Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1151:8a071bf52407
Finite intersection property to Compact done
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 18 Jan 2023 01:43:24 +0900 |
| parents | fe0129c40745 |
| children | e1c6719a8c38 |
| 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∅ |
| 1101 | 44 -- closed Set |
| 45 CS : HOD | |
| 1119 | 46 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where |
| 47 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) | |
| 48 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) | |
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49 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L |
| 1108 | 50 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy ) |
| 1122 | 51 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L |
| 52 cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) (sym *iso) xy ) | |
| 53 CS∋L : CS ∋ L | |
| 1123 | 54 CS∋L = ⟪ subst (λ k → k ⊆ L) (sym *iso) (λ x → x) , subst (λ k → odef OS (& k)) (sym lem0) OS∋od∅ ⟫ where |
| 55 lem0 : L ＼ * (& L) ≡ od∅ | |
| 56 lem0 = subst (λ k → L ＼ k ≡ od∅) (sym *iso) L＼L=0 | |
| 1122 | 57 --- we may add |
| 58 -- OS∋L : OS ∋ L | |
| 431 | 59 |
| 482 | 60 open Topology |
| 431 | 61 |
| 1122 | 62 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → A ⊆ L → HOD |
| 1150 | 63 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 ) } |
| 64 ; odmax = & L ; <odmax = odef∧< } | |
| 1122 | 65 |
| 1123 | 66 ClL : {L : HOD} → (top : Topology L) → {f : L ⊆ L } → Cl top L f ≡ L |
| 1150 | 67 ClL {L} top {f} = ==→o≡ ( record { eq→ = λ {x} ic |
| 1142 | 68 → subst (λ k → odef k x) *iso ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) (sym *iso) ( λ x → x))) |
| 69 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } ) | |
| 1123 | 70 |
| 1119 | 71 -- Subbase P |
| 72 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 73 |
| 74 data Subbase (P : HOD) : Ordinal → Set n where | |
| 75 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 76 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 77 | |
| 1119 | 78 -- |
| 1150 | 79 -- if y is in a Subbase, some element of P contains it |
| 1119 | 80 |
| 1111 | 81 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 82 sbp P {x} (gi {y} px) = x | |
| 83 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 84 |
| 1111 | 85 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 86 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
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87 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) |
| 1107 | 88 |
| 1119 | 89 -- An open set generate from a base |
| 90 -- | |
| 1115 | 91 -- OS = { U ⊂ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊂ U } |
| 1114 | 92 |
| 1115 | 93 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 94 field |
| 1150 | 95 b : Ordinal |
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96 u⊂L : * u ⊆ L |
| 1114 | 97 sb : Subbase P b |
| 98 b⊆u : * b ⊆ * u | |
| 99 bx : odef (* b) x | |
| 1150 | 100 x⊆L : odef L x |
| 1119 | 101 x⊆L = u⊂L (b⊆u bx) |
| 1114 | 102 |
| 1115 | 103 SO : (L P : HOD) → HOD |
| 1119 | 104 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 |
| 105 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 106 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 107 |
| 1111 | 108 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 109 field |
| 1122 | 110 P⊆PL : P ⊆ Power L |
| 1116 | 111 -- we may need these if OS ∋ L is necessary |
| 112 -- p : {x : HOD} → L ∋ x → HOD | |
| 113 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx | |
| 114 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x | |
| 1110 | 115 |
| 1111 | 116 GeneratedTopogy : (L P : HOD) → IsSubBase L P → Topology L |
| 1115 | 117 GeneratedTopogy L P isb = record { OS = SO L P ; OS⊆PL = tp00 |
| 1122 | 118 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 119 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1150 | 120 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x )) |
| 1115 | 121 tp00 : SO L P ⊆ Power L |
| 122 tp00 {u} ou x ux with ou ux | |
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123 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊂L (b⊆u bx) |
| 1115 | 124 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1150 | 125 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊂L = subst (λ k → k ⊆ L) (sym *iso) ul |
| 1116 | 126 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 127 px : odef (* (& p)) x |
| 128 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 129 qx : odef (* (& q)) x | |
| 130 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 131 b : Ordinal | |
| 132 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 133 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1150 | 134 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 | 135 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 136 tp11 = Base.b⊆u (op px) | |
| 137 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 138 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 139 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1116 | 140 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) |
| 1150 | 141 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1116 | 142 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) |
| 143 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x | |
| 144 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ | |
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145 ul : (p ∩ q) ⊆ L |
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146 ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊂L (op px)) (pz pq) ) where |
| 1116 | 147 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 148 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) | |
| 1115 | 149 tp02 : { q : HOD} → q ⊂ SO L P → SO L P ∋ Union q |
| 150 tp02 {q} q⊂O {x} ux with subst (λ k → odef k x) *iso ux | |
| 151 ... | record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx | |
| 1150 | 152 ... | 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 | 153 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where |
| 154 tp05 : Union q ⊆ L | |
| 155 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with proj2 q⊂O qy yx | |
| 1150 | 156 ... | record { b = b ; u⊂L = u⊂L ; sb = sb ; b⊆u = b⊆u ; bx = bx } |
| 1116 | 157 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) |
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158 tp04 : Union q ⊆ L |
| 1150 | 159 tp04 = tp05 |
| 1116 | 160 tp06 : * b ⊆ Union q |
| 1150 | 161 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } |
| 1110 | 162 |
| 1142 | 163 -- Product Topology |
| 164 | |
| 165 open ZFProduct | |
| 166 | |
| 1150 | 167 -- Product Topology is not |
| 1142 | 168 -- ZFP (OS TP) (OS TQ) (box) |
| 169 | |
| 170 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where | |
| 171 field | |
| 172 p q : Ordinal | |
| 173 op : odef (OS TP) p | |
| 174 prod : x ≡ & (ZFP (* p) Q ) | |
| 175 | |
| 176 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where | |
| 177 field | |
| 178 p q : Ordinal | |
| 179 oq : odef (OS TQ) q | |
| 180 prod : x ≡ & (ZFP P (* q )) | |
| 181 | |
| 182 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 | |
| 183 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 | |
| 184 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) | |
| 185 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 186 ... | 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 | |
| 187 tp03 : odef P a | |
| 188 tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 189 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 | |
| 190 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) | |
| 191 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 192 ... | 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 | |
| 193 tp03 : odef Q b | |
| 194 tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 195 | |
| 196 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 197 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 | |
| 198 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 199 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 200 |
| 201 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 202 ProductTopology {P} {Q} TP TQ = GeneratedTopogy (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } | |
| 203 | |
| 1107 | 204 -- covers |
| 1101 | 205 |
| 1120 | 206 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 207 field |
| 1120 | 208 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 209 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 210 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 211 |
| 212 open _covers_ | |
| 431 | 213 |
| 214 -- Finite Intersection Property | |
| 215 | |
| 1120 | 216 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 217 field |
| 1150 | 218 limit : {X : Ordinal } → * X ⊆ CS top |
| 1120 | 219 → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal |
| 1150 | 220 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 221 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 1143 | 222 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 223 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 224 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 225 → {x : Ordinal } → odef (* X) x | |
| 1143 | 226 → odef L (limit CX fip) |
| 227 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 | 228 |
| 229 -- Compact | |
| 230 | |
| 1119 | 231 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1150 | 232 fin-e : {x : Ordinal } → * x ⊆ S → Finite-∪ S x |
| 1119 | 233 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 431 | 234 |
| 1120 | 235 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 236 field |
| 1120 | 237 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 238 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 239 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 240 |
| 241 -- FIP is Compact | |
| 242 | |
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243 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 244 FIP→Compact {L} top fip with trio< (& L) o∅ |
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245 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 246 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 247 -- L is empty | |
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248 fip02 : {x : Ordinal } → ¬ odef L x |
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249 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 250 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
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251 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 252 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1150 | 253 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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254 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 255 -- set of coset of X |
| 256 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal | |
| 257 CX {X} ox = & ( Replace' (* X) (λ z xz → L ＼ z )) | |
| 1150 | 258 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1143 | 259 CCX {X} os {x} ox with subst (λ k → odef k x) *iso ox |
| 260 ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 261 fip07 : z ≡ & (L ＼ * x) | |
| 262 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 | |
| 263 fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 1150 | 264 fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) *iso not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ |
| 1143 | 265 ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) |
| 266 fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 267 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 | |
| 268 fip10 : ¬ (odef (L ＼ * z) w) | |
| 269 fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 270 fip06 : odef (OS top) (& (L ＼ * x)) | |
| 271 fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 272 fip05 : * x ⊆ L | |
| 273 fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso ) xw ) | |
| 274 -- | |
| 275 -- X covres L means Intersection of (CX X) contains nothing | |
| 276 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP ) | |
| 277 -- it means there is a finite cover | |
| 278 -- | |
| 1150 | 279 record CFIP (X x : Ordinal) : Set n where |
| 1143 | 280 field |
| 1150 | 281 is-CS : * x ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 282 sx : Subbase (* x) o∅ | |
| 283 Cex : (X : Ordinal ) → HOD | |
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284 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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285 fip05 : {y : Ordinal} → CFIP X y → y o< osuc (& (Replace' (* X) (λ z xz → L ＼ z))) |
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286 fip05 {y} cf = subst₂ (λ j k → j o< osuc k ) &iso refl ( ⊆→o≤ ( CFIP.is-CS cf ) ) |
| 1150 | 287 fip00 : {X : Ordinal } → * X ⊆ OS top → * X covers L → ¬ ( Cex X =h= od∅ ) |
| 288 fip00 {X} ox oc cex=0 = ⊥-elim (fip09 fip25 fip20) where | |
| 1148 | 289 -- CX is finite intersection |
| 290 fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x | |
| 1150 | 291 fip02 {C} {x} C<CX sc with trio< x o∅ |
| 1148 | 292 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 293 ... | tri> ¬a ¬b c = c | |
| 294 ... | 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 | 295 fip10 : * C ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 296 fip10 {w} cw = subst (λ k → odef k w) *iso ( C<CX cw ) | |
| 297 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) | |
| 1148 | 298 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) ))) |
| 1150 | 299 fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) )) )) (sym *iso) |
| 300 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 | 301 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 302 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 303 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 304 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 305 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1150 | 306 fip22 = subst (λ k → odef k (& ( L ＼ * y ))) (sym *iso) record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 307 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 308 fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) | |
| 1150 | 309 fip09 : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) |
| 1148 | 310 fip09 {z} Lz nc = nc ( P∋cover oc Lz ) (subst (λ k → odef (* (cover oc Lz)) k) refl (isCover oc _ )) |
| 1121 | 311 cex : {X : Ordinal } → * X ⊆ OS top → * X covers L → Ordinal |
| 1150 | 312 cex {X} ox oc = & ( ODC.minimal O (Cex X) (fip00 ox oc)) |
| 313 CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → CFIP X (cex ox oc) | |
| 314 CXfip {X} ox oc = ODC.x∋minimal O (Cex X) (fip00 ox oc) | |
| 1149 | 315 -- |
| 1121 | 316 -- this defines finite cover |
| 1120 | 317 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1121 | 318 finCover {X} ox oc = & ( Replace' (* (cex ox oc)) (λ z xz → L ＼ z )) |
| 1150 | 319 -- create Finite-∪ from cex |
| 1120 | 320 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1150 | 321 isFinite {X} xo xcp = fip30 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp)) where |
| 322 fip30 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y → Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z ))) | |
| 323 fip30 x y x⊆cs (gi sb) = fip31 where | |
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324 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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325 fip32 {w} record { z = z ; az = xz ; x=ψz = x=ψz } with x⊆cs xz |
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326 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
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327 fip34 : * z1 ⊆ L |
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328 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
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329 fip33 : z1 ≡ w |
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330 fip33 = begin |
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331 z1 ≡⟨ sym &iso ⟩ |
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332 & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ |
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333 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ |
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334 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ |
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335 & (L ＼ * z) ≡⟨ sym x=ψz ⟩ |
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336 w ∎ where open ≡-Reasoning |
| 1150 | 337 fip31 : Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z))) |
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338 fip31 = fin-e (subst (λ k → k ⊆ * X ) (sym *iso) fip32 ) |
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339 fip30 x yz x⊆cs (g∩ {y} {z} sy sz) = fip35 where |
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340 fip35 : Finite-∪ (* X) (& (Replace' (* x) (λ z₁ xz → L ＼ z₁))) |
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341 fip35 = subst (λ k → Finite-∪ (* X) k) |
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342 (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 | 343 -- is also a cover |
| 344 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
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345 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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346 ( fip40 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp))) where |
| 1150 | 347 -- record { cover = λ {x} Lx → ? ; P∋cover = ? ; isCover = ? } |
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348 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
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349 fip45 {L} {a} {b} {x} Lab with ODC.∋-p O b (* x) |
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350 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ |
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351 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ |
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352 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) |
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353 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where |
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354 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal |
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355 fip44 {x} Lab with fip45 {L} {a} {b} Lab |
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356 ... | case1 La = cover ca La |
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357 ... | case2 Lb = cover cb Lb |
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358 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) |
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359 fip46 {x} Lab with fip45 {L} {a} {b} Lab |
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360 ... | case1 La = P∋cover ca La |
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361 ... | case2 Lb = P∋cover cb Lb |
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362 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x |
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363 fip47 {x} Lab with fip45 {L} {a} {b} Lab |
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364 ... | case1 La = isCover ca La |
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365 ... | case2 Lb = isCover cb Lb |
| 1150 | 366 fip40 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y |
| 367 → (Replace' (* x) (λ z xz → L ＼ z )) covers (L ＼ * y ) | |
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368 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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369 ( fip43 {_} {L} {* a} {* b} fip41 fip42 ) where |
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370 fip41 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * a) |
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371 fip41 = fip40 x a x⊆r sa |
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372 fip42 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * b) |
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373 fip42 = fip40 x b x⊆r sb |
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374 fip40 x y x⊆r (gi sb) with x⊆r sb |
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375 ... | record { z = z ; az = az ; x=ψz = x=ψz } = record { cover = fip51 ; P∋cover = fip53 ; isCover = fip50 }where |
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376 fip51 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → Ordinal |
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377 fip51 {w} Lyw = z |
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378 fip52 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* X) z |
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379 fip52 {w} Lyw = az |
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380 fip55 : * z ⊆ L |
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381 fip55 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az)) wz1 |
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382 fip56 : * z ≡ L ＼ * y |
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383 fip56 = begin |
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384 * z ≡⟨ sym (L＼Lx=x fip55 ) ⟩ |
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385 L ＼ ( L ＼ * z ) ≡⟨ cong (λ k → L ＼ k) (sym *iso) ⟩ |
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386 L ＼ * ( & ( L ＼ * z )) ≡⟨ cong (λ k → L ＼ * k) (sym x=ψz) ⟩ |
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387 L ＼ * y ∎ where open ≡-Reasoning |
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388 fip53 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (Replace' (* x) (λ z₁ xz → L ＼ z₁)) z |
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389 fip53 {w} Lyw = record { z = _ ; az = sb ; x=ψz = fip54 } where |
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390 fip54 : z ≡ & ( L ＼ * y ) |
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391 fip54 = begin |
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392 z ≡⟨ sym &iso ⟩ |
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393 & (* z) ≡⟨ cong (&) fip56 ⟩ |
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394 & (L ＼ * y ) |
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395 ∎ where open ≡-Reasoning |
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396 fip50 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* z) w |
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397 fip50 {w} Lyw = subst (λ k → odef k w ) (sym fip56) Lyw |
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398 |
| 1120 | 399 |
| 431 | 400 |
| 1150 | 401 -- some day |
| 402 Compact→FIP : Set (suc n) | |
| 403 Compact→FIP = {L : HOD} → (top : Topology L ) → Compact top → FIP top | |
| 431 | 404 |
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405 -- existence of Ultra Filter |
| 431 | 406 |
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407 open Filter |
| 1102 | 408 |
| 431 | 409 -- Ultra Filter has limit point |
| 410 | |
| 1150 | 411 record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter {L} {P} LP ) |
| 1142 | 412 (FL : filter F ∋ P) (ultra : ultra-filter F ) : Set (suc (suc n)) where |
| 1102 | 413 field |
| 414 limit : Ordinal | |
| 415 P∋limit : odef P limit | |
| 416 is-limit : {o : Ordinal} → odef (OS TP) o → odef (* o) limit → (* o) ⊆ filter F | |
| 417 | |
| 431 | 418 -- FIP is UFL |
| 419 | |
| 1150 | 420 UFLP→FIP : {P : HOD} (TP : Topology P) → {L : HOD} (LP : L ⊆ Power P ) → |
| 1142 | 421 ( (F : Filter {L} {P} LP ) (FP : filter F ∋ P) (UF : ultra-filter F ) → UFLP TP LP F FP UF ) → FIP TP |
| 422 UFLP→FIP {P} TP {L} LP uflp = record { limit = uf00 ; is-limit = {!!} } where | |
| 1143 | 423 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 424 fip {X} CSX = {C x : Ordinal} → * C ⊆ * X → Subbase (* C) x → o∅ o< x | |
| 1150 | 425 F : {X : Ordinal} → (CSX : * X ⊆ CS TP ) → fip {X} CSX → Filter {L} {P} LP |
| 1142 | 426 F = ? |
| 1143 | 427 uf00 : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1150 | 428 uf00 {X} CSX fip = UFLP.limit ( uflp (F CSX fip) ? (F→ultra LP ? ? (F CSX fip) ? ? ? ) ) |
| 1142 | 429 |
| 1150 | 430 -- some day |
| 431 FIP→UFLP : Set (suc (suc n)) | |
| 432 FIP→UFLP = {P : HOD} (TP : Topology P) → FIP TP | |
| 1142 | 433 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (FP : filter F ∋ P) (UF : ultra-filter F ) → UFLP {P} TP {L} LP F FP UF |
| 1102 | 434 |
| 1124 | 435 -- product topology of compact topology is compact |
| 431 | 436 |
| 1142 | 437 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1150 | 438 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) LPQ uflp ) where |
| 1142 | 439 L = pbase TP TQ |
| 440 LPQ = pbase⊆PL TP TQ | |
| 1150 | 441 -- Product of UFL has limit point |
| 1142 | 442 uflp : (F : Filter {pbase TP TQ} LPQ) (LF : filter F ∋ ZFP P Q) (UF : ultra-filter F) |
| 443 → UFLP (ProductTopology TP TQ) LPQ F LF UF | |
| 1150 | 444 uflp F LF UF = record { limit = & < ? , {!!} > ; P∋limit = {!!} ; is-limit = {!!} } |