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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 02 Jun 2023 08:52:13 +0900 |
| parents | 37f28f427661 |
| children | 2c34f2b554cf |
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{-# OPTIONS --allow-unsolved-metas #-} open import Level open import Ordinals module Tychonoff {n : Level } (O : Ordinals {n}) where open import logic open _∧_ open _∨_ open Bool import OD open import Relation.Nullary open import Data.Empty open import Relation.Binary.Core open import Relation.Binary.Definitions 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 import filter O open import ZProduct O open import Topology O -- open import maximum-filter O open Filter open Topology -- -- Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) -- → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) -- -- ULFP : Compact <=> Every ultra filter F have a limit i.e. open sets which contains the limit (Neighbor limit) is in F -- -- Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where -- -- FP FQ : create projections of a filter F, so it is ULFP -- -- Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) -- -- the product of each limits must be a limit of ultra filter F -- -- its neighbor is in F, because we can decompose Neighbors nei into subbase of Product Topology which is a open set of P ∋ p or Q ∋ q -- so (P x q) ∋ limit ∨ (q x P) ∋ limit. P x q ⊆ nei , so nei ∋ limit -- -- uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) -- → UFLP (ProductTopology TP TQ) F UF -- -- QED. -- FIP is UFL -- filter Base record FBase (P : HOD ) (X : Ordinal ) (u : Ordinal) : Set n where field b x : Ordinal b⊆X : * b ⊆ * X sb : Subbase (* b) x u⊆P : * u ⊆ P x⊆u : * x ⊆ * u record UFLP {P : HOD} (TP : Topology P) (F : Filter {Power P} {P} (λ x → x) ) (ultra : ultra-filter F ) : Set (suc (suc n)) where field limit : Ordinal P∋limit : odef P limit is-limit : {v : Ordinal} → Neighbor TP limit v → filter F ∋ (* v) -- If any ultrafilter has a limit such that all its neighbors -- are within the filter, it possesses the finite intersection -- property (FIP). -- The finite intersection property defines a filter, and through Zorn's lemma, -- we can maximize it to obtain an ultrafilter. -- If the limit of the filter is not contained within a closed -- set 'p' in the FIP, then it must be in the complement of 'p' -- (P \ p). Since this complement is open and contains the limit, -- it is included in the ultrafilter. However, this implies that -- both 'p' and its complement (P \ p) are present in the filter, -- which contradicts the proper characteristic of the ultrafilter, -- meaning that the filter contains no empty set. -- UFLP→FIP : {P : HOD} (TP : Topology P) → ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP UFLP→FIP {P} TP uflp with trio< (& P) o∅ ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) ... | tri≈ ¬a P=0 ¬c = record { limit = λ CX fip → o∅ ; is-limit = fip03 } where -- P is empty ( this case cannot happen because ulfp → 0 < P ) fip02 : {x : Ordinal } → ¬ odef P x fip02 {x} Px = ⊥-elim ( o<¬≡ (sym P=0) (∈∅< Px) ) fip03 : {X : Ordinal} (CX : * X ⊆ CS TP) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) {x : Ordinal} → odef (* X) x → odef (* x) o∅ -- empty P case -- if 0 < X then 0 < x ∧ P ∋ xfrom fip -- if 0 ≡ X then ¬ odef X x fip03 {X} CX fip {x} xx with trio< o∅ X ... | tri< 0<X ¬b ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) P=0)) o∅≡od∅ ) (sym &iso) ( cs⊆L TP (subst (λ k → odef (CS TP) k ) (sym &iso) (CX xx)) xe ))) where 0<x : o∅ o< x 0<x = fip (gi xx ) e : HOD -- we have an element of x e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) xe : odef (* x) (& e) xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin * X ≡⟨ cong (*) (sym 0=X) ⟩ * o∅ ≡⟨ o∅≡od∅ ⟩ od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) ... | tri> ¬a ¬b 0<P = record { limit = λ CSX fip → limit CSX fip ; is-limit = uf01 } where fip : {X : Ordinal} → * X ⊆ CS TP → Set n fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb -- filter base is not empty, it is necessary to maximize fip filter nc : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD nc {X} 0<X CSX fip = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) -- an element of X N∋nc :{X : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → odef (N CSX fip) (& (nc 0<X CSX fip) ) N∋nc {X} 0<X CSX fip = record { b = X ; x = & e ; b⊆X = λ x → x ; sb = gi Xe ; u⊆P = nn02 ; x⊆u = λ x → x } where e : HOD -- we have an element of X e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) Xe : odef (* X) (& e) Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) nn02 : * (& (nc 0<X CSX fip)) ⊆ P nn02 = subst (λ k → k ⊆ P ) (sym *iso) (cs⊆L TP (CSX Xe ) ) 0<PP : o∅ o< & (Power P) -- Power P contaisn od∅, so it is not empty 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where nn00 : odef (Power P) o∅ nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt ... | x<0 = ⊥-elim ( ¬x<0 x<0) -- -- FIP defines a filter -- F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = subst (λ k → k ⊆ P) (sym *iso) f10 ; x⊆u = λ {z} xp → subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where f10 : q ⊆ P f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) f2 {p} {q} Np Nq Xpq = record { b = & Np+Nq ; x = & xp∧xq ; b⊆X = f20 ; sb = sbpq ; u⊆P = p∩q⊆p ; x⊆u = f22 } where p∩q⊆p : * (& (p ∩ q)) ⊆ P p∩q⊆p {x} pqx = subst (λ k → odef P k) &iso (power→ P _ Xpq (subst₂ (λ j k → odef j k ) *iso (sym &iso) pqx )) Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) f20 : * (& Np+Nq) ⊆ * X f20 {x} npq with subst (λ k → odef k x) *iso npq ... | case1 np = FBase.b⊆X Np np ... | case2 nq = FBase.b⊆X Nq nq f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq → ⟪ subst (λ k → odef k w) *iso ( FBase.x⊆u Np (proj1 xpq)) , subst (λ k → odef k w) *iso ( FBase.x⊆u Nq (proj2 xpq)) ⟫ ) -- -- it contains no empty sets becase it is a finite intersection ( Subbase (* X) x ) -- proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip {X} CSX) → ¬ (N CSX fip ∋ od∅) proper {X} CSX fip record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = u⊆P ; x⊆u = x⊆u } = o≤> x≤0 (fip (fp00 _ _ b⊆X sb)) where x≤0 : x o< osuc o∅ x≤0 = subst₂ (λ j k → j o< osuc k) &iso (trans (cong (&) *iso) ord-od∅ ) (⊆→o≤ (x⊆u)) fp00 : (b x : Ordinal) → * b ⊆ * X → Subbase (* b) x → Subbase (* X) x fp00 b y b<X (gi by ) = gi ( b<X by ) fp00 b _ b<X (g∩ {y} {z} sy sz ) = g∩ (fp00 _ _ b<X sy) (fp00 _ _ b<X sz) -- -- then we have maximum ultra filter ( Zorn lemma ) -- to debug this file, commet out the maximum filter and open import -- otherwise the check requires a minute -- maxf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) maxf {X} 0<X CSX fp = ? -- F→Maximum {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc 0<X CSX fp) (proper CSX fp) mf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) mf {X} 0<X CSX fp = ? -- MaximumFilter.mf (maxf 0<X CSX fp) ultraf : {X : Ordinal} → (0<X : o∅ o< X ) → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf 0<X CSX fp) ultraf {X} 0<X CSX fp = ? -- F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc 0<X CSX fp) (proper CSX fp) -- -- so it has a limit as a limit of FIP -- limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal limit {X} CSX fp with trio< o∅ X ... | tri< 0<X ¬b ¬c = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) ... | tri≈ ¬a 0=X ¬c = o∅ ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) -- -- the limit is an limit of entire elements of X -- uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) uf01 {X} CSX fp {x} xx with trio< o∅ X ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬a (subst (λ k → o∅ o< k) &iso ( ∈∅< xx ))) -- 0<X limit is in * x or P ＼ * x ... | tri< 0<X ¬b ¬c with ∨L＼X {P} {* x} {UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))} (UFLP.P∋limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))) ... | case1 lt = lt -- odef (* x) y ... | case2 nlxy = ⊥-elim (MaximumFilter.proper (maxf 0<X CSX fp) uf11 ) where -- -- if (* x) do not conatins a limit, P ＼ * x contains it, (P ＼ * x) is open so it is the maxf ( UFLP.is-limit ) -- UFLP contains (* x) and P ＼ * x, it contains od∅, contradicts the proper -- y = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) x⊆P : * x ⊆ P x⊆P = cs⊆L TP (CSX (subst (λ k → odef (* X) k) (sym &iso) xx)) uf10 : odef (P ＼ * x ) y uf10 = nlxy uf03 : Neighbor TP y (& (P ＼ * x )) uf03 = record { u = _ ; ou = P＼CS=OS TP (CSX (subst (λ k → odef (* X) k ) (sym &iso) xx)) ; ux = subst (λ k → odef k y) (sym *iso) uf10 ; v⊆P = λ {z} xz → proj1 (subst(λ k → odef k z) *iso xz ) ; u⊆v = λ x → x } uf07 : * (& (* x , * x)) ⊆ * X uf07 {y} lt with subst (λ k → odef k y) *iso lt ... | case1 refl = subst (λ k → odef (* X) k ) (sym &iso) xx ... | case2 refl = subst (λ k → odef (* X) k ) (sym &iso) xx uf05 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) x uf05 = MaximumFilter.F⊆mf (maxf 0<X CSX fp) record { b = & (* x , * x) ; b⊆X = uf07 ; sb = gi (subst (λ k → odef k x) (sym *iso) (case1 (sym &iso)) ) ; u⊆P = x⊆P ; x⊆u = λ x → x } uf061 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (* (& (P ＼ * x )))) uf061 = UFLP.is-limit ( uflp (mf 0<X CSX fp) (ultraf 0<X CSX fp)) {& (P ＼ * x)} uf03 -- uf06 (same as uf061) have yellow if zorn lemma is not imported uf06 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (P ＼ * x )) uf06 = subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k) &iso (UFLP.is-limit ( uflp (mf 0<X CSX fp) (ultraf 0<X CSX fp)) {& (P ＼ * x)} uf03 ) uf13 : & ((* x) ∩ (P ＼ * x )) ≡ o∅ uf13 = subst₂ (λ j k → j ≡ k ) refl ord-od∅ (cong (&) ( ==→o≡ record { eq→ = uf14 ; eq← = λ {x} lt → ⊥-elim (¬x<0 lt) } ) ) where uf14 : {y : Ordinal} → odef (* x ∩ (P ＼ * x)) y → odef od∅ y uf14 {y} ⟪ xy , ⟪ Px , ¬xy ⟫ ⟫ = ⊥-elim ( ¬xy xy ) uf12 : odef (Power P) (& ((* x) ∩ (P ＼ * x ))) uf12 z pz with subst (λ k → odef k z) *iso pz ... | ⟪ xz , ⟪ Pz , ¬xz ⟫ ⟫ = Pz uf11 : filter (MaximumFilter.mf (maxf 0<X CSX fp)) ∋ od∅ uf11 = subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k ) (trans uf13 (sym ord-od∅)) ( filter2 (MaximumFilter.mf (maxf 0<X CSX fp)) (subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k) (sym &iso) uf05) uf06 uf12 ) x⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → x ⊆ Cl TP x x⊆Clx {P} TP {x} x<p {y} xy = ⟪ x<p xy , (λ c csc x<c → x<c xy ) ⟫ P⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → Cl TP x ⊆ P P⊆Clx {P} TP {x} x<p {y} xy = proj1 xy -- -- Finite intersection property implies that any ultra filter have a limit, that is, neighbors of the limit is in the filter. -- -- An ultra filter F is given. Take a closure of a filter. It is closed and it has finite intersection property, because F is porper. -- So it has a limit as a FIP. If a neighbor p which contains the limit, p or P \ p is in the ultra filter. -- If it is in P \ p, it cannot contains the limit, contradiction. -- FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF FIP→UFLP {P} TP fip F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 ; P∋limit = ufl10 ; is-limit = ufl00 } where F∋P : odef (filter F) (& P) F∋P with ultra-filter.ultra UF {od∅} (λ z az → ⊥-elim (¬x<0 (subst (λ k → odef k z) *iso az)) ) (λ z az → proj1 (subst (λ k → odef k z) *iso az ) ) ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where fl20 : (P ＼ Ord o∅) =h= P fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } 0<P : o∅ o< & P 0<P with trio< o∅ (& P) ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋P) ) ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) -- -- take closure of given filter elements -- CF : HOD CF = Replace (filter F) (λ x → Cl TP x ) {P} record { ≤COD = λ {z} {y} lt → proj1 lt } CF⊆CS : CF ⊆ CS TP CF⊆CS {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef (CS TP) k) (sym x=ψz) (CS∋Cl TP (* z)) -- -- it is set of closed set and has FIP ( F is proper ) -- ufl08 : {z : Ordinal} → odef (Power P) (& (Cl TP (* z))) ufl08 {z} w zw with subst (λ k → odef k w ) *iso zw ... | t = proj1 t fx→px : {x : Ordinal} → odef (filter F) x → Power P ∋ * x fx→px {x} fx z xz = f⊆L F fx _ (subst (λ k → odef k z) *iso xz ) F∋sb : {x : Ordinal} → Subbase CF x → odef (filter F) x F∋sb {x} (gi record { z = z ; az = az ; x=ψz = x=ψz }) = ufl07 where ufl09 : * z ⊆ P ufl09 {y} zy = f⊆L F az _ zy ufl07 : odef (filter F) x ufl07 = subst (λ k → odef (filter F) k) &iso ( filter1 F (subst (λ k → odef (Power P) k) (trans (sym x=ψz) (sym &iso)) ufl08 ) (subst (λ k → odef (filter F) k) (sym &iso) az) (subst (λ k → * z ⊆ k ) (trans (sym *iso) (sym (cong (*) x=ψz)) ) (x⊆Clx TP {* z} ufl09 ) )) F∋sb (g∩ {x} {y} sx sy) = filter2 F (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sx)) (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sy)) (λ z xz → fx→px (F∋sb sx) _ (subst (λ k → odef k _) (sym *iso) (proj1 (subst (λ k → odef k z) *iso xz) ))) ufl01 : {x : Ordinal} → Subbase (* (& CF)) x → o∅ o< x ufl01 {x} sb = ufl04 where ufl04 : o∅ o< x ufl04 with trio< o∅ x ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim ( ultra-filter.proper UF (subst (λ k → odef (filter F) k) ( begin x ≡⟨ sym b ⟩ o∅ ≡⟨ sym ord-od∅ ⟩ & od∅ ∎ ) (F∋sb (subst (λ k → Subbase k x) *iso sb )) )) where open ≡-Reasoning ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) ufl10 : odef P (FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01) ufl10 = FIP.L∋limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 {& P} ufl11 where ufl11 : odef (* (& CF)) (& P) ufl11 = subst (λ k → odef k (& P)) (sym *iso) record { z = _ ; az = F∋P ; x=ψz = sym (cong (&) (trans (cong (Cl TP) *iso) (ClL TP))) } -- -- so we have a limit -- limit : Ordinal limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 -- -- Neigbor of limit ⊆ Filter -- -- if odef (* X) x, Cl TP x contains limit (ufl02) -- in (nei : Neighbor TP limit v) , there is an open Set u which contains the limit -- F contains u or P ＼ u because F is ultra -- if F ∋ u, then F ∋ v from u ⊆ v which is a propetery of Neighbor -- if F ∋ P ＼ u, it is a closed set (Cl (P ＼ u) ≡ P ＼ u) and it does not contains the limit -- this contradicts ufl02 pp : {v : Ordinal} → (nei : Neighbor TP limit v ) → Power P ∋ (* (Neighbor.u nei)) pp {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; u⊆v = u⊆v } z pz = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) ou ) (subst (λ k → odef k z) *iso pz ) ufl00 : {v : Ordinal} → Neighbor TP limit v → filter F ∋ * v ufl00 {v} nei with ultra-filter.ultra UF (pp nei ) (NEG P (pp nei )) ... | case1 fu = subst (λ k → odef (filter F) k) &iso ( filter1 F (subst (λ k → odef (Power P) k ) (sym &iso) px) fu (subst (λ k → u ⊆ k ) (sym *iso) (Neighbor.u⊆v nei))) where u = * (Neighbor.u nei) px : Power P ∋ * v px z vz = Neighbor.v⊆P nei (subst (λ k → odef k z) *iso vz ) ... | case2 nfu = ⊥-elim ( ¬P\u∋limit P\u∋limit ) where u = * (Neighbor.u nei) P\u : HOD P\u = P ＼ u P\u∋limit : odef P\u limit P\u∋limit = subst (λ k → odef k limit) *iso ( ufl02 (subst (λ k → odef k (& P\u)) (sym *iso) ufl03 )) where ufl04 : & P\u ≡ & (Cl TP (* (& P\u))) ufl04 = cong (&) (sym (trans (cong (Cl TP) *iso) (CS∋x→Clx=x TP (P＼OS=CS TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou nei) ))))) ufl03 : odef CF (& P\u ) ufl03 = record { z = _ ; az = nfu ; x=ψz = ufl04 } ¬P\u∋limit : ¬ odef P\u limit ¬P\u∋limit ⟪ Pl , nul ⟫ = nul ( Neighbor.ux nei ) -- product topology of compact topology is compact import Axiom.Extensionality.Propositional postulate f-extensionality : { n m : Level} → Axiom.Extensionality.Propositional.Extensionality n m open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- FilterQP : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) -- → Filter {Power (ZFP Q P)} {ZFP Q P} (λ x → x) -- FilterQP {P} {Q} F = record { filter = ? ; f⊆L = ? ; filter1 = ? ; filter2 = ? } -- -- projection-of-filter : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) -- → Filter {Power P} {P} (λ x → x) -- projection-of-filter = ? -- -- projection-of-ultra-filter : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) -- → ultra-filter (projection-of-filter F) -- projection-of-ultra-filter = ? -- -- We have UFLP both in P and Q. Given an ultra filter F on P x Q. It has limits on P and Q because a projection of ultra filter -- is a ultra filter. Show the product of the limits is a limit of P x Q. A neighbor of P x Q contains subbase of P x Q, -- which is either inverse projection x of P or Q. The x in in projection of F, because of UFLP. So it is in F, because of the -- property of the filter. -- Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) → UFLP TP F UF uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) → UFLP TQ F UF uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF -- Product of UFL has a limit point uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) → UFLP (ProductTopology TP TQ) F UF uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where F∋PQ : odef (filter F) (& (ZFP P Q)) F∋PQ with ultra-filter.ultra UF {od∅} (λ z az → ⊥-elim (¬x<0 (subst (λ k → odef k z) *iso az)) ) (λ z az → proj1 (subst (λ k → odef k z) *iso az ) ) ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where fl20 : (ZFP P Q ＼ Ord o∅) =h= ZFP P Q fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } 0<PQ : o∅ o< & (ZFP P Q) 0<PQ with trio< o∅ (& (ZFP P Q)) ... | tri< a ¬b ¬c = a ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋PQ) ) ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) apq : HOD apq = ODC.minimal O (ZFP P Q) (0<P→ne 0<PQ) is-apq : ZFP P Q ∋ apq is-apq = ODC.x∋minimal O (ZFP P Q) (0<P→ne 0<PQ) ap : odef P ( zπ1 is-apq ) ap = zp1 is-apq aq : odef Q ( zπ2 is-apq ) aq = zp2 is-apq F⊆pxq : {x : HOD } → filter F ∋ x → x ⊆ ZFP P Q F⊆pxq {x} fx {y} xy = f⊆L F fx y (subst (λ k → odef k y) (sym *iso) xy) --- --- FP is a P-projection of F, which is a ultra filter --- isP→PxQ : {x : HOD} → (x⊆P : x ⊆ P ) → ZFP x Q ⊆ ZFP P Q isP→PxQ {x} x⊆P (ab-pair p q) = ab-pair (x⊆P p) q fx→px : {x : HOD } → filter F ∋ x → HOD fx→px {x} fx = Replace' x ( λ y xy → * (zπ1 (F⊆pxq fx xy) )) {P} record { ≤COD = λ {x} lt {y} ly → ? } fx→px1 : {p : HOD } {q : Ordinal } → odef Q q → (fp : filter F ∋ ZFP p Q ) → fx→px fp ≡ p fx→px1 {p} {q} qq fp = ==→o≡ record { eq→ = ty20 ; eq← = ty22 } where ty21 : {a b : Ordinal } → (pz : odef p a) → (qz : odef Q b) → ZFProduct P Q (& (* (& < * a , * b >))) ty21 pz qz = F⊆pxq fp (subst (odef (ZFP p Q)) (sym &iso) (ab-pair pz qz )) ty32 : {a b : Ordinal } → (pz : odef p a) → (qz : odef Q b) → zπ1 (ty21 pz qz) ≡ a ty32 {a} {b} pz qz = ty33 (ty21 pz qz) (cong (&) *iso) where ty33 : {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ1 p ≡ a ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq)) ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) a=c) ty20 : {x : Ordinal} → odef (fx→px fp) x → odef p x ty20 {x} record { z = _ ; az = ab-pair {a} {b} pz qz ; x=ψz = x=ψz } = subst (λ k → odef p k) a=x pz where ty24 : * x ≡ * a ty24 = begin * x ≡⟨ cong (*) x=ψz ⟩ _ ≡⟨ *iso ⟩ * (zπ1 (F⊆pxq fp (subst (odef (ZFP p Q)) (sym &iso) (ab-pair pz qz)))) ≡⟨ cong (*) (ty32 pz qz) ⟩ * a ∎ where open ≡-Reasoning a=x : a ≡ x a=x = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (sym ty24)) ty22 : {x : Ordinal} → odef p x → odef (fx→px fp) x ty22 {x} px = record { z = _ ; az = ab-pair px qq ; x=ψz = subst₂ (λ j k → j ≡ k) &iso refl (cong (&) ty12 ) } where ty12 : * x ≡ * (zπ1 (F⊆pxq fp (subst (odef (ZFP p Q)) (sym &iso) (ab-pair px qq )))) ty12 = begin * x ≡⟨ sym (cong (*) (ty32 px qq )) ⟩ * (zπ1 (F⊆pxq fp (subst (odef (ZFP p Q)) (sym &iso) (ab-pair px qq )))) ∎ where open ≡-Reasoning -- Projection of F FPSet : HOD FPSet = Replace' (filter F) (λ x fx → Replace' x ( λ y xy → * (zπ1 (F⊆pxq fx xy) )) ? ) ? -- Projection ⁻¹ F (which is in F) is in FPSet FPSet∋zpq : {q : HOD} → q ⊆ P → filter F ∋ ZFP q Q → FPSet ∋ q FPSet∋zpq {q} q⊆P fq = record { z = _ ; az = fq ; x=ψz = sym (cong (&) ty10) } where -- brain damaged one ty11 : {y : HOD} {xy : (* (& (ZFP q Q))) ∋ y } → * (zπ1 ( (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ≡ * (zπ1 ( (F⊆pxq fq (subst (λ k → odef k (& y)) *iso xy) ))) ty11 {y} {xy} = cong (λ k → * (zπ1 k)) ( HE.≅-to-≡ (∋-irr {ZFP P Q} a b )) where a = F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy b = F⊆pxq fq (subst (λ k → odef k (& y)) *iso xy) ty10 : Replace' (* (& (ZFP q Q))) (λ y xy → * (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ? ≡ q ty10 = begin Replace' (* (& (ZFP q Q))) (λ y xy → * (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ? ≡⟨ cong (λ k → Replace' (* (& (ZFP q Q))) k ? ) (f-extensionality (λ y → (f-extensionality (λ xy → ty11 {y} {xy})))) ⟩ Replace' (* (& (ZFP q Q))) (λ y xy → * (zπ1 (F⊆pxq fq (subst (λ k → odef k (& y)) *iso xy) ))) ? ≡⟨ Replace'-iso _ ? ? ? ⟩ Replace' (ZFP q Q) ( λ y xy → * (zπ1 (F⊆pxq fq xy) )) ? ≡⟨ refl ⟩ fx→px fq ≡⟨ fx→px1 aq fq ⟩ q ∎ where open ≡-Reasoning FPSet⊆PP : FPSet ⊆ Power P FPSet⊆PP {x} record { z = z ; az = fz ; x=ψz = x=ψz } w xw with subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso) xw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k) (sym (trans x=ψz1 &iso)) (zp1 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fz) (subst (λ k → odef (* z) k) (sym &iso) az1)) ) X=F1 : (x : Ordinal) (p : HOD) (fx : odef (filter F) x) → Set n X=F1 x p fx = & p ≡ & (Replace' (* x) (λ y xy → * (zπ1 (f⊆L F (subst (odef (filter F)) (sym &iso) fx) (& y) (subst (λ k → OD.def (HOD.od k) (& y)) (sym *iso) xy)))) ? ) x⊆pxq : {x : Ordinal} {p : HOD} (fx : odef (filter F) x) → X=F1 x p fx → * x ⊆ ZFP p Q x⊆pxq {x} {p} fx x=ψz {w} xw with F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) xw ... | ab-pair {a} {b} pw qw = ab-pair ty08 qw where ty21 : ZFProduct P Q (& (* (& < * a , * b >))) ty21 = subst (λ k → ZFProduct P Q k) (cong & (sym *iso)) (ab-pair pw qw) ty32 : ZFProduct P Q (& (* (& < * a , * b >))) ty32 = f⊆L F (subst (odef (filter F)) (sym &iso) fx) (& (* (& < * a , * b >))) (subst (λ k → odef k (& (* (& < * a , * b >)))) (sym *iso) (subst (odef (* x)) (sym &iso) xw)) ty07 : odef (* x) (& < * a , * b >) ty07 = xw ty08 : odef p a ty08 = subst (λ k → odef k a ) (subst₂ (λ j k → j ≡ k) *iso *iso (sym (cong (*) x=ψz))) record { z = _ ; az = xw ; x=ψz = sym (trans &iso (ty33 ty32 (cong (&) *iso ))) } where ty33 : {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ1 p ≡ a ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq)) ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) a=c) p⊆P : {x : Ordinal} {p : HOD} (fx : odef (filter F) x) → X=F1 x p fx → p ⊆ P p⊆P {x} {p} fx x=ψz {w} pw with subst (λ k → odef k w) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) x=ψz)) pw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k) (sym (trans x=ψz1 &iso)) (zp1 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) (subst (λ k → odef (* x) k) (sym &iso) az1 )) ) FP : Filter {Power P} {P} (λ x → x) FP = record { filter = FPSet ; f⊆L = FPSet⊆PP ; filter1 = ty01 ; filter2 = ty02 } where ty01 : {p q : HOD} → Power P ∋ q → FPSet ∋ p → p ⊆ q → FPSet ∋ q ty01 {p} {q} Pq record { z = x ; az = fx ; x=ψz = x=ψz } p⊆q = FPSet∋zpq q⊆P (ty10 ty05 ty06 ) where -- p ≡ (Replace' (* x) (λ y xy → (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) fx) xy)) -- x = ( px , qx ) , px ⊆ q ty03 : Power (ZFP P Q) ∋ ZFP q Q ty03 z zpq = isP→PxQ {* (& q)} (Pq _) ( subst (λ k → odef k z ) (trans *iso (cong (λ k → ZFP k Q) (sym *iso))) zpq ) q⊆P : q ⊆ P q⊆P {w} qw = Pq _ (subst (λ k → odef k w ) (sym *iso) qw ) ty05 : filter F ∋ ZFP p Q ty05 = filter1 F (λ z wz → isP→PxQ (p⊆P fx x=ψz) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fx) (x⊆pxq fx x=ψz) ty06 : ZFP p Q ⊆ ZFP q Q ty06 (ab-pair wp wq ) = ab-pair (p⊆q wp) wq ty10 : filter F ∋ ZFP p Q → ZFP p Q ⊆ ZFP q Q → filter F ∋ ZFP q Q ty10 fzp zp⊆zq = filter1 F ty03 fzp zp⊆zq ty02 : {p q : HOD} → FPSet ∋ p → FPSet ∋ q → Power P ∋ (p ∩ q) → FPSet ∋ (p ∩ q) ty02 {p} {q} record { z = zp ; az = fzp ; x=ψz = x=ψzp } record { z = zq ; az = fzq ; x=ψz = x=ψzq } Ppq = FPSet∋zpq (λ {z} xz → Ppq z (subst (λ k → odef k z) (sym *iso) xz )) ty50 where ty54 : Power (ZFP P Q) ∋ (ZFP p Q ∩ ZFP q Q) ty54 z xz = subst (λ k → ZFProduct P Q k ) (zp-iso pqz) (ab-pair pqz1 pqz2 ) where pqz : odef (ZFP (p ∩ q) Q) z pqz = subst (λ k → odef k z ) (trans *iso (sym (proj1 ZFP∩) )) xz pqz1 : odef P (zπ1 pqz) pqz1 = p⊆P fzp x=ψzp (proj1 (zp1 pqz)) pqz2 : odef Q (zπ2 pqz) pqz2 = zp2 pqz ty53 : filter F ∋ ZFP p Q ty53 = filter1 F (λ z wz → isP→PxQ (p⊆P fzp x=ψzp) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fzp ) (x⊆pxq fzp x=ψzp) ty52 : filter F ∋ ZFP q Q ty52 = filter1 F (λ z wz → isP→PxQ (p⊆P fzq x=ψzq) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fzq ) (x⊆pxq fzq x=ψzq) ty51 : filter F ∋ ( ZFP p Q ∩ ZFP q Q ) ty51 = filter2 F ty53 ty52 ty54 ty50 : filter F ∋ ZFP (p ∩ q) Q ty50 = subst (λ k → filter F ∋ k ) (sym (proj1 ZFP∩)) ty51 UFP : ultra-filter FP UFP = record { proper = ty61 ; ultra = ty60 } where ty61 : ¬ (FPSet ∋ od∅) ty61 record { z = z ; az = az ; x=ψz = x=ψz } = ultra-filter.proper UF ty62 where ty63 : {x : Ordinal} → ¬ odef (* z) x ty63 {x} zx with x⊆pxq az x=ψz zx ... | ab-pair xp xq = ¬x<0 xp ty62 : odef (filter F) (& od∅) ty62 = subst (λ k → odef (filter F) k ) (trans (sym &iso) (cong (&) (¬x∋y→x≡od∅ ty63)) ) az ty60 : {p : HOD} → Power P ∋ p → Power P ∋ (P ＼ p) → (FPSet ∋ p) ∨ (FPSet ∋ (P ＼ p)) ty60 {p} Pp Pnp with ultra-filter.ultra UF {ZFP p Q} (λ z xz → isP→PxQ (λ {x} lt → Pp _ (subst (λ k → odef k x) (sym *iso) lt)) (subst (λ k → odef k z) *iso xz)) (λ z xz → proj1 (subst (λ k → odef k z) *iso xz )) ... | case1 fp = case1 (FPSet∋zpq (λ {z} xz → Pp z (subst (λ k → odef k z) (sym *iso) xz )) fp ) ... | case2 fnp = case2 (FPSet∋zpq (λ pp → proj1 pp) (subst (λ k → odef (filter F) k) (cong (&) (proj1 ZFP\Q)) fnp )) uflp : UFLP TP FP UFP uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP -- FPSet is in Projection ⁻¹ F FPSet⊆F : {x : Ordinal } → odef FPSet x → odef (filter F) (& (ZFP (* x) Q)) FPSet⊆F {x} record { z = z ; az = az ; x=ψz = x=ψz } = filter1 F ty80 (subst (λ k → odef (filter F) k) (sym &iso) az) ty71 where Rx : HOD Rx = Replace' (* z) (λ y xy → * (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) az) xy))) ? RxQ∋z : * z ⊆ ZFP Rx Q RxQ∋z {w} zw = subst (λ k → ZFProduct Rx Q k ) ty70 ( ab-pair record { z = w ; az = zw ; x=ψz = refl } (zp2 b )) where a = F⊆pxq (subst (odef (filter F)) (sym &iso) az) (subst (odef (* z)) (sym &iso) zw) b = subst (λ k → odef (ZFP P Q) k ) (sym &iso) ( f⊆L F az w zw ) ty73 : & (* (zπ1 a)) ≡ zπ1 b ty73 = begin & (* (zπ1 a)) ≡⟨ &iso ⟩ zπ1 a ≡⟨ cong zπ1 (HE.≅-to-≡ (∋-irr {ZFP _ _ } a b)) ⟩ zπ1 b ∎ where open ≡-Reasoning ty70 : & < * (& (* (zπ1 a))) , * (zπ2 b) > ≡ w ty70 with zp-iso (subst (λ k → odef (ZFP P Q) k) (sym &iso) (f⊆L F az _ zw )) ... | eq = trans (cong₂ (λ j k → & < * j , k > ) ty73 refl ) (trans eq &iso ) ty71 : * z ⊆ ZFP (* x) Q ty71 = subst (λ k → * z ⊆ ZFP k Q) ty72 RxQ∋z where ty72 : Rx ≡ * x ty72 = begin Rx ≡⟨ sym *iso ⟩ * (& Rx) ≡⟨ cong (*) (sym x=ψz ) ⟩ * x ∎ where open ≡-Reasoning ty80 : Power (ZFP P Q) ∋ ZFP (* x) Q ty80 y yx = isP→PxQ ty81 (subst (λ k → odef k y ) *iso yx ) where ty81 : * x ⊆ P ty81 {w} xw with subst (λ k → odef k w) (trans (cong (*) x=ψz ) *iso ) xw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k) (sym ty84) ty87 where a = f⊆L F (subst (odef (filter F)) (sym &iso) az) (& (* z1)) (subst (λ k → odef k (& (* z1))) (sym *iso) (subst (odef (* z)) (sym &iso) az1)) b = subst (λ k → odef (ZFP P Q) k ) (sym &iso) (f⊆L F az _ az1 ) ty87 : odef P (zπ1 b) ty87 = zp1 b ty84 : w ≡ (zπ1 b) ty84 = begin w ≡⟨ trans x=ψz1 &iso ⟩ zπ1 a ≡⟨ cong zπ1 (HE.≅-to-≡ (∋-irr {ZFP _ _ } a b )) ⟩ zπ1 b ∎ where open ≡-Reasoning -- copy and pasted, sorry -- isQ→PxQ : {x : HOD} → (x⊆Q : x ⊆ Q ) → ZFP P x ⊆ ZFP P Q isQ→PxQ {x} x⊆Q (ab-pair p q) = ab-pair p (x⊆Q q) fx→qx : {x : HOD } → filter F ∋ x → HOD fx→qx {x} fx = Replace' x ( λ y xy → * (zπ2 (F⊆pxq fx xy) )) ? fx→qx1 : {q : HOD } {p : Ordinal } → odef P p → (fq : filter F ∋ ZFP P q ) → fx→qx fq ≡ q fx→qx1 {q} {p} qq fq = ==→o≡ record { eq→ = ty20 ; eq← = ty22 } where ty21 : {a b : Ordinal } → (qz : odef q b) → (pz : odef P a) → ZFProduct P Q (& (* (& < * a , * b >))) ty21 qz pz = F⊆pxq fq (subst (odef (ZFP P q)) (sym &iso) (ab-pair pz qz )) ty32 : {a b : Ordinal } → (qz : odef q b) → (pz : odef P a) → zπ2 (ty21 qz pz) ≡ b ty32 {a} {b} pz qz = ty33 (ty21 pz qz) (cong (&) *iso) where ty33 : {a b x : Ordinal } ( q : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ2 q ≡ b ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq)) ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) b=d) ty20 : {x : Ordinal} → odef (fx→qx fq) x → odef q x ty20 {x} record { z = _ ; az = ab-pair {a} {b} pz qz ; x=ψz = x=ψz } = subst (λ k → odef q k) b=x qz where ty24 : * x ≡ * b ty24 = begin * x ≡⟨ cong (*) x=ψz ⟩ _ ≡⟨ *iso ⟩ * (zπ2 (F⊆pxq fq (subst (odef (ZFP P q)) (sym &iso) (ab-pair pz qz)))) ≡⟨ cong (*) (ty32 qz pz) ⟩ * b ∎ where open ≡-Reasoning b=x : b ≡ x b=x = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (sym ty24)) ty22 : {x : Ordinal} → odef q x → odef (fx→qx fq) x ty22 {x} qx = record { z = _ ; az = ab-pair qq qx ; x=ψz = subst₂ (λ j k → j ≡ k) &iso refl (cong (&) ty12 ) } where ty12 : * x ≡ * (zπ2 (F⊆pxq fq (subst (odef (ZFP P q)) (sym &iso) (ab-pair qq qx )))) ty12 = begin * x ≡⟨ sym (cong (*) (ty32 qx qq )) ⟩ * (zπ2 (F⊆pxq fq (subst (odef (ZFP P q)) (sym &iso) (ab-pair qq qx )))) ∎ where open ≡-Reasoning FQSet : HOD FQSet = Replace' (filter F) (λ x fx → Replace' x ( λ y xy → * (zπ2 (F⊆pxq fx xy) )) ? ) ? FQSet∋zpq : {q : HOD} → q ⊆ Q → filter F ∋ ZFP P q → FQSet ∋ q FQSet∋zpq {q} q⊆P fq = record { z = _ ; az = fq ; x=ψz = sym (cong (&) ty10) } where -- brain damaged one ty11 : {y : HOD} {xy : (* (& (ZFP P q ))) ∋ y } → * (zπ2 ( (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ≡ * (zπ2 ( (F⊆pxq fq (subst (λ k → odef k (& y)) *iso xy) ))) ty11 {y} {xy} = cong (λ k → * (zπ2 k)) ( HE.≅-to-≡ (∋-irr {ZFP P Q} a b )) where a = F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy b = F⊆pxq fq (subst (λ k → odef k (& y)) *iso xy) ty10 : Replace' (* (& (ZFP P q ))) (λ y xy → * (zπ2 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ? ≡ q ty10 = begin Replace' (* (& (ZFP P q))) (λ y xy → * (zπ2 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy))) ? ≡⟨ Replace'-iso _ ? ? ? ⟩ Replace' (ZFP P q ) ( λ y xy → * (zπ2 (F⊆pxq fq xy) )) ? ≡⟨ refl ⟩ fx→qx fq ≡⟨ fx→qx1 ap fq ⟩ q ∎ where open ≡-Reasoning FQSet⊆PP : FQSet ⊆ Power Q FQSet⊆PP {x} record { z = z ; az = fz ; x=ψz = x=ψz } w xw with subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso) xw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef Q k) (sym (trans x=ψz1 &iso)) (zp2 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fz) (subst (λ k → odef (* z) k) (sym &iso) az1)) ) X=F2 : (x : Ordinal) (q : HOD) (fx : odef (filter F) x) → Set n X=F2 x q fx = & q ≡ & (Replace' (* x) (λ y xy → * (zπ2 (f⊆L F (subst (odef (filter F)) (sym &iso) fx) (& y) (subst (λ k → OD.def (HOD.od k) (& y)) (sym *iso) xy)))) ? ) x⊆qxq : {x : Ordinal} {q : HOD} (fx : odef (filter F) x) → X=F2 x q fx → * x ⊆ ZFP P q x⊆qxq {x} {p} fx x=ψz {w} xw with F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) xw ... | ab-pair {a} {b} pw qw = ab-pair pw ty08 where ty21 : ZFProduct P Q (& (* (& < * a , * b >))) ty21 = subst (λ k → ZFProduct P Q k) (cong & (sym *iso)) (ab-pair pw qw) ty32 : ZFProduct P Q (& (* (& < * a , * b >))) ty32 = f⊆L F (subst (odef (filter F)) (sym &iso) fx) (& (* (& < * a , * b >))) (subst (λ k → odef k (& (* (& < * a , * b >)))) (sym *iso) (subst (odef (* x)) (sym &iso) xw)) ty07 : odef (* x) (& < * a , * b >) ty07 = xw ty08 : odef p b ty08 = subst (λ k → odef k b ) (subst₂ (λ j k → j ≡ k) *iso *iso (sym (cong (*) x=ψz))) record { z = _ ; az = xw ; x=ψz = sym (trans &iso (ty33 ty32 (cong (&) *iso ))) } where ty33 : {a b x : Ordinal } ( p : ZFProduct P Q x ) → x ≡ & < * a , * b > → zπ2 p ≡ b ty33 {a} {b} (ab-pair {c} {d} zp zq) eq with prod-≡ (subst₂ (λ j k → j ≡ k) *iso *iso (cong (*) eq)) ... | ⟪ a=c , b=d ⟫ = subst₂ (λ j k → j ≡ k) &iso &iso (cong (&) b=d) q⊆Q : {x : Ordinal} {p : HOD} (fx : odef (filter F) x) → X=F2 x p fx → p ⊆ Q q⊆Q {x} {p} fx x=ψz {w} pw with subst (λ k → odef k w) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) x=ψz)) pw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef Q k) (sym (trans x=ψz1 &iso)) (zp2 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) (subst (λ k → odef (* x) k) (sym &iso) az1 )) ) FQ : Filter {Power Q} {Q} (λ x → x) FQ = record { filter = FQSet ; f⊆L = FQSet⊆PP ; filter1 = ty01 ; filter2 = ty02 } where ty01 : {p q : HOD} → Power Q ∋ q → FQSet ∋ p → p ⊆ q → FQSet ∋ q ty01 {p} {q} Pq record { z = x ; az = fx ; x=ψz = x=ψz } p⊆q = FQSet∋zpq q⊆P (ty10 ty05 ty06 ) where -- p ≡ (Replace' (* x) (λ y xy → (zπ2 (F⊆qxq (subst (odef (filter F)) (sym &iso) fx) xy)) -- x = ( px , qx ) , qx ⊆ q ty03 : Power (ZFP P Q) ∋ ZFP P q ty03 z zpq = isQ→PxQ {* (& q)} (Pq _) ( subst (λ k → odef k z ) (trans *iso (cong (λ k → ZFP P k ) (sym *iso))) zpq ) q⊆P : q ⊆ Q q⊆P {w} qw = Pq _ (subst (λ k → odef k w ) (sym *iso) qw ) ty05 : filter F ∋ ZFP P p ty05 = filter1 F (λ z wz → isQ→PxQ (q⊆Q fx x=ψz) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fx) (x⊆qxq fx x=ψz) ty06 : ZFP P p ⊆ ZFP P q ty06 (ab-pair wp wq ) = ab-pair wp (p⊆q wq) ty10 : filter F ∋ ZFP P p → ZFP P p ⊆ ZFP P q → filter F ∋ ZFP P q ty10 fzp zp⊆zq = filter1 F ty03 fzp zp⊆zq ty02 : {p q : HOD} → FQSet ∋ p → FQSet ∋ q → Power Q ∋ (p ∩ q) → FQSet ∋ (p ∩ q) ty02 {p} {q} record { z = zp ; az = fzp ; x=ψz = x=ψzp } record { z = zq ; az = fzq ; x=ψz = x=ψzq } Ppq = FQSet∋zpq (λ {z} xz → Ppq z (subst (λ k → odef k z) (sym *iso) xz )) ty50 where ty54 : Power (ZFP P Q) ∋ (ZFP P p ∩ ZFP P q ) ty54 z xz = subst (λ k → ZFProduct P Q k ) (zp-iso pqz) (ab-pair pqz1 pqz2 ) where pqz : odef (ZFP P (p ∩ q) ) z pqz = subst (λ k → odef k z ) (trans *iso (sym (proj2 ZFP∩) )) xz pqz1 : odef P (zπ1 pqz) pqz1 = zp1 pqz pqz2 : odef Q (zπ2 pqz) pqz2 = q⊆Q fzp x=ψzp (proj1 (zp2 pqz)) ty53 : filter F ∋ ZFP P p ty53 = filter1 F (λ z wz → isQ→PxQ (q⊆Q fzp x=ψzp) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fzp ) (x⊆qxq fzp x=ψzp) ty52 : filter F ∋ ZFP P q ty52 = filter1 F (λ z wz → isQ→PxQ (q⊆Q fzq x=ψzq) (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fzq ) (x⊆qxq fzq x=ψzq) ty51 : filter F ∋ ( ZFP P p ∩ ZFP P q ) ty51 = filter2 F ty53 ty52 ty54 ty50 : filter F ∋ ZFP P (p ∩ q) ty50 = subst (λ k → filter F ∋ k ) (sym (proj2 ZFP∩)) ty51 UFQ : ultra-filter FQ UFQ = record { proper = ty61 ; ultra = ty60 } where ty61 : ¬ (FQSet ∋ od∅) ty61 record { z = z ; az = az ; x=ψz = x=ψz } = ultra-filter.proper UF ty62 where ty63 : {x : Ordinal} → ¬ odef (* z) x ty63 {x} zx with x⊆qxq az x=ψz zx ... | ab-pair xp xq = ¬x<0 xq ty62 : odef (filter F) (& od∅) ty62 = subst (λ k → odef (filter F) k ) (trans (sym &iso) (cong (&) (¬x∋y→x≡od∅ ty63)) ) az ty60 : {p : HOD} → Power Q ∋ p → Power Q ∋ (Q ＼ p) → (FQSet ∋ p) ∨ (FQSet ∋ (Q ＼ p)) ty60 {q} Pp Pnp with ultra-filter.ultra UF {ZFP P q} (λ z xz → isQ→PxQ (λ {x} lt → Pp _ (subst (λ k → odef k x) (sym *iso) lt)) (subst (λ k → odef k z) *iso xz)) (λ z xz → proj1 (subst (λ k → odef k z) *iso xz )) ... | case1 fq = case1 (FQSet∋zpq (λ {z} xz → Pp z (subst (λ k → odef k z) (sym *iso) xz )) fq ) ... | case2 fnp = case2 (FQSet∋zpq (λ pp → proj1 pp) (subst (λ k → odef (filter F) k) (cong (&) (proj2 ZFP\Q)) fnp )) -- FQSet is in Projection ⁻¹ F FQSet⊆F : {x : Ordinal } → odef FQSet x → odef (filter F) (& (ZFP P (* x) )) FQSet⊆F {x} record { z = z ; az = az ; x=ψz = x=ψz } = filter1 F ty80 (subst (λ k → odef (filter F) k) (sym &iso) az) ty71 where Rx : HOD Rx = Replace' (* z) (λ y xy → * (zπ2 (F⊆pxq (subst (odef (filter F)) (sym &iso) az) xy))) ? PxRx∋z : * z ⊆ ZFP P Rx PxRx∋z {w} zw = subst (λ k → ZFProduct P Rx k ) ty70 ( ab-pair (zp1 b) record { z = w ; az = zw ; x=ψz = refl } ) where a = F⊆pxq (subst (odef (filter F)) (sym &iso) az) (subst (odef (* z)) (sym &iso) zw) b = subst (λ k → odef (ZFP P Q) k ) (sym &iso) ( f⊆L F az w zw ) ty73 : & (* (zπ2 a)) ≡ zπ2 b ty73 = begin & (* (zπ2 a)) ≡⟨ &iso ⟩ zπ2 a ≡⟨ cong zπ2 (HE.≅-to-≡ (∋-irr {ZFP _ _ } a b)) ⟩ zπ2 b ∎ where open ≡-Reasoning ty70 : & < * (zπ1 b) , * (& (* (zπ2 a))) > ≡ w ty70 with zp-iso (subst (λ k → odef (ZFP P Q) k) (sym &iso) (f⊆L F az _ zw )) ... | eq = trans (cong₂ (λ j k → & < j , * k > ) refl ty73 ) (trans eq &iso ) ty71 : * z ⊆ ZFP P (* x) ty71 = subst (λ k → * z ⊆ ZFP P k ) ty72 PxRx∋z where ty72 : Rx ≡ * x ty72 = begin Rx ≡⟨ sym *iso ⟩ * (& Rx) ≡⟨ cong (*) (sym x=ψz ) ⟩ * x ∎ where open ≡-Reasoning ty80 : Power (ZFP P Q) ∋ ZFP P (* x) ty80 y yx = isQ→PxQ ty81 (subst (λ k → odef k y ) *iso yx ) where ty81 : * x ⊆ Q ty81 {w} xw with subst (λ k → odef k w) (trans (cong (*) x=ψz ) *iso ) xw ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef Q k) (sym ty84) ty87 where a = f⊆L F (subst (odef (filter F)) (sym &iso) az) (& (* z1)) (subst (λ k → odef k (& (* z1))) (sym *iso) (subst (odef (* z)) (sym &iso) az1)) b = subst (λ k → odef (ZFP P Q) k ) (sym &iso) (f⊆L F az _ az1 ) ty87 : odef Q (zπ2 b) ty87 = zp2 b ty84 : w ≡ (zπ2 b) ty84 = begin w ≡⟨ trans x=ψz1 &iso ⟩ zπ2 a ≡⟨ cong zπ2 (HE.≅-to-≡ (∋-irr {ZFP _ _ } a b )) ⟩ zπ2 b ∎ where open ≡-Reasoning uflq : UFLP TQ FQ UFQ uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) Pf = ab-pair (UFLP.P∋limit uflp) (UFLP.P∋limit uflq) isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → filter F ∋ * v isL {v} nei = filter1 F (λ z xz → Neighbor.v⊆P nei (subst (λ k → odef k z) *iso xz)) (subst (λ k → odef (filter F) k) (sym &iso) (F∋base pqb b∋l )) bpq⊆v where -- -- Product Topolgy's open set contains a subbase which is an element of ZPF p Q or ZPF P q -- Neighbor of limit contains an open set which conatins a limit -- every point of an open set is covered by a subbase -- so there is a subbase which contains a limit, the subbase is an element of projection of a filter (P or Q) TPQ = ProductTopology TP TQ lim = & < * (UFLP.limit uflp) , * (UFLP.limit uflq) > bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u nei) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) bpq = Neighbor.ou nei (Neighbor.ux nei) b∋l : odef (* (Base.b bpq)) lim b∋l = Base.bx bpq pqb : Subbase (pbase TP TQ) (Base.b bpq ) pqb = Base.sb bpq pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u nei ) bpq⊆v : * (Base.b bpq) ⊆ * v bpq⊆v {x} bx = Neighbor.u⊆v nei (pqb⊆opq bx) pqb⊆opq = Base.b⊆u bpq F∋base : {b : Ordinal } → Subbase (pbase TP TQ) b → odef (* b) lim → odef (filter F) b F∋base {b} (gi (case1 px)) bl = subst (λ k → odef (filter F) k) (sym (BaseP.prod px)) f∋b where -- -- subbase of product topology which includes lim is in FP, so in F -- isl00 : odef (ZFP (* (BaseP.p px)) Q) lim isl00 = subst (λ k → odef k lim ) (trans (cong (*) (BaseP.prod px)) *iso ) bl px∋limit : odef (* (BaseP.p px)) (UFLP.limit uflp) px∋limit = isl01 isl00 refl where isl01 : {x : Ordinal } → odef (ZFP (* (BaseP.p px)) Q) x → x ≡ lim → odef (* (BaseP.p px)) (UFLP.limit uflp) isl01 (ab-pair {a} {b} bx qx) ab=lim = subst (λ k → odef (* (BaseP.p px)) k) a=lim bx where a=lim : a ≡ UFLP.limit uflp a=lim = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj1 ( prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ab=lim) ) ))) fp∋b : filter FP ∋ * (BaseP.p px) fp∋b = UFLP.is-limit uflp record { u = _ ; ou = BaseP.op px ; ux = px∋limit ; v⊆P = λ {x} lt → os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) (BaseP.op px)) lt ; u⊆v = λ x → x } f∋b : odef (filter F) (& (ZFP (* (BaseP.p px)) Q)) f∋b = FPSet⊆F (subst (λ k → odef (filter FP) k ) &iso fp∋b ) F∋base {b} (gi (case2 qx)) bl = subst (λ k → odef (filter F) k) (sym (BaseQ.prod qx)) f∋b where isl00 : odef (ZFP P (* (BaseQ.q qx))) lim isl00 = subst (λ k → odef k lim ) (trans (cong (*) (BaseQ.prod qx)) *iso ) bl qx∋limit : odef (* (BaseQ.q qx)) (UFLP.limit uflq) qx∋limit = isl01 isl00 refl where isl01 : {x : Ordinal } → odef (ZFP P (* (BaseQ.q qx)) ) x → x ≡ lim → odef (* (BaseQ.q qx)) (UFLP.limit uflq) isl01 (ab-pair {a} {b} px bx) ab=lim = subst (λ k → odef (* (BaseQ.q qx)) k) b=lim bx where b=lim : b ≡ UFLP.limit uflq b=lim = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj2 ( prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ab=lim) ) ))) fp∋b : filter FQ ∋ * (BaseQ.q qx) fp∋b = UFLP.is-limit uflq record { u = _ ; ou = BaseQ.oq qx ; ux = qx∋limit ; v⊆P = λ {x} lt → os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) (BaseQ.oq qx)) lt ; u⊆v = λ x → x } f∋b : odef (filter F) (& (ZFP P (* (BaseQ.q qx)) )) f∋b = FQSet⊆F (subst (λ k → odef (filter FQ) k ) &iso fp∋b ) F∋base (g∩ {x} {y} b1 b2) bl = F∋x∩y where -- filter contains finite intersection fb01 : odef (filter F) x fb01 = F∋base b1 (proj1 (subst (λ k → odef k lim) *iso bl)) fb02 : odef (filter F) y fb02 = F∋base b2 (proj2 (subst (λ k → odef k lim) *iso bl)) F∋x∩y : odef (filter F) (& (* x ∩ * y)) F∋x∩y = filter2 F (subst (λ k → odef (filter F) k) (sym &iso) fb01) (subst (λ k → odef (filter F) k) (sym &iso) fb02) (CAP (ZFP P Q) (subst (λ k → odef (Power (ZFP P Q)) k) (sym &iso) (f⊆L F fb01)) (subst (λ k → odef (Power (ZFP P Q)) k) (sym &iso) (f⊆L F fb02)))