Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1160:2479884b35b2
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 20 Jan 2023 17:02:09 +0900 |
| parents | adba530ce1f0 |
| children | b45925515f77 |
| rev | line source |
|---|---|
| 431 | 1 open import Level |
| 2 open import Ordinals | |
| 3 module Topology {n : Level } (O : Ordinals {n}) where | |
| 4 | |
| 5 open import zf | |
| 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 | |
| 27 open Ordinals.IsNext isNext | |
| 28 open OrdUtil O | |
| 29 open ODUtil O | |
| 30 | |
| 31 import ODC | |
| 32 open ODC O | |
| 33 | |
| 1102 | 34 open import filter O |
| 1101 | 35 open import OPair O |
| 36 | |
| 482 | 37 record Topology ( L : HOD ) : Set (suc n) where |
| 431 | 38 field |
| 39 OS : HOD | |
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40 OS⊆PL : OS ⊆ Power L |
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41 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) |
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42 o∪ : { P : HOD } → P ⊂ OS → OS ∋ Union P |
| 1122 | 43 OS∋od∅ : OS ∋ od∅ |
| 1160 | 44 --- we may add |
| 45 -- OS∋L : OS ∋ L | |
| 1101 | 46 -- closed Set |
| 47 CS : HOD | |
| 1119 | 48 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where |
| 49 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) | |
| 50 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) | |
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51 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L |
| 1108 | 52 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy ) |
| 1122 | 53 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L |
| 54 cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) (sym *iso) xy ) | |
| 55 CS∋L : CS ∋ L | |
| 1123 | 56 CS∋L = ⟪ subst (λ k → k ⊆ L) (sym *iso) (λ x → x) , subst (λ k → odef OS (& k)) (sym lem0) OS∋od∅ ⟫ where |
| 57 lem0 : L ＼ * (& L) ≡ od∅ | |
| 58 lem0 = subst (λ k → L ＼ k ≡ od∅) (sym *iso) L＼L=0 | |
| 1154 | 59 CS⊆PL : CS ⊆ Power L |
| 60 CS⊆PL {x} Cx y xy = proj1 Cx xy | |
| 1160 | 61 P＼CS=OS : {cs : HOD} → CS ∋ cs → OS ∋ ( L ＼ cs ) |
| 62 P＼CS=OS {cs} ⟪ cs⊆L , olcs ⟫ = subst (λ k → odef OS k) (cong (λ k → & ( L ＼ k)) *iso) olcs | |
| 63 P＼OS=CS : {cs : HOD} → OS ∋ cs → CS ∋ ( L ＼ cs ) | |
| 64 P＼OS=CS {os} oos = ⟪ subst (λ k → k ⊆ L) (sym *iso) proj1 | |
| 65 , subst (λ k → odef OS k) (cong (&) (trans (sym (L＼Lx=x (os⊆L oos))) (cong (λ k → L ＼ k) (sym *iso)) )) oos ⟫ | |
| 431 | 66 |
| 482 | 67 open Topology |
| 431 | 68 |
| 1122 | 69 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → A ⊆ L → HOD |
| 1150 | 70 Cl {L} top A A⊆L = record { od = record { def = λ x → odef L x ∧ ( (c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x ) } |
| 71 ; odmax = & L ; <odmax = odef∧< } | |
| 1122 | 72 |
| 1123 | 73 ClL : {L : HOD} → (top : Topology L) → {f : L ⊆ L } → Cl top L f ≡ L |
| 1150 | 74 ClL {L} top {f} = ==→o≡ ( record { eq→ = λ {x} ic |
| 1142 | 75 → subst (λ k → odef k x) *iso ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) (sym *iso) ( λ x → x))) |
| 76 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } ) | |
| 1123 | 77 |
| 1160 | 78 record NAO {L : HOD} (top : Topology L) (A : HOD) (A⊆L : A ⊆ L) (x : Ordinal) : Set n where |
| 79 field | |
| 80 ox : odef (OS top) x | |
| 81 na : ¬ ( A ⊆ * x) | |
| 82 | |
| 83 CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → (A⊆L : A ⊆ L) → CS top ∋ Cl top A A⊆L | |
| 84 CS∋Cl {L} top A A⊆L = ⟪ (λ {x} cx → proj1 ( subst (λ k → odef k x) *iso cx )) , ? ⟫ where | |
| 85 NCl : HOD | |
| 86 NCl = record { od = record { def = λ x → NAO top A A⊆L x } ; odmax = & (OS top) ; <odmax = λ lt → odef< (NAO.ox lt) } | |
| 87 NCl⊂OS : NCl ⊂ OS top | |
| 88 NCl⊂OS = ⟪ ? , ? ⟫ | |
| 89 ccl00 : (L ＼ (Cl top A A⊆L)) ≡ Union NCl | |
| 90 ccl00 = ==→o≡ ( record { eq→ = ? ; eq← = ? } ) | |
| 91 | |
| 1119 | 92 -- Subbase P |
| 93 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 94 |
| 95 data Subbase (P : HOD) : Ordinal → Set n where | |
| 96 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 97 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 98 | |
| 1119 | 99 -- |
| 1150 | 100 -- if y is in a Subbase, some element of P contains it |
| 1119 | 101 |
| 1111 | 102 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 103 sbp P {x} (gi {y} px) = x | |
| 104 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 105 |
| 1111 | 106 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 107 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
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108 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) |
| 1107 | 109 |
| 1155 | 110 sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x |
| 111 sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) | |
| 112 sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) | |
| 113 | |
| 1119 | 114 -- An open set generate from a base |
| 115 -- | |
| 1115 | 116 -- OS = { U ⊂ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊂ U } |
| 1114 | 117 |
| 1115 | 118 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 119 field |
| 1150 | 120 b : Ordinal |
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121 u⊂L : * u ⊆ L |
| 1114 | 122 sb : Subbase P b |
| 123 b⊆u : * b ⊆ * u | |
| 124 bx : odef (* b) x | |
| 1150 | 125 x⊆L : odef L x |
| 1119 | 126 x⊆L = u⊂L (b⊆u bx) |
| 1114 | 127 |
| 1115 | 128 SO : (L P : HOD) → HOD |
| 1119 | 129 SO L P = record { od = record { def = λ u → {x : Ordinal } → odef (* u) x → Base L P u x } ; odmax = osuc (& L) ; <odmax = tp00 } where |
| 130 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 131 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 132 |
| 1111 | 133 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 134 field |
| 1122 | 135 P⊆PL : P ⊆ Power L |
| 1116 | 136 -- we may need these if OS ∋ L is necessary |
| 137 -- p : {x : HOD} → L ∋ x → HOD | |
| 138 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx | |
| 139 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x | |
| 1110 | 140 |
| 1152 | 141 InducedTopology : (L P : HOD) → IsSubBase L P → Topology L |
| 142 InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 | |
| 1122 | 143 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 144 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1150 | 145 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x )) |
| 1115 | 146 tp00 : SO L P ⊆ Power L |
| 147 tp00 {u} ou x ux with ou ux | |
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148 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊂L (b⊆u bx) |
| 1115 | 149 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1150 | 150 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) ul |
| 1116 | 151 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 152 px : odef (* (& p)) x |
| 153 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 154 qx : odef (* (& q)) x | |
| 155 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 156 b : Ordinal | |
| 157 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 158 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1150 | 159 tp08 = subst₂ (λ j k → j ⊆ k ) (sym *iso) (sym *iso) (⊆∩-dist {(* (Base.b (op px)) ∩ * (Base.b (oq qx)))} {p} {q} tp09 tp10 ) where |
| 1116 | 160 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 161 tp11 = Base.b⊆u (op px) | |
| 162 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 163 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 164 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1116 | 165 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) |
| 1150 | 166 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1116 | 167 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) |
| 168 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x | |
| 169 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ | |
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170 ul : (p ∩ q) ⊆ L |
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171 ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊂L (op px)) (pz pq) ) where |
| 1116 | 172 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 173 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) | |
| 1115 | 174 tp02 : { q : HOD} → q ⊂ SO L P → SO L P ∋ Union q |
| 175 tp02 {q} q⊂O {x} ux with subst (λ k → odef k x) *iso ux | |
| 176 ... | record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx | |
| 1150 | 177 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) tp04 |
| 1116 | 178 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where |
| 179 tp05 : Union q ⊆ L | |
| 180 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx | |
| 1150 | 181 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } |
| 1116 | 182 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) |
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183 tp04 : Union q ⊆ L |
| 1150 | 184 tp04 = tp05 |
| 1116 | 185 tp06 : * b ⊆ Union q |
| 1150 | 186 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } |
| 1110 | 187 |
| 1142 | 188 -- Product Topology |
| 189 | |
| 190 open ZFProduct | |
| 191 | |
| 1150 | 192 -- Product Topology is not |
| 1142 | 193 -- ZFP (OS TP) (OS TQ) (box) |
| 194 | |
| 195 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where | |
| 196 field | |
| 197 p q : Ordinal | |
| 198 op : odef (OS TP) p | |
| 199 prod : x ≡ & (ZFP (* p) Q ) | |
| 200 | |
| 201 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where | |
| 202 field | |
| 203 p q : Ordinal | |
| 204 oq : odef (OS TQ) q | |
| 205 prod : x ≡ & (ZFP P (* q )) | |
| 206 | |
| 207 pbase⊆PL : {P Q : HOD} → (TP : Topology P) → (TQ : Topology Q) → {x : Ordinal } → BaseP TP Q x ∨ BaseQ P TQ x → odef (Power (ZFP P Q)) x | |
| 208 pbase⊆PL {P} {Q} TP TQ {z} (case1 record { p = p ; q = q ; op = op ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where | |
| 209 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) | |
| 210 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 211 ... | ab-pair {a} {b} pa qb = ZFP→ (subst (λ k → odef P k ) (sym &iso) tp03 ) (subst (λ k → odef Q k ) (sym &iso) qb ) where | |
| 212 tp03 : odef P a | |
| 213 tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 214 pbase⊆PL {P} {Q} TP TQ {z} (case2 record { p = p ; q = q ; oq = oq ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where | |
| 215 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) | |
| 216 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 217 ... | ab-pair {a} {b} pa qb = ZFP→ (subst (λ k → odef P k ) (sym &iso) pa ) (subst (λ k → odef Q k ) (sym &iso) tp03 ) where | |
| 218 tp03 : odef Q b | |
| 219 tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 220 | |
| 221 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 222 pbase {P} {Q} TP TQ = record { od = record { def = λ x → BaseP TP Q x ∨ BaseQ P TQ x } ; odmax = & (Power (ZFP P Q)) ; <odmax = tp00 } where | |
| 223 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 224 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 225 |
| 226 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 1152 | 227 ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } |
| 1142 | 228 |
| 1152 | 229 -- covers ( q ⊆ Union P ) |
| 1101 | 230 |
| 1120 | 231 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 232 field |
| 1120 | 233 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 234 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 235 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 236 |
| 237 open _covers_ | |
| 431 | 238 |
| 239 -- Finite Intersection Property | |
| 240 | |
| 1120 | 241 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 242 field |
| 1150 | 243 limit : {X : Ordinal } → * X ⊆ CS top |
| 1120 | 244 → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal |
| 1150 | 245 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 246 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 1143 | 247 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 248 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 249 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 250 → {x : Ordinal } → odef (* X) x | |
| 1143 | 251 → odef L (limit CX fip) |
| 252 L∋limit {X} CX fip {x} xx = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX xx)) (is-limit CX fip xx) | |
| 431 | 253 |
| 254 -- Compact | |
| 255 | |
| 1119 | 256 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1150 | 257 fin-e : {x : Ordinal } → * x ⊆ S → Finite-∪ S x |
| 1119 | 258 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 431 | 259 |
| 1120 | 260 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 261 field |
| 1120 | 262 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 263 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 264 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 265 |
| 266 -- FIP is Compact | |
| 267 | |
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268 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 269 FIP→Compact {L} top fip with trio< (& L) o∅ |
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270 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 271 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 272 -- L is empty | |
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273 fip02 : {x : Ordinal } → ¬ odef L x |
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274 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 275 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
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276 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 277 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1150 | 278 fip00 {X} xo xcp = fin-e ( λ {x} 0x → ⊥-elim (¬x<0 (subst (λ k → odef k x) o∅≡od∅ 0x) ) ) |
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279 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 280 -- set of coset of X |
| 281 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal | |
| 282 CX {X} ox = & ( Replace' (* X) (λ z xz → L ＼ z )) | |
| 1150 | 283 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1143 | 284 CCX {X} os {x} ox with subst (λ k → odef k x) *iso ox |
| 285 ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 286 fip07 : z ≡ & (L ＼ * x) | |
| 287 fip07 = subst₂ (λ j k → j ≡ k) &iso (cong (λ k → & ( L ＼ k )) (cong (*) (sym x=ψz))) ( cong (&) ( ==→o≡ record { eq→ = fip09 ; eq← = fip08 } )) where | |
| 288 fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 1150 | 289 fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) *iso not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ |
| 1143 | 290 ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) |
| 291 fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 292 fip09 {w} zw = ⟪ os⊆L top (os (subst (λ k → odef (* X) k) (sym &iso) az)) zw , subst (λ k → ¬ odef k w) (sym *iso) fip10 ⟫ where | |
| 293 fip10 : ¬ (odef (L ＼ * z) w) | |
| 294 fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 295 fip06 : odef (OS top) (& (L ＼ * x)) | |
| 296 fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 297 fip05 : * x ⊆ L | |
| 298 fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso ) xw ) | |
| 299 -- | |
| 300 -- X covres L means Intersection of (CX X) contains nothing | |
| 1152 | 301 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) |
| 1143 | 302 -- it means there is a finite cover |
| 303 -- | |
| 1150 | 304 record CFIP (X x : Ordinal) : Set n where |
| 1143 | 305 field |
| 1150 | 306 is-CS : * x ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 307 sx : Subbase (* x) o∅ | |
| 308 Cex : (X : Ordinal ) → HOD | |
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309 Cex X = record { od = record { def = λ x → CFIP X x } ; odmax = osuc (& (Replace' (* X) (λ z xz → L ＼ z))) ; <odmax = fip05 } where |
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310 fip05 : {y : Ordinal} → CFIP X y → y o< osuc (& (Replace' (* X) (λ z xz → L ＼ z))) |
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311 fip05 {y} cf = subst₂ (λ j k → j o< osuc k ) &iso refl ( ⊆→o≤ ( CFIP.is-CS cf ) ) |
| 1150 | 312 fip00 : {X : Ordinal } → * X ⊆ OS top → * X covers L → ¬ ( Cex X =h= od∅ ) |
| 313 fip00 {X} ox oc cex=0 = ⊥-elim (fip09 fip25 fip20) where | |
| 1148 | 314 -- CX is finite intersection |
| 315 fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x | |
| 1150 | 316 fip02 {C} {x} C<CX sc with trio< x o∅ |
| 1148 | 317 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 318 ... | tri> ¬a ¬b c = c | |
| 319 ... | tri≈ ¬a b ¬c = ⊥-elim (¬x<0 ( _==_.eq→ cex=0 record { is-CS = fip10 ; sx = subst (λ k → Subbase (* C) k) b sc } )) where | |
| 1150 | 320 fip10 : * C ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 321 fip10 {w} cw = subst (λ k → odef k w) *iso ( C<CX cw ) | |
| 322 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) | |
| 1148 | 323 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) ))) |
| 1150 | 324 fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) )) )) (sym *iso) |
| 325 record { z = cover oc (x∋minimal L (0<P→ne 0<L)) ; az = P∋cover oc (x∋minimal L (0<P→ne 0<L)) ; x=ψz = refl } | |
| 1148 | 326 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 327 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 328 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 329 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 330 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1150 | 331 fip22 = subst (λ k → odef k (& ( L ＼ * y ))) (sym *iso) record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 332 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 333 fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) | |
| 1150 | 334 fip09 : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) |
| 1148 | 335 fip09 {z} Lz nc = nc ( P∋cover oc Lz ) (subst (λ k → odef (* (cover oc Lz)) k) refl (isCover oc _ )) |
| 1121 | 336 cex : {X : Ordinal } → * X ⊆ OS top → * X covers L → Ordinal |
| 1152 | 337 cex {X} ox oc = & ( ODC.minimal O (Cex X) (fip00 ox oc)) -- this will be the finite cover |
| 1150 | 338 CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → CFIP X (cex ox oc) |
| 339 CXfip {X} ox oc = ODC.x∋minimal O (Cex X) (fip00 ox oc) | |
| 1149 | 340 -- |
| 1121 | 341 -- this defines finite cover |
| 1120 | 342 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1121 | 343 finCover {X} ox oc = & ( Replace' (* (cex ox oc)) (λ z xz → L ＼ z )) |
| 1150 | 344 -- create Finite-∪ from cex |
| 1120 | 345 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1150 | 346 isFinite {X} xo xcp = fip30 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp)) where |
| 347 fip30 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y → Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z ))) | |
| 348 fip30 x y x⊆cs (gi sb) = fip31 where | |
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349 fip32 : Replace' (* x) (λ z xz → L ＼ z) ⊆ * X -- x⊆cs :* x ⊆ Replace' (* X) (λ z₁ xz → L ＼ z₁) , x=ψz : w ≡ & (L ＼ * z) , odef (* x) z |
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350 fip32 {w} record { z = z ; az = xz ; x=ψz = x=ψz } with x⊆cs xz |
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351 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
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352 fip34 : * z1 ⊆ L |
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353 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
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354 fip33 : z1 ≡ w |
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355 fip33 = begin |
| 1152 | 356 z1 ≡⟨ sym &iso ⟩ |
| 357 & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ | |
| 358 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ | |
| 359 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ | |
| 360 & (L ＼ * z) ≡⟨ sym x=ψz ⟩ | |
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361 w ∎ where open ≡-Reasoning |
| 1150 | 362 fip31 : Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z))) |
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363 fip31 = fin-e (subst (λ k → k ⊆ * X ) (sym *iso) fip32 ) |
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364 fip30 x yz x⊆cs (g∩ {y} {z} sy sz) = fip35 where |
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365 fip35 : Finite-∪ (* X) (& (Replace' (* x) (λ z₁ xz → L ＼ z₁))) |
| 1152 | 366 fip35 = subst (λ k → Finite-∪ (* X) k) |
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367 (cong (&) (subst (λ k → (k ∪ k ) ≡ (Replace' (* x) (λ z₁ xz → L ＼ z₁)) ) (sym *iso) x∪x≡x )) ( fin-∪ (fip30 _ _ x⊆cs sy) (fip30 _ _ x⊆cs sz) ) |
| 1120 | 368 -- is also a cover |
| 369 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
| 1152 | 370 isCover1 {X} xo xcp = subst₂ (λ j k → j covers k ) (sym *iso) (subst (λ k → L ＼ k ≡ L) (sym o∅≡od∅) L＼0=L) |
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371 ( fip40 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp))) where |
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372 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
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373 fip45 {L} {a} {b} {x} Lab with ODC.∋-p O b (* x) |
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374 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ |
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375 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ |
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376 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) |
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377 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where |
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378 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal |
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379 fip44 {x} Lab with fip45 {L} {a} {b} Lab |
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380 ... | case1 La = cover ca La |
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381 ... | case2 Lb = cover cb Lb |
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382 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) |
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383 fip46 {x} Lab with fip45 {L} {a} {b} Lab |
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384 ... | case1 La = P∋cover ca La |
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385 ... | case2 Lb = P∋cover cb Lb |
| 1152 | 386 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x |
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387 fip47 {x} Lab with fip45 {L} {a} {b} Lab |
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388 ... | case1 La = isCover ca La |
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389 ... | case2 Lb = isCover cb Lb |
| 1150 | 390 fip40 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y |
| 391 → (Replace' (* x) (λ z xz → L ＼ z )) covers (L ＼ * y ) | |
| 1152 | 392 fip40 x .(& (* _ ∩ * _)) x⊆r (g∩ {a} {b} sa sb) = subst (λ k → (Replace' (* x) (λ z xz → L ＼ z)) covers ( L ＼ k ) ) (sym *iso) |
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393 ( fip43 {_} {L} {* a} {* b} fip41 fip42 ) where |
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394 fip41 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * a) |
| 1152 | 395 fip41 = fip40 x a x⊆r sa |
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396 fip42 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * b) |
| 1152 | 397 fip42 = fip40 x b x⊆r sb |
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398 fip40 x y x⊆r (gi sb) with x⊆r sb |
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399 ... | record { z = z ; az = az ; x=ψz = x=ψz } = record { cover = fip51 ; P∋cover = fip53 ; isCover = fip50 }where |
| 1152 | 400 fip51 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → Ordinal |
| 401 fip51 {w} Lyw = z | |
| 402 fip52 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* X) z | |
| 403 fip52 {w} Lyw = az | |
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404 fip55 : * z ⊆ L |
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405 fip55 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az)) wz1 |
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406 fip56 : * z ≡ L ＼ * y |
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407 fip56 = begin |
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408 * z ≡⟨ sym (L＼Lx=x fip55 ) ⟩ |
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409 L ＼ ( L ＼ * z ) ≡⟨ cong (λ k → L ＼ k) (sym *iso) ⟩ |
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410 L ＼ * ( & ( L ＼ * z )) ≡⟨ cong (λ k → L ＼ * k) (sym x=ψz) ⟩ |
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411 L ＼ * y ∎ where open ≡-Reasoning |
| 1152 | 412 fip53 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (Replace' (* x) (λ z₁ xz → L ＼ z₁)) z |
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413 fip53 {w} Lyw = record { z = _ ; az = sb ; x=ψz = fip54 } where |
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414 fip54 : z ≡ & ( L ＼ * y ) |
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415 fip54 = begin |
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416 z ≡⟨ sym &iso ⟩ |
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417 & (* z) ≡⟨ cong (&) fip56 ⟩ |
| 1152 | 418 & (L ＼ * y ) |
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419 ∎ where open ≡-Reasoning |
| 1152 | 420 fip50 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* z) w |
| 421 fip50 {w} Lyw = subst (λ k → odef k w ) (sym fip56) Lyw | |
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422 |
| 1158 | 423 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top |
| 424 Compact→FIP = ? | |
| 431 | 425 |
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426 -- existence of Ultra Filter |
| 431 | 427 |
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428 open Filter |
| 1102 | 429 |
| 431 | 430 -- Ultra Filter has limit point |
| 431 | |
| 1159 | 432 record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where |
| 433 field | |
| 434 u : Ordinal | |
| 435 ou : odef (OS TP) u | |
| 436 ux : odef (* u) x | |
| 437 v⊆P : * v ⊆ P | |
| 438 o⊆u : * u ⊆ * v | |
| 439 | |
| 1150 | 440 record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter {L} {P} LP ) |
| 1155 | 441 (ultra : ultra-filter F ) : Set (suc (suc n)) where |
| 1102 | 442 field |
| 443 limit : Ordinal | |
| 444 P∋limit : odef P limit | |
| 1159 | 445 is-limit : {v : Ordinal} → Neighbor TP limit v → filter F ⊆ (* v) |
| 1102 | 446 |
| 431 | 447 -- FIP is UFL |
| 448 | |
| 1159 | 449 -- filter Base |
| 450 record FBase (P : HOD )(X : Ordinal ) (u : Ordinal) : Set n where | |
| 1153 | 451 field |
| 1154 | 452 b x : Ordinal |
| 1155 | 453 b⊆X : * b ⊆ * X |
| 454 sb : Subbase (* b) x | |
| 1158 | 455 u⊆P : * u ⊆ P |
| 1154 | 456 x⊆u : * x ⊆ * u |
| 1155 | 457 |
| 458 open import maximum-filter O | |
| 1153 | 459 |
| 1154 | 460 UFLP→FIP : {P : HOD} (TP : Topology P) → |
| 1155 | 461 ( {L : HOD} (LP : L ⊆ Power P ) → (F : Filter {L} {P} LP ) (UF : ultra-filter F ) → UFLP TP LP F UF ) → FIP TP |
| 1154 | 462 UFLP→FIP {P} TP uflp = record { limit = uf00 ; is-limit = {!!} } where |
| 1143 | 463 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 1153 | 464 fip {X} CSX = {u x : Ordinal} → * u ⊆ * X → Subbase (* u) x → o∅ o< x |
| 1154 | 465 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
| 1159 | 466 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
| 467 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } | |
| 1158 | 468 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
| 1159 | 469 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
| 1158 | 470 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 471 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 472 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 473 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = | |
| 474 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 → | |
| 475 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where | |
| 476 f10 : q ⊆ P | |
| 477 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 478 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 479 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 | |
| 480 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 481 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 | 482 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 483 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 484 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 485 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 486 f20 : * (& Np+Nq) ⊆ * X |
| 487 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 488 ... | case1 np = FBase.b⊆X Np np |
| 489 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 490 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 491 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq | |
| 1159 | 492 → ⟪ 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)) ⟫ ) |
| 1155 | 493 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ¬ (N CSX fp ∋ od∅) |
| 494 proper = ? | |
| 1158 | 495 CAP : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) |
| 496 CAP {X} CSX fp {p} {q} Pp Pq x pqx with subst (λ k → odef k x ) *iso pqx | |
| 497 ... | ⟪ px , qx ⟫ = Pp _ (subst (λ k → odef k x) (sym *iso) px ) | |
| 498 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) | |
| 499 maxf {X} CSX fp = F→Maximum {Power P} {P} (λ x → x) (CAP CSX fp) (F CSX fp) ? ? (proper CSX fp) | |
| 1143 | 500 uf00 : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1158 | 501 uf00 {X} CSX fp = UFLP.limit ( uflp (λ x → x) |
| 1155 | 502 ( MaximumFilter.mf (maxf CSX fp) ) |
| 1158 | 503 (F→ultra {Power P} {P} (λ x → x) (CAP CSX fp) (F CSX fp) ? ? (proper CSX fp))) |
| 1142 | 504 |
| 1158 | 505 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 506 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (UF : ultra-filter F ) → UFLP {P} TP {L} LP F UF | |
| 1159 | 507 FIP→UFLP {P} TP fip {L} LP F UF = record { limit = FIP.limit fip ? ? ; P∋limit = ? ; is-limit = ? } where |
| 1160 | 508 CF : HOD |
| 509 CF = Replace' (filter F) (λ x fx → Cl TP x (λ {w} xw → LP (f⊆L F fx) _ (subst (λ k → odef k w) (sym *iso) xw )) ) | |
| 510 CF⊆CS : CF ⊆ CS TP | |
| 511 CF⊆CS {x} record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ subst (λ k → k ⊆ P) (trans (sym *iso) (sym (cong (*) x=ψz))) uf01 , uf02 ⟫ where | |
| 512 uf01 : (Cl TP (* z) (λ {w} xw → LP (f⊆L F (subst (odef (filter F)) (sym &iso) az)) w (subst (λ k → odef k w) (sym *iso) xw))) ⊆ P | |
| 513 uf01 = ? | |
| 514 uf02 : odef (OS TP) (& (P ＼ * x)) | |
| 515 uf02 = ? | |
| 1159 | 516 ufl00 : {v : Ordinal} → Neighbor TP (FIP.limit fip ? ?) v → filter F ⊆ * v |
| 517 ufl00 {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; o⊆u = o⊆u } = ? | |
| 1102 | 518 |
| 1124 | 519 -- product topology of compact topology is compact |
| 431 | 520 |
| 1142 | 521 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 522 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1159 | 523 uflP : {L : HOD} → (LP : L ⊆ Power P) (F : Filter {L} LP) (UF : ultra-filter F) |
| 1158 | 524 → UFLP TP LP F UF |
| 1159 | 525 uflP {L} LP F UF = FIP→UFLP TP (Compact→FIP TP CP) LP F UF |
| 526 uflQ : {L : HOD} → (LP : L ⊆ Power Q) (F : Filter {L} LP) (UF : ultra-filter F) | |
| 1158 | 527 → UFLP TQ LP F UF |
| 1159 | 528 uflQ {L} LP F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) LP F UF |
| 529 -- Product of UFL has limit point | |
| 530 uflPQ : {L : HOD} → (LPQ : L ⊆ Power (ZFP P Q )) (F : Filter {L} LPQ) (UF : ultra-filter F) | |
| 1158 | 531 → UFLP (ProductTopology TP TQ) LPQ F UF |
| 1159 | 532 uflPQ {L} LPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , {!!} > ; P∋limit = {!!} ; is-limit = {!!} } where |
| 533 LP : HOD -- proj1 of LPQ | |
| 534 LP = Proj1PP L (Power P) (Power Q) | |
| 535 LPP : LP ⊆ Power P | |
| 536 LPP {x} ⟪ Px , p1 ⟫ = Px | |
| 537 FP : Filter {LP} {P} LPP | |
| 538 FP = record { filter = Proj1PP (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where | |
| 539 ty00 : Proj1PP (filter F) (Power P) (Power Q) ⊆ LP | |
| 540 ty00 {x} ⟪ PPx , ppf ⟫ = ⟪ PPx , ( λ y → record { pq = PProj1.pq (ppf y) | |
| 541 ; opq = PProj1.opq (ppf y) ; Lpq = f⊆L F (PProj1.Lpq (ppf y)) ; x=pi1 = PProj1.x=pi1 (ppf y) } ) ⟫ | |
| 542 UFP : ultra-filter FP | |
| 543 UFP = record { proper = ? ; ultra = ? } | |
| 1158 | 544 uflp : UFLP TP LPP FP UFP |
| 1159 | 545 uflp = FIP→UFLP TP (Compact→FIP TP CP) LPP FP UFP |
| 1154 | 546 |
| 547 | |
| 548 |