) → {y : Ordinal } → odef (filter F0) y → (mx : MaximumFilter {L} {P} LP F0 ) → ultra-filter ( MaximumFilter.mf mx ) max→ultra {L} {P} LP CAP F0 {y} mfy mx = record { proper = MaximumFilter.proper mx ; ultra = ultra } where mf = MaximumFilter.mf mx ultra : {p : HOD} → L ∋ p → L ∋ (P ＼ p) → (filter mf ∋ p) ∨ (filter mf ∋ (P ＼ p)) ultra {p} Lp Lnp with ODC.∋-p O (filter mf) p ... | yes y = case1 y ... | no np = case2 (subst (λ k → k ∋ (P ＼ p)) F=mf F∋P-p) where F : HOD F = record { od = record { def = λ x → odef L x ∧ Fp {L} {P} LP F0 mx (& p) x } ; odmax = & L ; <odmax = λ lt → odef< (proj1 lt) } mu01 : {r : HOD} {q : HOD} → L ∋ q → F ∋ r → r ⊆ q → F ∋ q mu01 {r} {q} Lq ⟪ Lr , record { y = y ; mfy = mfy ; y-p⊂x = y-p⊂x } ⟫ r⊆q = ⟪ Lq , record { y = y ; mfy = mfy ; y-p⊂x = mu03 } ⟫ where mu05 : (* y ＼ p) ⊆ r mu05 = subst₂ (λ j k → (* y ＼ j ) ⊆ k ) *iso *iso y-p⊂x mu04 : (* y ＼ * (& p)) ⊆ * (& q) mu04 {x} ⟪ yx , npx ⟫ = subst (λ k → odef k x ) (sym *iso) (r⊆q (mu05 ⟪ yx , (λ px1 → npx (subst (λ k → odef k x) (sym *iso) px1 )) ⟫ ) ) mu03 : (* y ＼ * (& p)) ⊆ * (& q) mu03 = mu04 mu02 : {r : HOD} {q : HOD} → F ∋ r → F ∋ q → L ∋ (r ∩ q) → F ∋ (r ∩ q) mu02 {r} {q} ⟪ Lr , record { y = ry ; mfy = mfry ; y-p⊂x = ry-p⊂x } ⟫ ⟪ Lq , record { y = qy ; mfy = mfqy ; y-p⊂x = qy-p⊂x } ⟫ Lrq = ⟪ Lrq , record { y = & (* qy ∩ * ry) ; mfy = mu20 ; y-p⊂x = mu22 } ⟫ where mu21 : L ∋ (* qy ∩ * ry) mu21 = CAP (subst (λ k → odef L k ) (sym &iso) (f⊆L mf mfqy)) (subst (λ k → odef L k ) (sym &iso) (f⊆L mf mfry)) mu20 : odef (filter mf) (& (* qy ∩ * ry)) mu20 = filter2 mf (subst (λ k → odef (filter mf) k) (sym &iso) mfqy) (subst (λ k → odef (filter mf) k) (sym &iso) mfry) mu21 mu24 : ((* qy ∩ * ry) ＼ * (& p)) ⊆ (r ∩ q) mu24 {x} ⟪ qry , npx ⟫ = ⟪ subst (λ k → odef k x) *iso ( ry-p⊂x ⟪ proj2 qry , npx ⟫ ) , subst (λ k → odef k x) *iso ( qy-p⊂x ⟪ proj1 qry , npx ⟫ ) ⟫ mu23 : ((* qy ∩ * ry) ＼ * (& p) ) ⊆ (r ∩ q) mu23 = mu24 mu22 : (* (& (* qy ∩ * ry)) ＼ * (& p)) ⊆ * (& (r ∩ q)) mu22 = subst₂ (λ j k → (j ＼ * (& p)) ⊆ k ) (sym *iso) (sym *iso) mu23 FisFilter : Filter {L} {P} LP FisFilter = record { filter = F ; f⊆L = λ {x} lt → proj1 lt ; filter1 = mu01 ; filter2 = mu02 } FisGreater : {x : Ordinal } → odef (filter (MaximumFilter.mf mx)) x → odef (filter FisFilter ) x FisGreater {x} mfx = ⟪ f⊆L mf mfx , record { y = x ; mfy = mfx ; y-p⊂x = mu03 } ⟫ where mu03 : (* x ＼ * (& p)) ⊆ * x mu03 {z} ⟪ xz , _ ⟫ = xz F∋P-p : F ∋ (P ＼ p ) F∋P-p = ⟪ Lnp , record { y = y ; mfy = mxy ; y-p⊂x = mu30 } ⟫ where mxy : odef (filter (MaximumFilter.mf mx)) y mxy = MaximumFilter.F⊆mf mx mfy mu30 : (* y ＼ * (& p)) ⊆ * (& (P ＼ p)) mu30 {z} ⟪ yz , ¬pz ⟫ = subst (λ k → odef k z) (sym *iso) ( ⟪ Pz , (λ pz → ¬pz (subst (λ k → odef k z) (sym *iso) pz )) ⟫ ) where Pz : odef P z Pz = LP (f⊆L mf mxy ) _ yz FisProper : ¬ (filter FisFilter ∋ od∅) -- if F contains p, p is in mf which contract np FisProper ⟪ L0 , record { y = z ; mfy = mfz ; y-p⊂x = z-p⊂x } ⟫ = ⊥-elim ( np (filter1 mf Lp (subst (λ k → odef (filter mf) k) (sym &iso) mfz) mu31) ) where mu31 : * z ⊆ p mu31 {x} zx with ODC.decp O (odef p x) ... | yes px = px ... | no npx = ⊥-elim ( ¬x<0 (subst (λ k → odef k x) *iso (z-p⊂x ⟪ zx , (λ px → npx (subst (λ k → odef k x) *iso px) ) ⟫ ) ) ) mu40 = (MaximumFilter.is-maximum mx) F=mf : F ≡ filter mf F=mf with osuc-≡< ( ⊆→o≤ FisGreater ) ... | case1 eq = &≡&→≡ (sym eq) ... | case2 lt = ⊥-elim ( MaximumFilter.is-maximum mx FisFilter FisProper ⟪ lt , FisGreater ⟫ ) open _==_ ultra→max : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (U : Filter {L} {P} LP) → ultra-filter U → MaximumFilter LP U ultra→max {L} {P} LP NG CAP U u = record { mf = U ; F⊆mf = λ x → x ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where is-maximum : (F : Filter {L} {P} LP) → (¬ (filter F ∋ od∅)) → (U⊂F : filter U ⊂ filter F ) → ⊥ is-maximum F Prop ⟪ U<F , U⊆F ⟫ = Prop f0 where GT : HOD GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = & L ; <odmax = um02 } where um02 : {y : Ordinal } → odef (filter F) y ∧ (¬ odef (filter U) y) → y o< & L um02 {y} Fy = odef< ( f⊆L F (proj1 Fy ) ) GT≠∅ : ¬ (GT =h= od∅) GT≠∅ eq = ⊥-elim (U≠F ( ==→o≡ ((⊆→= {filter U} {filter F}) U⊆F (U-F=∅→F⊆U {filter F} {filter U} gt01)))) where U≠F : ¬ ( filter U ≡ filter F ) U≠F eq = o<¬≡ (cong (&) eq) U<F gt01 : (x : Ordinal) → ¬ ( odef (filter F) x ∧ (¬ odef (filter U) x)) gt01 x not = ¬x<0 ( eq→ eq not ) p : HOD p = ODC.minimal O GT GT≠∅ ¬U∋p : ¬ ( filter U ∋ p ) ¬U∋p = proj2 (ODC.x∋minimal O GT GT≠∅) L∋p : L ∋ p L∋p = f⊆L F ( proj1 (ODC.x∋minimal O GT GT≠∅)) um00 : ¬ odef (filter U) (& p) um00 = proj2 (ODC.x∋minimal O GT GT≠∅) L∋-p : L ∋ ( P ＼ p ) L∋-p = NG L∋p U∋-p : filter U ∋ ( P ＼ p ) U∋-p with ultra-filter.ultra u {p} L∋p L∋-p ... | case1 ux = ⊥-elim ( ¬U∋p ux ) ... | case2 u-x = u-x F∋p : filter F ∋ p F∋p = proj1 (ODC.x∋minimal O GT GT≠∅) F∋-p : filter F ∋ ( P ＼ p ) F∋-p = U⊆F U∋-p f0 : filter F ∋ od∅ f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP L∋p L∋-p) ) -- if there is a filter , there is a ultra filter under the axiom of choise -- Zorn Lemma import zorn open import Relation.Binary PO : IsStrictPartialOrder _≡_ _⊂_ PO = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; trans = λ {a} {b} {c} → trans-⊂ {a} {b} {c} ; irrefl = λ x=y x<y → o<¬≡ (cong (&) x=y) (proj1 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 ) } } open zorn O _⊂_ PO open import Relation.Binary.Structures record IsFilter { L P : HOD } (LP : L ⊆ Power P) (filter : Ordinal ) : Set n where field f⊆L : (* filter) ⊆ L filter1 : { p q : Ordinal } → odef L q → odef (* filter) p → (* p) ⊆ (* q) → odef (* filter) q filter2 : { p q : Ordinal } → odef (* filter) p → odef (* filter) q → odef L (& ((* p) ∩ (* q))) → odef (* filter) (& ((* p) ∩ (* q))) proper : ¬ (odef (* filter ) o∅) -- all filter contains F F⊆X : { L P : HOD } (LP : L ⊆ Power P) → Filter {L} {P} LP → HOD F⊆X {L} {P} LP F = record { od = record { def = λ x → IsFilter {L} {P} LP x ∧ ( filter F ⊆ * x) } ; odmax = osuc (& L) ; <odmax = λ {x} lt → fx00 lt } where fx00 : {x : Ordinal } → IsFilter LP x ∧ filter F ⊆ * x → x o< osuc (& L) fx00 {x} lt = subst (λ k → k o< osuc (& L)) &iso ( ⊆→o≤ (IsFilter.f⊆L (proj1 lt )) ) F→Maximum : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F : Filter {L} {P} LP) → o∅ o< & L → {y : Ordinal } → odef (filter F) y → (¬ (filter F ∋ od∅)) → MaximumFilter {L} {P} LP F F→Maximum {L} {P} LP NEG CAP F 0<L {y} 0<F Fprop = record { mf = mf ; F⊆mf = ? ; proper = subst (λ k → ¬ ( odef (filter mf ) k)) (sym ord-od∅) ( IsFilter.proper imf) ; is-maximum = {!!} } where FX : HOD FX = F⊆X {L} {P} LP F oF = & (filter F) mf11 : { p q : Ordinal } → odef L q → odef (* oF) p → (* p) ⊆ (* q) → odef (* oF) q mf11 {p} {q} Lq Fp p⊆q = subst₂ (λ j k → odef j k ) (sym *iso) &iso ( filter1 F (subst (λ k → odef L k) (sym &iso) Lq) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) p⊆q ) mf12 : { p q : Ordinal } → odef (* oF) p → odef (* oF) q → odef L (& ((* p) ∩ (* q))) → odef (* oF) (& ((* p) ∩ (* q))) mf12 {p} {q} Fp Fq Lpq = subst (λ k → odef k (& ((* p) ∩ (* q))) ) (sym *iso) ( filter2 F (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fq) Lpq) FX∋F : odef FX (& (filter F)) FX∋F = ⟪ record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) (f⊆L F) ; filter1 = mf11 ; filter2 = mf12 ; proper = subst₂ (λ j k → ¬ (odef j k) ) (sym *iso) ord-od∅ Fprop } , subst (λ k → filter F ⊆ k ) (sym *iso) ( λ x → x ) ⟫ SUP⊆ : (B : HOD) → B ⊆ FX → IsTotalOrderSet B → SUP FX B SUP⊆ B B⊆FX OB with trio< (& B) o∅ ... | tri< a ¬b ¬c = ⊥-elim (¬x<0 a ) ... | tri≈ ¬a b ¬c = record { sup = filter F ; isSUP = record { ax = FX∋F ; x≤sup = λ {y} zy → ⊥-elim (o<¬≡ (sym b) (∈∅< zy)) } } ... | tri> ¬a ¬b 0<B = record { sup = Union B ; isSUP = record { ax = mf13 ; x≤sup = ? } } where mf20 : HOD mf20 = ODC.minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B ))) mf18 : odef B (& mf20 ) mf18 = ODC.x∋minimal O B (λ eq → (o<¬≡ (cong (&) (sym (==→o≡ eq))) (subst (λ k → k o< & B) (sym ord-od∅) 0<B ))) mf16 : Union B ⊆ L mf16 record { owner = b ; ao = Bb ; ox = bx } = IsFilter.f⊆L ( proj1 ( B⊆FX Bb )) bx mf17 : {p q : Ordinal} → odef L q → odef (* (& (Union B))) p → * p ⊆ * q → odef (* (& (Union B))) q mf17 {p} {q} Lq bp p⊆q with subst (λ k → odef k p ) *iso bp ... | record { owner = owner ; ao = ao ; ox = ox } = subst (λ k → odef k q) (sym *iso) record { owner = owner ; ao = ao ; ox = IsFilter.filter1 mf30 Lq ox p⊆q } where mf30 : IsFilter {L} {P} LP owner mf30 = proj1 ( B⊆FX ao ) mf31 : {p q : Ordinal} → odef (* (& (Union B))) p → odef (* (& (Union B))) q → odef L (& ((* p) ∩ (* q))) → odef (* (& (Union B))) (& ((* p) ∩ (* q))) mf31 {p} {q} bp bq Lpq with subst (λ k → odef k p ) *iso bp | subst (λ k → odef k q ) *iso bq ... | record { owner = bp ; ao = Bbp ; ox = bbp } | record { owner = bq ; ao = Bbq ; ox = bbq } with OB (subst (λ k → odef B k) (sym &iso) Bbp) (subst (λ k → odef B k) (sym &iso) Bbq) ... | tri< bp⊂bq ¬b ¬c = ? ... | tri≈ ¬a bq=bp ¬c = ? ... | tri> ¬a ¬b bq⊂bp = ? mf14 : IsFilter LP (& (Union B)) mf14 = record { f⊆L = subst (λ k → k ⊆ L) (sym *iso) mf16 ; filter1 = mf17 ; filter2 = mf31 ; proper = ? } mf15 : filter F ⊆ Union B mf15 {x} Fx = record { owner = & mf20 ; ao = mf18 ; ox = subst (λ k → odef k x) (sym *iso) (mf22 Fx) } where mf22 : odef (filter F) x → odef mf20 x mf22 Fx = subst (λ k → odef k x) *iso ( proj2 (B⊆FX mf18) Fx ) mf13 : odef FX (& (Union B)) mf13 = ⟪ mf14 , subst (λ k → filter F ⊆ k ) (sym *iso) mf15 ⟫ mx : Maximal FX mx = Zorn-lemma (∈∅< FX∋F) SUP⊆ imf : IsFilter {L} {P} LP (& (Maximal.maximal mx)) imf = proj1 (Maximal.as mx) mf00 : (* (& (Maximal.maximal mx))) ⊆ L mf00 = IsFilter.f⊆L imf mf01 : { p q : HOD } → L ∋ q → (* (& (Maximal.maximal mx))) ∋ p → p ⊆ q → (* (& (Maximal.maximal mx))) ∋ q mf01 {p} {q} Lq Fq p⊆q = IsFilter.filter1 imf Lq Fq (λ {x} lt → subst (λ k → odef k x) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso lt ) )) mf02 : { p q : HOD } → (* (& (Maximal.maximal mx))) ∋ p → (* (& (Maximal.maximal mx))) ∋ q → L ∋ (p ∩ q) → (* (& (Maximal.maximal mx))) ∋ (p ∩ q) mf02 {p} {q} Fp Fq Lpq = subst₂ (λ j k → odef (* (& (Maximal.maximal mx))) (& (j ∩ k ))) *iso *iso ( IsFilter.filter2 imf Fp Fq (subst₂ (λ j k → odef L (& (j ∩ k ))) (sym *iso) (sym *iso) Lpq )) mf : Filter {L} {P} LP mf = record { filter = * (& (Maximal.maximal mx)) ; f⊆L = mf00 ; filter1 = mf01 ; filter2 = mf02 } F→ultra : {L P : HOD} (LP : L ⊆ Power P) → (NEG : {p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → (CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) → (F : Filter {L} {P} LP) → (0<L : o∅ o< & L) → {y : Ordinal} → (0<F : odef (filter F) y) → (proper : ¬ (filter F ∋ od∅)) → ultra-filter ( MaximumFilter.mf (F→Maximum {L} {P} LP NEG CAP F 0<L 0<F proper )) F→ultra {L} {P} LP NEG CAP F 0<L 0<F proper = max→ultra {L} {P} LP CAP F 0<F (F→Maximum {L} {P} LP NEG CAP F 0<L 0<F proper )