Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Tychonoff.agda @ 1332:87df366f85f3
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 13 Jun 2023 12:28:04 +0900 |
| parents | 47d3cc596d68 |
| children | ecfc24f53df4 |
| 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 logic | |
| 7 open _∧_ | |
| 8 open _∨_ | |
| 9 open Bool | |
| 10 | |
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11 import OD |
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12 open import Relation.Nullary |
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13 open import Data.Empty |
| 431 | 14 open import Relation.Binary.Core |
| 1143 | 15 open import Relation.Binary.Definitions |
| 431 | 16 open import Relation.Binary.PropositionalEquality |
| 1124 | 17 import BAlgebra |
| 18 open BAlgebra O | |
| 431 | 19 open inOrdinal O |
| 20 open OD O | |
| 21 open OD.OD | |
| 22 open ODAxiom odAxiom | |
| 23 import OrdUtil | |
| 24 import ODUtil | |
| 25 open Ordinals.Ordinals O | |
| 26 open Ordinals.IsOrdinals isOrdinal | |
| 1300 | 27 -- open Ordinals.IsNext isNext |
| 431 | 28 open OrdUtil O |
| 29 open ODUtil O | |
| 30 | |
| 31 import ODC | |
| 32 open ODC O | |
| 33 | |
| 1102 | 34 open import filter O |
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35 open import filter-util O |
| 1218 | 36 open import ZProduct O |
| 1170 | 37 open import Topology O |
| 1293 | 38 -- open import maximum-filter O |
| 431 | 39 |
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40 open Filter |
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41 open Topology |
| 1169 | 42 |
| 1237 | 43 -- |
| 44 -- Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) | |
| 45 -- → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) | |
| 46 -- | |
| 47 -- ULFP : Compact <=> Every ultra filter F have a limit i.e. open sets which contains the limit (Neighbor limit) is in F | |
| 48 -- | |
| 49 -- Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where | |
| 50 -- | |
| 51 -- FP FQ : create projections of a filter F, so it is ULFP | |
| 52 -- | |
| 53 -- Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) | |
| 54 -- | |
| 55 -- the product of each limits must be a limit of ultra filter F | |
| 56 -- | |
| 57 -- 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 | |
| 58 -- so (P x q) ∋ limit ∨ (q x P) ∋ limit. P x q ⊆ nei , so nei ∋ limit | |
| 59 -- | |
| 60 -- uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) | |
| 61 -- → UFLP (ProductTopology TP TQ) F UF | |
| 62 -- | |
| 63 -- QED. | |
| 64 | |
| 431 | 65 -- FIP is UFL |
| 66 | |
| 1159 | 67 -- filter Base |
| 1205 | 68 record FBase (P : HOD ) (X : Ordinal ) (u : Ordinal) : Set n where |
| 1153 | 69 field |
| 1161 | 70 b x : Ordinal |
| 1155 | 71 b⊆X : * b ⊆ * X |
| 1161 | 72 sb : Subbase (* b) x |
| 1158 | 73 u⊆P : * u ⊆ P |
| 1154 | 74 x⊆u : * x ⊆ * u |
| 1155 | 75 |
| 1170 | 76 record UFLP {P : HOD} (TP : Topology P) (F : Filter {Power P} {P} (λ x → x) ) |
| 77 (ultra : ultra-filter F ) : Set (suc (suc n)) where | |
| 78 field | |
| 79 limit : Ordinal | |
| 80 P∋limit : odef P limit | |
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81 is-limit : {v : Ordinal} → Neighbor TP limit v → filter F ∋ (* v) |
| 1165 | 82 |
| 1283 | 83 -- If any ultrafilter has a limit such that all its neighbors |
| 84 -- are within the filter, it possesses the finite intersection | |
| 85 -- property (FIP). | |
| 86 | |
| 87 -- The finite intersection property defines a filter, and through Zorn's lemma, | |
| 88 -- we can maximize it to obtain an ultrafilter. | |
| 89 -- If the limit of the filter is not contained within a closed | |
| 90 -- set 'p' in the FIP, then it must be in the complement of 'p' | |
| 91 -- (P \ p). Since this complement is open and contains the limit, | |
| 92 -- it is included in the ultrafilter. However, this implies that | |
| 93 -- both 'p' and its complement (P \ p) are present in the filter, | |
| 94 -- which contradicts the proper characteristic of the ultrafilter, | |
| 95 -- meaning that the filter contains no empty set. | |
| 96 | |
| 1279 | 97 -- |
| 1161 | 98 UFLP→FIP : {P : HOD} (TP : Topology P) → |
| 1169 | 99 ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP |
| 1163 | 100 UFLP→FIP {P} TP uflp with trio< (& P) o∅ |
| 101 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 1201 | 102 ... | tri≈ ¬a P=0 ¬c = record { limit = λ CX fip → o∅ ; is-limit = fip03 } where |
| 1237 | 103 -- P is empty ( this case cannot happen because ulfp → 0 < P ) |
| 1163 | 104 fip02 : {x : Ordinal } → ¬ odef P x |
| 1201 | 105 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym P=0) (∈∅< Px) ) |
| 106 fip03 : {X : Ordinal} (CX : * X ⊆ CS TP) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) {x : Ordinal} → | |
| 107 odef (* X) x → odef (* x) o∅ | |
| 108 -- empty P case | |
| 109 -- if 0 < X then 0 < x ∧ P ∋ xfrom fip | |
| 110 -- if 0 ≡ X then ¬ odef X x | |
| 111 fip03 {X} CX fip {x} xx with trio< o∅ X | |
| 112 ... | tri< 0<X ¬b ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) P=0)) o∅≡od∅ ) (sym &iso) | |
| 113 ( cs⊆L TP (subst (λ k → odef (CS TP) k ) (sym &iso) (CX xx)) xe ))) where | |
| 114 0<x : o∅ o< x | |
| 115 0<x = fip (gi xx ) | |
| 116 e : HOD -- we have an element of x | |
| 117 e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 118 xe : odef (* x) (& e) | |
| 119 xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 120 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin | |
| 121 * X ≡⟨ cong (*) (sym 0=X) ⟩ | |
| 122 * o∅ ≡⟨ o∅≡od∅ ⟩ | |
| 123 od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning | |
| 124 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1204 | 125 ... | tri> ¬a ¬b 0<P = record { limit = λ CSX fip → limit CSX fip ; is-limit = uf01 } where |
| 1143 | 126 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 1187 | 127 fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x |
| 1154 | 128 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
| 1161 | 129 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
| 1159 | 130 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } |
| 1158 | 131 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
| 1159 | 132 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
| 1237 | 133 -- filter base is not empty, it is necessary to maximize fip filter |
| 1204 | 134 nc : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD |
| 1205 | 135 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 |
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136 N∋nc :{X : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) |
| 1204 | 137 → (fip : fip CSX) → odef (N CSX fip) (& (nc 0<X CSX fip) ) |
| 1205 | 138 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 | 139 e : HOD -- we have an element of X |
| 140 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 141 Xe : odef (* X) (& e) | |
| 142 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
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143 nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) |
| 1205 | 144 nn02 : * (& (nc 0<X CSX fip)) ⊆ P |
| 145 nn02 = subst (λ k → k ⊆ P ) (sym *iso) (cs⊆L TP (CSX Xe ) ) | |
| 146 | |
| 1237 | 147 0<PP : o∅ o< & (Power P) -- Power P contaisn od∅, so it is not empty |
| 1201 | 148 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where |
| 149 nn00 : odef (Power P) o∅ | |
| 150 nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt | |
| 151 ... | x<0 = ⊥-elim ( ¬x<0 x<0) | |
| 1174 | 152 -- |
| 153 -- FIP defines a filter | |
| 154 -- | |
| 1158 | 155 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 156 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 157 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 1161 | 158 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
| 159 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 | 160 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
| 161 f10 : q ⊆ P | |
| 162 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 163 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 164 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 | |
| 165 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 166 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 | 167 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 168 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 169 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 170 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 171 f20 : * (& Np+Nq) ⊆ * X |
| 172 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 173 ... | case1 np = FBase.b⊆X Np np |
| 174 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 175 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 1161 | 176 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
| 1159 | 177 → ⟪ 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 | 178 -- |
| 1207 | 179 -- it contains no empty sets becase it is a finite intersection ( Subbase (* X) x ) |
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180 -- |
| 1207 | 181 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip {X} CSX) → ¬ (N CSX fip ∋ od∅) |
| 182 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 | |
| 183 x≤0 : x o< osuc o∅ | |
| 184 x≤0 = subst₂ (λ j k → j o< osuc k) &iso (trans (cong (&) *iso) ord-od∅ ) (⊆→o≤ (x⊆u)) | |
| 185 fp00 : (b x : Ordinal) → * b ⊆ * X → Subbase (* b) x → Subbase (* X) x | |
| 186 fp00 b y b<X (gi by ) = gi ( b<X by ) | |
| 187 fp00 b _ b<X (g∩ {y} {z} sy sz ) = g∩ (fp00 _ _ b<X sy) (fp00 _ _ b<X sz) | |
| 1174 | 188 -- |
| 1237 | 189 -- then we have maximum ultra filter ( Zorn lemma ) |
| 1239 | 190 -- to debug this file, commet out the maximum filter and open import |
| 1237 | 191 -- otherwise the check requires a minute |
| 1174 | 192 -- |
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193 maxf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
| 1293 | 194 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) |
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195 mf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) |
| 1293 | 196 mf {X} 0<X CSX fp = ? -- MaximumFilter.mf (maxf 0<X CSX fp) |
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197 ultraf : {X : Ordinal} → (0<X : o∅ o< X ) → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf 0<X CSX fp) |
| 1293 | 198 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) |
| 1174 | 199 -- |
| 1279 | 200 -- so it has a limit as a limit of FIP |
| 1174 | 201 -- |
| 1170 | 202 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1203 | 203 limit {X} CSX fp with trio< o∅ X |
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204 ... | tri< 0<X ¬b ¬c = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) |
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205 ... | tri≈ ¬a 0=X ¬c = o∅ |
| 1203 | 206 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1174 | 207 -- |
| 1201 | 208 -- the limit is an limit of entire elements of X |
| 1174 | 209 -- |
| 1170 | 210 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) |
| 1204 | 211 uf01 {X} CSX fp {x} xx with trio< o∅ X |
| 1206 | 212 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1205 | 213 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬a (subst (λ k → o∅ o< k) &iso ( ∈∅< xx ))) |
| 1237 | 214 -- 0<X limit is in * x or P ＼ * x |
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215 ... | tri< 0<X ¬b ¬c with ∨L＼X {P} {* x} {UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))} |
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216 (UFLP.P∋limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))) |
| 1206 | 217 ... | case1 lt = lt -- odef (* x) y |
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218 ... | case2 nlxy = ⊥-elim (MaximumFilter.proper (maxf 0<X CSX fp) uf11 ) where |
| 1237 | 219 -- |
| 220 -- if (* x) do not conatins a limit, P ＼ * x contains it, (P ＼ * x) is open so it is the maxf ( UFLP.is-limit ) | |
| 221 -- UFLP contains (* x) and P ＼ * x, it contains od∅, contradicts the proper | |
| 222 -- | |
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223 y = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) |
| 1207 | 224 x⊆P : * x ⊆ P |
| 225 x⊆P = cs⊆L TP (CSX (subst (λ k → odef (* X) k) (sym &iso) xx)) | |
| 1206 | 226 uf10 : odef (P ＼ * x ) y |
| 227 uf10 = nlxy | |
| 228 uf03 : Neighbor TP y (& (P ＼ * x )) | |
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229 uf03 = record { u = _ ; ou = P＼CS=OS TP (CSX (subst (λ k → odef (* X) k ) (sym &iso) xx)) |
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230 ; ux = subst (λ k → odef k y) (sym *iso) uf10 |
| 1206 | 231 ; v⊆P = λ {z} xz → proj1 (subst(λ k → odef k z) *iso xz ) |
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232 ; u⊆v = λ x → x } |
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233 uf07 : * (& (* x , * x)) ⊆ * X |
| 1207 | 234 uf07 {y} lt with subst (λ k → odef k y) *iso lt |
| 235 ... | case1 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
| 236 ... | case2 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
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237 uf05 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) x |
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238 uf05 = MaximumFilter.F⊆mf (maxf 0<X CSX fp) record { b = & (* x , * x) ; b⊆X = uf07 |
| 1207 | 239 ; sb = gi (subst (λ k → odef k x) (sym *iso) (case1 (sym &iso)) ) ; u⊆P = x⊆P ; x⊆u = λ x → x } |
| 1213 | 240 uf061 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (* (& (P ＼ * x )))) |
| 241 uf061 = UFLP.is-limit ( uflp (mf 0<X CSX fp) (ultraf 0<X CSX fp)) {& (P ＼ * x)} uf03 | |
| 242 -- uf06 (same as uf061) have yellow if zorn lemma is not imported | |
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243 uf06 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (P ＼ * x )) |
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244 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 ) |
| 1206 | 245 uf13 : & ((* x) ∩ (P ＼ * x )) ≡ o∅ |
| 1207 | 246 uf13 = subst₂ (λ j k → j ≡ k ) refl ord-od∅ (cong (&) ( ==→o≡ record { eq→ = uf14 ; eq← = λ {x} lt → ⊥-elim (¬x<0 lt) } ) ) where |
| 247 uf14 : {y : Ordinal} → odef (* x ∩ (P ＼ * x)) y → odef od∅ y | |
| 248 uf14 {y} ⟪ xy , ⟪ Px , ¬xy ⟫ ⟫ = ⊥-elim ( ¬xy xy ) | |
| 1206 | 249 uf12 : odef (Power P) (& ((* x) ∩ (P ＼ * x ))) |
| 1207 | 250 uf12 z pz with subst (λ k → odef k z) *iso pz |
| 251 ... | ⟪ xz , ⟪ Pz , ¬xz ⟫ ⟫ = Pz | |
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252 uf11 : filter (MaximumFilter.mf (maxf 0<X CSX fp)) ∋ od∅ |
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253 uf11 = subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k ) (trans uf13 (sym ord-od∅)) |
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254 ( 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 ) |
| 1142 | 255 |
| 1208 | 256 x⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → x ⊆ Cl TP x |
| 257 x⊆Clx {P} TP {x} x<p {y} xy = ⟪ x<p xy , (λ c csc x<c → x<c xy ) ⟫ | |
| 258 P⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → Cl TP x ⊆ P | |
| 259 P⊆Clx {P} TP {x} x<p {y} xy = proj1 xy | |
| 260 | |
| 1279 | 261 -- |
| 262 -- Finite intersection property implies that any ultra filter have a limit, that is, neighbors of the limit is in the filter. | |
| 263 -- | |
| 264 -- 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. | |
| 265 -- 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. | |
| 266 -- If it is in P \ p, it cannot contains the limit, contradiction. | |
| 267 -- | |
| 1158 | 268 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 1169 | 269 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF |
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270 FIP→UFLP {P} TP fip F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1209 | 271 ; P∋limit = ufl10 ; is-limit = ufl00 } where |
| 272 F∋P : odef (filter F) (& P) | |
| 273 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 ) ) | |
| 274 ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) | |
| 275 ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where | |
| 276 fl20 : (P ＼ Ord o∅) =h= P | |
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277 fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } |
| 1209 | 278 0<P : o∅ o< & P |
| 279 0<P with trio< o∅ (& P) | |
| 280 ... | tri< a ¬b ¬c = a | |
| 281 ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋P) ) | |
| 282 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1174 | 283 -- |
| 284 -- take closure of given filter elements | |
| 285 -- | |
| 1160 | 286 CF : HOD |
| 1293 | 287 CF = Replace (filter F) (λ x → Cl TP x ) {P} record { ≤COD = λ {z} {y} lt → proj1 lt } |
| 1160 | 288 CF⊆CS : CF ⊆ CS TP |
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289 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 | 290 -- |
| 291 -- it is set of closed set and has FIP ( F is proper ) | |
| 292 -- | |
| 1208 | 293 ufl08 : {z : Ordinal} → odef (Power P) (& (Cl TP (* z))) |
| 294 ufl08 {z} w zw with subst (λ k → odef k w ) *iso zw | |
| 295 ... | t = proj1 t | |
| 296 fx→px : {x : Ordinal} → odef (filter F) x → Power P ∋ * x | |
| 297 fx→px {x} fx z xz = f⊆L F fx _ (subst (λ k → odef k z) *iso xz ) | |
| 298 F∋sb : {x : Ordinal} → Subbase CF x → odef (filter F) x | |
| 299 F∋sb {x} (gi record { z = z ; az = az ; x=ψz = x=ψz }) = ufl07 where | |
| 300 ufl09 : * z ⊆ P | |
| 301 ufl09 {y} zy = f⊆L F az _ zy | |
| 302 ufl07 : odef (filter F) x | |
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303 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 ) |
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304 (subst (λ k → odef (filter F) k) (sym &iso) az) |
| 1208 | 305 (subst (λ k → * z ⊆ k ) (trans (sym *iso) (sym (cong (*) x=ψz)) ) (x⊆Clx TP {* z} ufl09 ) )) |
| 306 F∋sb (g∩ {x} {y} sx sy) = filter2 F (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sx)) | |
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307 (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sy)) |
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308 (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 | 309 ufl01 : {x : Ordinal} → Subbase (* (& CF)) x → o∅ o< x |
| 1208 | 310 ufl01 {x} sb = ufl04 where |
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311 ufl04 : o∅ o< x |
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312 ufl04 with trio< o∅ x |
| 1208 | 313 ... | tri< a ¬b ¬c = a |
| 314 ... | tri≈ ¬a b ¬c = ⊥-elim ( ultra-filter.proper UF (subst (λ k → odef (filter F) k) ( | |
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315 begin |
| 1208 | 316 x ≡⟨ sym b ⟩ |
| 317 o∅ ≡⟨ sym ord-od∅ ⟩ | |
| 318 & od∅ ∎ ) (F∋sb (subst (λ k → Subbase k x) *iso sb )) )) where open ≡-Reasoning | |
| 319 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1209 | 320 ufl10 : odef P (FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01) |
| 321 ufl10 = FIP.L∋limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 {& P} ufl11 where | |
| 322 ufl11 : odef (* (& CF)) (& P) | |
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323 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))) } |
| 1174 | 324 -- |
| 325 -- so we have a limit | |
| 326 -- | |
| 1170 | 327 limit : Ordinal |
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328 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1170 | 329 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit |
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330 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1174 | 331 -- |
| 332 -- Neigbor of limit ⊆ Filter | |
| 333 -- | |
| 1210 | 334 -- if odef (* X) x, Cl TP x contains limit (ufl02) |
| 335 -- in (nei : Neighbor TP limit v) , there is an open Set u which contains the limit | |
| 336 -- F contains u or P ＼ u because F is ultra | |
| 337 -- if F ∋ u, then F ∋ v from u ⊆ v which is a propetery of Neighbor | |
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338 -- if F ∋ P ＼ u, it is a closed set (Cl (P ＼ u) ≡ P ＼ u) and it does not contains the limit |
| 1210 | 339 -- this contradicts ufl02 |
| 340 pp : {v : Ordinal} → (nei : Neighbor TP limit v ) → Power P ∋ (* (Neighbor.u nei)) | |
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341 pp {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; u⊆v = u⊆v } z pz |
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342 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) ou ) (subst (λ k → odef k z) *iso pz ) |
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343 ufl00 : {v : Ordinal} → Neighbor TP limit v → filter F ∋ * v |
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344 ufl00 {v} nei with ultra-filter.ultra UF (pp nei ) (NEG P (pp nei )) |
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345 ... | case1 fu = subst (λ k → odef (filter F) k) &iso |
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346 ( 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 |
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347 u = * (Neighbor.u nei) |
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348 px : Power P ∋ * v |
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349 px z vz = Neighbor.v⊆P nei (subst (λ k → odef k z) *iso vz ) |
| 1210 | 350 ... | case2 nfu = ⊥-elim ( ¬P\u∋limit P\u∋limit ) where |
| 351 u = * (Neighbor.u nei) | |
| 352 P\u : HOD | |
| 353 P\u = P ＼ u | |
| 354 P\u∋limit : odef P\u limit | |
| 355 P\u∋limit = subst (λ k → odef k limit) *iso ( ufl02 (subst (λ k → odef k (& P\u)) (sym *iso) ufl03 )) where | |
| 356 ufl04 : & P\u ≡ & (Cl TP (* (& P\u))) | |
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357 ufl04 = cong (&) (sym (trans (cong (Cl TP) *iso) |
| 1210 | 358 (CS∋x→Clx=x TP (P＼OS=CS TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou nei) ))))) |
| 359 ufl03 : odef CF (& P\u ) | |
| 360 ufl03 = record { z = _ ; az = nfu ; x=ψz = ufl04 } | |
| 361 ¬P\u∋limit : ¬ odef P\u limit | |
| 362 ¬P\u∋limit ⟪ Pl , nul ⟫ = nul ( Neighbor.ux nei ) | |
| 1163 | 363 |
| 1124 | 364 -- product topology of compact topology is compact |
| 431 | 365 |
| 1218 | 366 import Axiom.Extensionality.Propositional |
| 367 postulate f-extensionality : { n m : Level} → Axiom.Extensionality.Propositional.Extensionality n m | |
| 368 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) | |
| 369 | |
| 1279 | 370 -- FilterQP : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) |
| 371 -- → Filter {Power (ZFP Q P)} {ZFP Q P} (λ x → x) | |
| 372 -- FilterQP {P} {Q} F = record { filter = ? ; f⊆L = ? ; filter1 = ? ; filter2 = ? } | |
| 373 -- | |
| 374 -- projection-of-filter : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) | |
| 375 -- → Filter {Power P} {P} (λ x → x) | |
| 376 -- projection-of-filter = ? | |
| 377 -- | |
| 378 -- projection-of-ultra-filter : {P Q : HOD } → (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) | |
| 379 -- → ultra-filter (projection-of-filter F) | |
| 380 -- projection-of-ultra-filter = ? | |
| 1274 | 381 |
| 1279 | 382 -- |
| 383 -- 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 | |
| 384 -- 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, | |
| 385 -- 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 | |
| 386 -- property of the filter. | |
| 387 -- | |
| 1142 | 388 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 389 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1169 | 390 uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) |
| 391 → UFLP TP F UF | |
| 392 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF | |
| 393 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) | |
| 394 → UFLP TQ F UF | |
| 395 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF | |
| 1201 | 396 -- Product of UFL has a limit point |
| 1169 | 397 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) |
| 398 → UFLP (ProductTopology TP TQ) F UF | |
| 399 uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where | |
| 1213 | 400 F∋PQ : odef (filter F) (& (ZFP P Q)) |
| 401 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 ) ) | |
| 402 ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) | |
| 403 ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where | |
| 404 fl20 : (ZFP P Q ＼ Ord o∅) =h= ZFP P Q | |
| 405 fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } | |
| 1218 | 406 0<PQ : o∅ o< & (ZFP P Q) |
| 407 0<PQ with trio< o∅ (& (ZFP P Q)) | |
| 1213 | 408 ... | tri< a ¬b ¬c = a |
| 409 ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋PQ) ) | |
| 410 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1218 | 411 apq : HOD |
| 412 apq = ODC.minimal O (ZFP P Q) (0<P→ne 0<PQ) | |
| 413 is-apq : ZFP P Q ∋ apq | |
| 414 is-apq = ODC.x∋minimal O (ZFP P Q) (0<P→ne 0<PQ) | |
| 415 ap : odef P ( zπ1 is-apq ) | |
| 416 ap = zp1 is-apq | |
| 417 aq : odef Q ( zπ2 is-apq ) | |
| 418 aq = zp2 is-apq | |
| 1223 | 419 F⊆pxq : {x : HOD } → filter F ∋ x → x ⊆ ZFP P Q |
| 420 F⊆pxq {x} fx {y} xy = f⊆L F fx y (subst (λ k → odef k y) (sym *iso) xy) | |
| 421 | |
| 1229 | 422 --- |
| 423 --- FP is a P-projection of F, which is a ultra filter | |
| 424 --- | |
| 1169 | 425 FP : Filter {Power P} {P} (λ x → x) |
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426 FP = Filter-Proj1 {P} {Q} is-apq F |
| 1161 | 427 UFP : ultra-filter FP |
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428 UFP = Filter-Proj1-UF {P} {Q} is-apq F UF |
| 1169 | 429 uflp : UFLP TP FP UFP |
| 430 uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP | |
| 1229 | 431 |
| 432 -- FPSet is in Projection ⁻¹ F | |
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433 FPSet⊆F1 : {x : Ordinal } → odef (filter FP) x → odef (filter F) (& (ZFP (* x) Q)) |
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434 FPSet⊆F1 {x} fpx = FPSet⊆F is-apq F fpx |
| 1154 | 435 |
| 1169 | 436 FQ : Filter {Power Q} {Q} (λ x → x) |
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437 FQ = Filter-Proj2 {P} {Q} is-apq F |
| 1166 | 438 UFQ : ultra-filter FQ |
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439 UFQ = Filter-Proj2-UF {P} {Q} is-apq F UF |
| 1229 | 440 |
| 441 -- FQSet is in Projection ⁻¹ F | |
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442 FQSet⊆F1 : {x : Ordinal } → odef (filter FQ) x → odef (filter F) (& (ZFP P (* x) )) |
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443 FQSet⊆F1 {x} fpx = FQSet⊆F is-apq F fpx |
| 1229 | 444 |
| 1169 | 445 uflq : UFLP TQ FQ UFQ |
| 446 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ | |
| 1154 | 447 |
| 1166 | 448 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 1229 | 449 Pf = ab-pair (UFLP.P∋limit uflp) (UFLP.P∋limit uflq) |
| 450 | |
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451 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → filter F ∋ * v |
| 1229 | 452 isL {v} nei = filter1 F (λ z xz → Neighbor.v⊆P nei (subst (λ k → odef k z) *iso xz)) |
| 453 (subst (λ k → odef (filter F) k) (sym &iso) (F∋base pqb b∋l )) bpq⊆v where | |
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454 -- |
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455 -- Product Topolgy's open set contains a subbase which is an element of ZPF p Q or ZPF P q |
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456 -- Neighbor of limit contains an open set which conatins a limit |
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457 -- every point of an open set is covered by a subbase |
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458 -- so there is a subbase which contains a limit, the subbase is an element of projection of a filter (P or Q) |
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459 TPQ = ProductTopology TP TQ |
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460 lim = & < * (UFLP.limit uflp) , * (UFLP.limit uflq) > |
| 1229 | 461 bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u nei) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 462 bpq = Neighbor.ou nei (Neighbor.ux nei) | |
| 463 b∋l : odef (* (Base.b bpq)) lim | |
| 464 b∋l = Base.bx bpq | |
| 1172 | 465 pqb : Subbase (pbase TP TQ) (Base.b bpq ) |
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466 pqb = Base.sb bpq |
| 1229 | 467 pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u nei ) |
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468 bpq⊆v : * (Base.b bpq) ⊆ * v |
| 1229 | 469 bpq⊆v {x} bx = Neighbor.u⊆v nei (pqb⊆opq bx) |
| 1173 | 470 pqb⊆opq = Base.b⊆u bpq |
| 1229 | 471 F∋base : {b : Ordinal } → Subbase (pbase TP TQ) b → odef (* b) lim → odef (filter F) b |
| 472 F∋base {b} (gi (case1 px)) bl = subst (λ k → odef (filter F) k) (sym (BaseP.prod px)) f∋b where | |
| 473 -- | |
| 474 -- subbase of product topology which includes lim is in FP, so in F | |
| 475 -- | |
| 476 isl00 : odef (ZFP (* (BaseP.p px)) Q) lim | |
| 477 isl00 = subst (λ k → odef k lim ) (trans (cong (*) (BaseP.prod px)) *iso ) bl | |
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478 px∋limit : odef (* (BaseP.p px)) (UFLP.limit uflp) |
| 1229 | 479 px∋limit = isl01 isl00 refl where |
| 480 isl01 : {x : Ordinal } → odef (ZFP (* (BaseP.p px)) Q) x → x ≡ lim → odef (* (BaseP.p px)) (UFLP.limit uflp) | |
| 481 isl01 (ab-pair {a} {b} bx qx) ab=lim = subst (λ k → odef (* (BaseP.p px)) k) a=lim bx where | |
| 482 a=lim : a ≡ UFLP.limit uflp | |
| 483 a=lim = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj1 ( prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ab=lim) ) ))) | |
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484 fp∋b : filter FP ∋ * (BaseP.p px) |
| 1229 | 485 fp∋b = UFLP.is-limit uflp record { u = _ ; ou = BaseP.op px ; ux = px∋limit |
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486 ; 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 } |
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487 f∋b : odef (filter F) (& (ZFP (* (BaseP.p px)) Q)) |
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488 f∋b = FPSet⊆F1 (subst (λ k → odef (filter FP) k ) &iso fp∋b ) |
| 1229 | 489 F∋base {b} (gi (case2 qx)) bl = subst (λ k → odef (filter F) k) (sym (BaseQ.prod qx)) f∋b where |
| 490 isl00 : odef (ZFP P (* (BaseQ.q qx))) lim | |
| 491 isl00 = subst (λ k → odef k lim ) (trans (cong (*) (BaseQ.prod qx)) *iso ) bl | |
| 492 qx∋limit : odef (* (BaseQ.q qx)) (UFLP.limit uflq) | |
| 493 qx∋limit = isl01 isl00 refl where | |
| 494 isl01 : {x : Ordinal } → odef (ZFP P (* (BaseQ.q qx)) ) x → x ≡ lim → odef (* (BaseQ.q qx)) (UFLP.limit uflq) | |
| 495 isl01 (ab-pair {a} {b} px bx) ab=lim = subst (λ k → odef (* (BaseQ.q qx)) k) b=lim bx where | |
| 496 b=lim : b ≡ UFLP.limit uflq | |
| 497 b=lim = subst₂ (λ j k → j ≡ k ) &iso &iso (cong (&) (proj2 ( prod-≡ (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) ab=lim) ) ))) | |
| 498 fp∋b : filter FQ ∋ * (BaseQ.q qx) | |
| 499 fp∋b = UFLP.is-limit uflq record { u = _ ; ou = BaseQ.oq qx ; ux = qx∋limit | |
| 500 ; 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 } | |
| 501 f∋b : odef (filter F) (& (ZFP P (* (BaseQ.q qx)) )) | |
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502 f∋b = FQSet⊆F1 (subst (λ k → odef (filter FQ) k ) &iso fp∋b ) |
| 1229 | 503 F∋base (g∩ {x} {y} b1 b2) bl = F∋x∩y where |
| 504 -- filter contains finite intersection | |
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505 fb01 : odef (filter F) x |
| 1229 | 506 fb01 = F∋base b1 (proj1 (subst (λ k → odef k lim) *iso bl)) |
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507 fb02 : odef (filter F) y |
| 1229 | 508 fb02 = F∋base b2 (proj2 (subst (λ k → odef k lim) *iso bl)) |
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509 F∋x∩y : odef (filter F) (& (* x ∩ * y)) |
| 1229 | 510 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) |
| 511 (CAP (ZFP P Q) (subst (λ k → odef (Power (ZFP P Q)) k) (sym &iso) (f⊆L F fb01)) | |
| 512 (subst (λ k → odef (Power (ZFP P Q)) k) (sym &iso) (f⊆L F fb02))) | |
| 1154 | 513 |
| 1170 | 514 |
| 515 | |
| 516 | |
| 517 | |
| 518 |