Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Tychonoff.agda @ 1208:151f4c971a50
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 03 Mar 2023 19:44:29 +0900 |
| parents | 56d501cf0318 |
| children | 09e4b32ece2e |
| rev | line source |
|---|---|
| 1175 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 431 | 2 open import Level |
| 3 open import Ordinals | |
| 1170 | 4 module Tychonoff {n : Level } (O : Ordinals {n}) where |
| 431 | 5 |
| 6 open import zf | |
| 7 open import logic | |
| 8 open _∧_ | |
| 9 open _∨_ | |
| 10 open Bool | |
| 11 | |
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fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12 import OD |
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384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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13 open import Relation.Nullary |
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384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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14 open import Data.Empty |
| 431 | 15 open import Relation.Binary.Core |
| 1143 | 16 open import Relation.Binary.Definitions |
| 431 | 17 open import Relation.Binary.PropositionalEquality |
| 1124 | 18 import BAlgebra |
| 19 open BAlgebra O | |
| 431 | 20 open inOrdinal O |
| 21 open OD O | |
| 22 open OD.OD | |
| 23 open ODAxiom odAxiom | |
| 24 import OrdUtil | |
| 25 import ODUtil | |
| 26 open Ordinals.Ordinals O | |
| 27 open Ordinals.IsOrdinals isOrdinal | |
| 28 open Ordinals.IsNext isNext | |
| 29 open OrdUtil O | |
| 30 open ODUtil O | |
| 31 | |
| 32 import ODC | |
| 33 open ODC O | |
| 34 | |
| 1102 | 35 open import filter O |
| 1101 | 36 open import OPair O |
| 1170 | 37 open import Topology O |
| 1201 | 38 -- open import maximum-filter O |
| 431 | 39 |
| 1170 | 40 open Filter |
| 41 open Topology | |
| 1169 | 42 |
| 431 | 43 -- FIP is UFL |
| 44 | |
| 1159 | 45 -- filter Base |
| 1205 | 46 record FBase (P : HOD ) (X : Ordinal ) (u : Ordinal) : Set n where |
| 1153 | 47 field |
| 1161 | 48 b x : Ordinal |
| 1155 | 49 b⊆X : * b ⊆ * X |
| 1161 | 50 sb : Subbase (* b) x |
| 1158 | 51 u⊆P : * u ⊆ P |
| 1154 | 52 x⊆u : * x ⊆ * u |
| 1155 | 53 |
| 1170 | 54 record UFLP {P : HOD} (TP : Topology P) (F : Filter {Power P} {P} (λ x → x) ) |
| 55 (ultra : ultra-filter F ) : Set (suc (suc n)) where | |
| 56 field | |
| 57 limit : Ordinal | |
| 58 P∋limit : odef P limit | |
| 59 is-limit : {v : Ordinal} → Neighbor TP limit v → (* v) ⊆ filter F | |
| 1165 | 60 |
| 1161 | 61 UFLP→FIP : {P : HOD} (TP : Topology P) → |
| 1169 | 62 ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP |
| 1163 | 63 UFLP→FIP {P} TP uflp with trio< (& P) o∅ |
| 64 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 1201 | 65 ... | tri≈ ¬a P=0 ¬c = record { limit = λ CX fip → o∅ ; is-limit = fip03 } where |
| 1163 | 66 -- P is empty |
| 67 fip02 : {x : Ordinal } → ¬ odef P x | |
| 1201 | 68 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym P=0) (∈∅< Px) ) |
| 69 fip03 : {X : Ordinal} (CX : * X ⊆ CS TP) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) {x : Ordinal} → | |
| 70 odef (* X) x → odef (* x) o∅ | |
| 71 -- empty P case | |
| 72 -- if 0 < X then 0 < x ∧ P ∋ xfrom fip | |
| 73 -- if 0 ≡ X then ¬ odef X x | |
| 74 fip03 {X} CX fip {x} xx with trio< o∅ X | |
| 75 ... | tri< 0<X ¬b ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) P=0)) o∅≡od∅ ) (sym &iso) | |
| 76 ( cs⊆L TP (subst (λ k → odef (CS TP) k ) (sym &iso) (CX xx)) xe ))) where | |
| 77 0<x : o∅ o< x | |
| 78 0<x = fip (gi xx ) | |
| 79 e : HOD -- we have an element of x | |
| 80 e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 81 xe : odef (* x) (& e) | |
| 82 xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 83 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin | |
| 84 * X ≡⟨ cong (*) (sym 0=X) ⟩ | |
| 85 * o∅ ≡⟨ o∅≡od∅ ⟩ | |
| 86 od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning | |
| 87 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1204 | 88 ... | tri> ¬a ¬b 0<P = record { limit = λ CSX fip → limit CSX fip ; is-limit = uf01 } where |
| 1143 | 89 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 1187 | 90 fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x |
| 1154 | 91 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
| 1161 | 92 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
| 1159 | 93 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } |
| 1158 | 94 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
| 1159 | 95 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
| 1205 | 96 -- filter base is not empty necessary for generating ultra filter |
| 1204 | 97 nc : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD |
| 1205 | 98 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 |
| 1204 | 99 N∋nc :{X : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) |
| 100 → (fip : fip CSX) → odef (N CSX fip) (& (nc 0<X CSX fip) ) | |
| 1205 | 101 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 |
| 1201 | 102 e : HOD -- we have an element of X |
| 103 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 104 Xe : odef (* X) (& e) | |
| 105 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 106 nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) | |
| 1205 | 107 nn02 : * (& (nc 0<X CSX fip)) ⊆ P |
| 108 nn02 = subst (λ k → k ⊆ P ) (sym *iso) (cs⊆L TP (CSX Xe ) ) | |
| 109 | |
| 1165 | 110 0<PP : o∅ o< & (Power P) |
| 1201 | 111 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where |
| 112 nn00 : odef (Power P) o∅ | |
| 113 nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt | |
| 114 ... | x<0 = ⊥-elim ( ¬x<0 x<0) | |
| 1174 | 115 -- |
| 116 -- FIP defines a filter | |
| 117 -- | |
| 1158 | 118 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 119 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 120 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 1161 | 121 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
| 122 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 → | |
| 1158 | 123 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
| 124 f10 : q ⊆ P | |
| 125 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 126 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 127 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 | |
| 128 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 129 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 )) | |
| 1159 | 130 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 131 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 132 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 133 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 134 f20 : * (& Np+Nq) ⊆ * X |
| 135 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 136 ... | case1 np = FBase.b⊆X Np np |
| 137 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 138 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 1161 | 139 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
| 1159 | 140 → ⟪ 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)) ⟫ ) |
| 1174 | 141 -- |
| 1207 | 142 -- it contains no empty sets becase it is a finite intersection ( Subbase (* X) x ) |
| 1174 | 143 -- |
| 1207 | 144 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip {X} CSX) → ¬ (N CSX fip ∋ od∅) |
| 145 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 | |
| 146 x≤0 : x o< osuc o∅ | |
| 147 x≤0 = subst₂ (λ j k → j o< osuc k) &iso (trans (cong (&) *iso) ord-od∅ ) (⊆→o≤ (x⊆u)) | |
| 148 fp00 : (b x : Ordinal) → * b ⊆ * X → Subbase (* b) x → Subbase (* X) x | |
| 149 fp00 b y b<X (gi by ) = gi ( b<X by ) | |
| 150 fp00 b _ b<X (g∩ {y} {z} sy sz ) = g∩ (fp00 _ _ b<X sy) (fp00 _ _ b<X sz) | |
| 1174 | 151 -- |
| 152 -- then we have maximum ultra filter | |
| 153 -- | |
| 1158 | 154 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
| 1201 | 155 maxf {X} CSX fp = ? -- F→Maximum {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp) |
| 1170 | 156 mf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) |
| 157 mf {X} CSX fp = MaximumFilter.mf (maxf CSX fp) | |
| 1204 | 158 ultraf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf CSX fp) |
| 159 ultraf {X} 0<X CSX fp = ? -- F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp) | |
| 1174 | 160 -- |
| 161 -- so i has a limit as a limit of UIP | |
| 162 -- | |
| 1170 | 163 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1203 | 164 limit {X} CSX fp with trio< o∅ X |
| 1204 | 165 ... | tri< 0<X ¬b ¬c = UFLP.limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp)) |
| 166 ... | tri≈ ¬a 0=X ¬c = o∅ | |
| 1203 | 167 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1174 | 168 -- |
| 1201 | 169 -- the limit is an limit of entire elements of X |
| 1174 | 170 -- |
| 1170 | 171 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) |
| 1204 | 172 uf01 {X} CSX fp {x} xx with trio< o∅ X |
| 1206 | 173 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1205 | 174 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬a (subst (λ k → o∅ o< k) &iso ( ∈∅< xx ))) |
| 1206 | 175 ... | tri< 0<X ¬b ¬c with ∨L＼X {P} {* x} {UFLP.limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp))} |
| 176 (UFLP.P∋limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp))) | |
| 177 ... | case1 lt = lt -- odef (* x) y | |
| 178 ... | case2 nlxy = ⊥-elim (MaximumFilter.proper (maxf CSX fp) uf11 ) where | |
| 179 y = UFLP.limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp)) | |
| 1207 | 180 x⊆P : * x ⊆ P |
| 181 x⊆P = cs⊆L TP (CSX (subst (λ k → odef (* X) k) (sym &iso) xx)) | |
| 1206 | 182 uf10 : odef (P ＼ * x ) y |
| 183 uf10 = nlxy | |
| 184 uf03 : Neighbor TP y (& (P ＼ * x )) | |
| 185 uf03 = record { u = _ ; ou = P＼CS=OS TP (CSX (subst (λ k → odef (* X) k ) (sym &iso) xx)) | |
| 186 ; ux = subst (λ k → odef k y) (sym *iso) uf10 | |
| 187 ; v⊆P = λ {z} xz → proj1 (subst(λ k → odef k z) *iso xz ) | |
| 188 ; u⊆v = λ x → x } | |
| 1207 | 189 uf07 : * (& (* x , * x)) ⊆ * X |
| 190 uf07 {y} lt with subst (λ k → odef k y) *iso lt | |
| 191 ... | case1 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
| 192 ... | case2 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
| 1206 | 193 uf05 : odef (filter (MaximumFilter.mf (maxf CSX fp))) x |
| 1207 | 194 uf05 = MaximumFilter.F⊆mf (maxf CSX fp) record { b = & (* x , * x) ; b⊆X = uf07 |
| 195 ; sb = gi (subst (λ k → odef k x) (sym *iso) (case1 (sym &iso)) ) ; u⊆P = x⊆P ; x⊆u = λ x → x } | |
| 1206 | 196 uf06 : odef (filter (MaximumFilter.mf (maxf CSX fp))) (& (P ＼ * x )) |
| 197 uf06 = UFLP.is-limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp)) uf03 (subst (λ k → odef k y) (sym *iso) uf10) | |
| 198 uf13 : & ((* x) ∩ (P ＼ * x )) ≡ o∅ | |
| 1207 | 199 uf13 = subst₂ (λ j k → j ≡ k ) refl ord-od∅ (cong (&) ( ==→o≡ record { eq→ = uf14 ; eq← = λ {x} lt → ⊥-elim (¬x<0 lt) } ) ) where |
| 200 uf14 : {y : Ordinal} → odef (* x ∩ (P ＼ * x)) y → odef od∅ y | |
| 201 uf14 {y} ⟪ xy , ⟪ Px , ¬xy ⟫ ⟫ = ⊥-elim ( ¬xy xy ) | |
| 1206 | 202 uf12 : odef (Power P) (& ((* x) ∩ (P ＼ * x ))) |
| 1207 | 203 uf12 z pz with subst (λ k → odef k z) *iso pz |
| 204 ... | ⟪ xz , ⟪ Pz , ¬xz ⟫ ⟫ = Pz | |
| 1206 | 205 uf11 : filter (MaximumFilter.mf (maxf CSX fp)) ∋ od∅ |
| 206 uf11 = subst (λ k → odef (filter (MaximumFilter.mf (maxf CSX fp))) k ) uf13 | |
| 207 ( filter2 (MaximumFilter.mf (maxf CSX fp)) uf05 uf06 uf12 ) | |
| 1142 | 208 |
| 1208 | 209 x⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → x ⊆ Cl TP x |
| 210 x⊆Clx {P} TP {x} x<p {y} xy = ⟪ x<p xy , (λ c csc x<c → x<c xy ) ⟫ | |
| 211 P⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → Cl TP x ⊆ P | |
| 212 P⊆Clx {P} TP {x} x<p {y} xy = proj1 xy | |
| 213 | |
| 1158 | 214 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 1169 | 215 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF |
| 1208 | 216 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 = ? ; is-limit = ufl00 } where |
| 1174 | 217 -- |
| 218 -- take closure of given filter elements | |
| 219 -- | |
| 1160 | 220 CF : HOD |
| 1188 | 221 CF = Replace (filter F) (λ x → Cl TP x ) |
| 1160 | 222 CF⊆CS : CF ⊆ CS TP |
| 1162 | 223 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)) |
| 1174 | 224 -- |
| 225 -- it is set of closed set and has FIP ( F is proper ) | |
| 226 -- | |
| 1208 | 227 ufl08 : {z : Ordinal} → odef (Power P) (& (Cl TP (* z))) |
| 228 ufl08 {z} w zw with subst (λ k → odef k w ) *iso zw | |
| 229 ... | t = proj1 t | |
| 230 fx→px : {x : Ordinal} → odef (filter F) x → Power P ∋ * x | |
| 231 fx→px {x} fx z xz = f⊆L F fx _ (subst (λ k → odef k z) *iso xz ) | |
| 232 F∋sb : {x : Ordinal} → Subbase CF x → odef (filter F) x | |
| 233 F∋sb {x} (gi record { z = z ; az = az ; x=ψz = x=ψz }) = ufl07 where | |
| 234 ufl09 : * z ⊆ P | |
| 235 ufl09 {y} zy = f⊆L F az _ zy | |
| 236 ufl07 : odef (filter F) x | |
| 237 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 ) | |
| 238 (subst (λ k → odef (filter F) k) (sym &iso) az) | |
| 239 (subst (λ k → * z ⊆ k ) (trans (sym *iso) (sym (cong (*) x=ψz)) ) (x⊆Clx TP {* z} ufl09 ) )) | |
| 240 F∋sb (g∩ {x} {y} sx sy) = filter2 F (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sx)) | |
| 241 (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sy)) | |
| 242 (λ z xz → fx→px (F∋sb sx) _ (subst (λ k → odef k _) (sym *iso) (proj1 (subst (λ k → odef k z) *iso xz) ))) | |
| 1187 | 243 ufl01 : {x : Ordinal} → Subbase (* (& CF)) x → o∅ o< x |
| 1208 | 244 ufl01 {x} sb = ufl04 where |
| 245 ufl04 : o∅ o< x | |
| 246 ufl04 with trio< o∅ x | |
| 247 ... | tri< a ¬b ¬c = a | |
| 248 ... | tri≈ ¬a b ¬c = ⊥-elim ( ultra-filter.proper UF (subst (λ k → odef (filter F) k) ( | |
| 249 begin | |
| 250 x ≡⟨ sym b ⟩ | |
| 251 o∅ ≡⟨ sym ord-od∅ ⟩ | |
| 252 & od∅ ∎ ) (F∋sb (subst (λ k → Subbase k x) *iso sb )) )) where open ≡-Reasoning | |
| 253 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1174 | 254 -- |
| 255 -- so we have a limit | |
| 256 -- | |
| 1170 | 257 limit : Ordinal |
| 1208 | 258 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1170 | 259 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit |
| 1208 | 260 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1174 | 261 -- |
| 262 -- Neigbor of limit ⊆ Filter | |
| 263 -- | |
| 1208 | 264 ufl03 : {f v : Ordinal } → odef (filter F) f → Neighbor TP limit v → ¬ ( * f ∩ * v ) =h= od∅ -- because limit is in CF |
| 1170 | 265 ufl03 {f} {v} ff nei fv=0 = ? |
| 266 pp : {v x : Ordinal} → Neighbor TP limit v → odef (* v) x → Power P ∋ (* x) | |
| 1208 | 267 pp {v} {x} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; u⊆v = u⊆v } vx z pz = v⊆P ? |
| 1170 | 268 ufl00 : {v : Ordinal} → Neighbor TP limit v → * v ⊆ filter F |
| 269 ufl00 {v} nei {x} fx with ultra-filter.ultra UF (pp nei fx) (NEG P (pp nei fx)) | |
| 270 ... | case1 fv = subst (λ k → odef (filter F) k) &iso fv | |
| 1171 | 271 ... | case2 nfv = ? -- will contradicts ufl03 |
| 1163 | 272 |
| 1124 | 273 -- product topology of compact topology is compact |
| 431 | 274 |
| 1142 | 275 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 276 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1169 | 277 uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) |
| 278 → UFLP TP F UF | |
| 279 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF | |
| 280 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) | |
| 281 → UFLP TQ F UF | |
| 282 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF | |
| 1201 | 283 -- Product of UFL has a limit point |
| 1169 | 284 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) |
| 285 → UFLP (ProductTopology TP TQ) F UF | |
| 286 uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where | |
| 287 FP : Filter {Power P} {P} (λ x → x) | |
| 1164 | 288 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
| 1169 | 289 ty00 : Proj1 (filter F) (Power P) (Power Q) ⊆ Power P |
| 290 ty00 {x} ⟪ PPx , ppf ⟫ = PPx | |
| 1161 | 291 UFP : ultra-filter FP |
| 1159 | 292 UFP = record { proper = ? ; ultra = ? } |
| 1169 | 293 uflp : UFLP TP FP UFP |
| 294 uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP | |
| 1154 | 295 |
| 1169 | 296 FQ : Filter {Power Q} {Q} (λ x → x) |
| 1166 | 297 FQ = record { filter = Proj2 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
| 1169 | 298 ty00 : Proj2 (filter F) (Power P) (Power Q) ⊆ Power Q |
| 299 ty00 {x} ⟪ QPx , ppf ⟫ = QPx | |
| 1166 | 300 UFQ : ultra-filter FQ |
| 301 UFQ = record { proper = ? ; ultra = ? } | |
| 1169 | 302 uflq : UFLP TQ FQ UFQ |
| 303 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ | |
| 1154 | 304 |
| 1166 | 305 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 306 Pf = ? | |
| 1171 | 307 pq⊆F : {p q : HOD} → Neighbor TP (& p) (UFLP.limit uflp) → Neighbor TP (& q) (UFLP.limit uflq) → ? ⊆ filter F |
| 1170 | 308 pq⊆F = ? |
| 309 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → * v ⊆ filter F | |
| 1173 | 310 isL {v} npq {x} fx = ? where |
| 1172 | 311 bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u npq) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 312 bpq = Neighbor.ou npq (Neighbor.ux npq) | |
| 313 pqb : Subbase (pbase TP TQ) (Base.b bpq ) | |
| 314 pqb = Base.sb bpq | |
| 1173 | 315 pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u npq ) |
| 316 pqb⊆opq = Base.b⊆u bpq | |
| 317 base⊆F : {b : Ordinal } → Subbase (pbase TP TQ) b → * b ⊆ * (Neighbor.u npq) → * b ⊆ filter F | |
| 318 base⊆F (gi (case1 px)) b⊆u {z} bz = fz where | |
| 319 -- F contains no od∅, because it projection contains no od∅ | |
| 320 -- x is an element of BaseP, which is a subset of Neighbor npq | |
| 321 -- x is also an elment of Proj1 F because Proj1 F has UFLP (uflp) | |
| 322 -- BaseP ∩ F is not empty | |
| 323 -- (Base P ∩ F) ⊆ F , (Base P ) ⊆ F , | |
| 324 il1 : odef (Power P) z ∧ ZProj1 (filter F) z | |
| 325 il1 = UFLP.is-limit uflp ? bz | |
| 326 nei1 : HOD | |
| 327 nei1 = Proj1 (* (Neighbor.u npq)) (Power P) (Power Q) | |
| 328 plimit : Ordinal | |
| 1174 | 329 plimit = UFLP.limit uflp |
| 1173 | 330 nproper : {b : Ordinal } → * b ⊆ nei1 → o∅ o< b |
| 331 nproper = ? | |
| 332 b∋z : odef nei1 z | |
| 333 b∋z = ? | |
| 334 bp : BaseP {P} TP Q z | |
| 1187 | 335 bp = record { p = ? ; op = ? ; prod = ? } |
| 1173 | 336 neip : {p : Ordinal } → ( bp : BaseP TP Q p ) → * p ⊆ filter F |
| 337 neip = ? | |
| 338 il2 : * z ⊆ ZFP (Power P) (Power Q) | |
| 339 il2 = ? | |
| 340 il3 : filter F ∋ ? | |
| 341 il3 = ? | |
| 342 fz : odef (filter F) z | |
| 343 fz = subst (λ k → odef (filter F) k) &iso (filter1 F ? ? ?) | |
| 344 base⊆F (gi (case2 qx)) b⊆u {z} bz = ? | |
| 345 base⊆F (g∩ b1 b2) b⊆u {z} bz = ? | |
| 1154 | 346 |
| 1170 | 347 |
| 348 | |
| 349 | |
| 350 | |
| 351 |