Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1167:fee9249a2f50
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 21 Jan 2023 18:30:23 +0900 |
| parents | 4e0a1f41910f |
| children | 938ada7fd66c |
| 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) |
| 1161 | 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 |
| 1161 | 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 ) | |
| 1161 | 64 P＼OS=CS {os} oos = ⟪ subst (λ k → k ⊆ L) (sym *iso) proj1 |
| 1160 | 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 |
| 1163 | 69 -- Closure ( Intersection of Closed Set which include A ) |
| 70 | |
| 1162 | 71 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → HOD |
| 72 Cl {L} top A = record { od = record { def = λ x → odef L x ∧ ( (c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x ) } | |
| 1150 | 73 ; odmax = & L ; <odmax = odef∧< } |
| 1122 | 74 |
| 1162 | 75 ClL : {L : HOD} → (top : Topology L) → Cl top L ≡ L |
| 76 ClL {L} top = ==→o≡ ( record { eq→ = λ {x} ic | |
| 1142 | 77 → subst (λ k → odef k x) *iso ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) (sym *iso) ( λ x → x))) |
| 78 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } ) | |
| 1123 | 79 |
| 1163 | 80 -- Closure is Closed Set |
| 81 | |
| 1162 | 82 CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → CS top ∋ Cl top A |
| 83 CS∋Cl {L} top A = subst (λ k → CS top ∋ k) (==→o≡ cc00) (P＼OS=CS top UOCl-is-OS) where | |
| 1163 | 84 OCl : HOD -- set of open set which it not contains A |
| 1162 | 85 OCl = record { od = record { def = λ o → odef (OS top) o ∧ ( A ⊆ (L ＼ * o) ) } ; odmax = & (OS top) ; <odmax = odef∧< } |
| 86 OCl⊆OS : OCl ⊆ OS top | |
| 87 OCl⊆OS ox = proj1 ox | |
| 88 UOCl-is-OS : OS top ∋ Union OCl | |
| 89 UOCl-is-OS = o∪ top OCl⊆OS | |
| 90 cc00 : (L ＼ Union OCl) =h= Cl top A | |
| 91 cc00 = record { eq→ = cc01 ; eq← = cc03 } where | |
| 92 cc01 : {x : Ordinal} → odef (L ＼ Union OCl) x → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) | |
| 93 cc01 {x} ⟪ Lx , nul ⟫ = ⟪ Lx , ( λ c cc ac → cc02 c cc ac nul ) ⟫ where | |
| 94 cc02 : (c : Ordinal) → odef (CS top) c → A ⊆ * c → ¬ odef (Union OCl) x → odef (* c) x | |
| 95 cc02 c cc ac nox with ODC.∋-p O (* c) (* x) | |
| 96 ... | yes y = subst (λ k → odef (* c) k) &iso y | |
| 97 ... | no ncx = ⊥-elim ( nox record { owner = & ( L ＼ * c) ; ao = ⟪ proj2 cc , cc07 ⟫ ; ox = subst (λ k → odef k x) (sym *iso) cc06 } ) where | |
| 98 cc06 : odef (L ＼ * c) x | |
| 99 cc06 = ⟪ Lx , subst (λ k → ¬ odef (* c) k) &iso ncx ⟫ | |
| 100 cc08 : * c ⊆ L | |
| 101 cc08 = cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) cc ) | |
| 102 cc07 : A ⊆ (L ＼ * (& (L ＼ * c))) | |
| 103 cc07 {z} az = subst (λ k → odef k z ) ( | |
| 104 begin * c ≡⟨ sym ( L＼Lx=x cc08 ) ⟩ | |
| 105 L ＼ (L ＼ * c) ≡⟨ cong (λ k → L ＼ k ) (sym *iso) ⟩ | |
| 106 L ＼ * (& (L ＼ * c)) ∎ ) ( ac az ) where open ≡-Reasoning | |
| 107 cc03 : {x : Ordinal} → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) → odef (L ＼ Union OCl) x | |
| 108 cc03 {x} ⟪ Lx , ccx ⟫ = ⟪ Lx , cc04 ⟫ where | |
| 1163 | 109 -- if x is in Cl A, it is in some c : CS, OCl says it is not , i.e. L ＼ o ∋ x, so it is in (L ＼ Union OCl) x |
| 1162 | 110 cc04 : ¬ odef (Union OCl) x |
| 111 cc04 record { owner = o ; ao = ⟪ oo , A⊆L-o ⟫ ; ox = ox } = proj2 ( subst (λ k → odef k x) *iso cc05) ox where | |
| 112 cc05 : odef (* (& (L ＼ * o))) x | |
| 113 cc05 = ccx (& (L ＼ * o)) (P＼OS=CS top (subst (λ k → odef (OS top) k) (sym &iso) oo)) (subst (λ k → A ⊆ k) (sym *iso) A⊆L-o) | |
| 1161 | 114 |
| 1160 | 115 |
| 1119 | 116 -- Subbase P |
| 117 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 118 |
| 119 data Subbase (P : HOD) : Ordinal → Set n where | |
| 120 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 121 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 122 | |
| 1119 | 123 -- |
| 1150 | 124 -- if y is in a Subbase, some element of P contains it |
| 1119 | 125 |
| 1111 | 126 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 127 sbp P {x} (gi {y} px) = x | |
| 128 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 129 |
| 1111 | 130 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 131 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
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132 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) |
| 1107 | 133 |
| 1155 | 134 sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x |
| 135 sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) | |
| 136 sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) | |
| 137 | |
| 1119 | 138 -- An open set generate from a base |
| 139 -- | |
| 1161 | 140 -- OS = { U ⊆ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊆ U } |
| 1114 | 141 |
| 1115 | 142 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 143 field |
| 1150 | 144 b : Ordinal |
| 1161 | 145 u⊆L : * u ⊆ L |
| 1114 | 146 sb : Subbase P b |
| 147 b⊆u : * b ⊆ * u | |
| 148 bx : odef (* b) x | |
| 1150 | 149 x⊆L : odef L x |
| 1161 | 150 x⊆L = u⊆L (b⊆u bx) |
| 1114 | 151 |
| 1115 | 152 SO : (L P : HOD) → HOD |
| 1119 | 153 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 |
| 154 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 155 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 156 |
| 1111 | 157 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 158 field |
| 1122 | 159 P⊆PL : P ⊆ Power L |
| 1116 | 160 -- we may need these if OS ∋ L is necessary |
| 161 -- p : {x : HOD} → L ∋ x → HOD | |
| 1161 | 162 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx |
| 1116 | 163 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x |
| 1110 | 164 |
| 1152 | 165 InducedTopology : (L P : HOD) → IsSubBase L P → Topology L |
| 166 InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 | |
| 1122 | 167 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 168 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1150 | 169 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x )) |
| 1115 | 170 tp00 : SO L P ⊆ Power L |
| 171 tp00 {u} ou x ux with ou ux | |
| 1161 | 172 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊆L (b⊆u bx) |
| 1115 | 173 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1161 | 174 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊆L = subst (λ k → k ⊆ L) (sym *iso) ul |
| 1116 | 175 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 176 px : odef (* (& p)) x |
| 177 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 178 qx : odef (* (& q)) x | |
| 179 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 180 b : Ordinal | |
| 181 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 182 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1150 | 183 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 | 184 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 185 tp11 = Base.b⊆u (op px) | |
| 186 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 187 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 188 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1116 | 189 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) |
| 1150 | 190 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1116 | 191 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) |
| 192 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x | |
| 193 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ | |
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194 ul : (p ∩ q) ⊆ L |
| 1161 | 195 ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊆L (op px)) (pz pq) ) where |
| 1116 | 196 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 197 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) | |
| 1161 | 198 tp02 : { q : HOD} → q ⊆ SO L P → SO L P ∋ Union q |
| 199 tp02 {q} q⊆O {x} ux with subst (λ k → odef k x) *iso ux | |
| 200 ... | record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 201 ... | 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 | 202 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where |
| 203 tp05 : Union q ⊆ L | |
| 1161 | 204 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx |
| 205 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } | |
| 1116 | 206 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) |
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207 tp04 : Union q ⊆ L |
| 1150 | 208 tp04 = tp05 |
| 1116 | 209 tp06 : * b ⊆ Union q |
| 1150 | 210 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } |
| 1110 | 211 |
| 1142 | 212 -- Product Topology |
| 213 | |
| 214 open ZFProduct | |
| 215 | |
| 1150 | 216 -- Product Topology is not |
| 1142 | 217 -- ZFP (OS TP) (OS TQ) (box) |
| 218 | |
| 219 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where | |
| 220 field | |
| 221 p q : Ordinal | |
| 222 op : odef (OS TP) p | |
| 223 prod : x ≡ & (ZFP (* p) Q ) | |
| 224 | |
| 225 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where | |
| 226 field | |
| 227 p q : Ordinal | |
| 228 oq : odef (OS TQ) q | |
| 229 prod : x ≡ & (ZFP P (* q )) | |
| 230 | |
| 231 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 | |
| 232 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 | |
| 233 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) | |
| 234 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 235 ... | 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 | |
| 236 tp03 : odef P a | |
| 237 tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 238 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 | |
| 239 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) | |
| 240 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 241 ... | 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 | |
| 242 tp03 : odef Q b | |
| 243 tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 244 | |
| 245 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 246 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 | |
| 247 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 248 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 249 |
| 250 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 1152 | 251 ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } |
| 1142 | 252 |
| 1152 | 253 -- covers ( q ⊆ Union P ) |
| 1101 | 254 |
| 1120 | 255 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 256 field |
| 1120 | 257 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 258 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 259 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 260 |
| 261 open _covers_ | |
| 431 | 262 |
| 263 -- Finite Intersection Property | |
| 264 | |
| 1120 | 265 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 266 field |
| 1150 | 267 limit : {X : Ordinal } → * X ⊆ CS top |
| 1120 | 268 → ( { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) → Ordinal |
| 1150 | 269 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 270 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 1143 | 271 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 272 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 273 → ( fip : { C : Ordinal } { x : Ordinal } → * C ⊆ * X → Subbase (* C) x → o∅ o< x ) | |
| 274 → {x : Ordinal } → odef (* X) x | |
| 1143 | 275 → odef L (limit CX fip) |
| 276 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 | 277 |
| 278 -- Compact | |
| 279 | |
| 1119 | 280 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1150 | 281 fin-e : {x : Ordinal } → * x ⊆ S → Finite-∪ S x |
| 1119 | 282 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 431 | 283 |
| 1120 | 284 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 285 field |
| 1120 | 286 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 287 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 288 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 289 |
| 290 -- FIP is Compact | |
| 291 | |
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292 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 293 FIP→Compact {L} top fip with trio< (& L) o∅ |
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294 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 295 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 296 -- L is empty | |
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297 fip02 : {x : Ordinal } → ¬ odef L x |
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298 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 299 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
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300 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 301 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1150 | 302 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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303 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 304 -- set of coset of X |
| 305 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal | |
| 306 CX {X} ox = & ( Replace' (* X) (λ z xz → L ＼ z )) | |
| 1150 | 307 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1143 | 308 CCX {X} os {x} ox with subst (λ k → odef k x) *iso ox |
| 309 ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 310 fip07 : z ≡ & (L ＼ * x) | |
| 311 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 | |
| 312 fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 1150 | 313 fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) *iso not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ |
| 1143 | 314 ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) |
| 315 fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 316 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 | |
| 317 fip10 : ¬ (odef (L ＼ * z) w) | |
| 318 fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 319 fip06 : odef (OS top) (& (L ＼ * x)) | |
| 320 fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 321 fip05 : * x ⊆ L | |
| 322 fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso ) xw ) | |
| 323 -- | |
| 324 -- X covres L means Intersection of (CX X) contains nothing | |
| 1152 | 325 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) |
| 1143 | 326 -- it means there is a finite cover |
| 327 -- | |
| 1150 | 328 record CFIP (X x : Ordinal) : Set n where |
| 1143 | 329 field |
| 1150 | 330 is-CS : * x ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 331 sx : Subbase (* x) o∅ | |
| 332 Cex : (X : Ordinal ) → HOD | |
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333 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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334 fip05 : {y : Ordinal} → CFIP X y → y o< osuc (& (Replace' (* X) (λ z xz → L ＼ z))) |
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335 fip05 {y} cf = subst₂ (λ j k → j o< osuc k ) &iso refl ( ⊆→o≤ ( CFIP.is-CS cf ) ) |
| 1150 | 336 fip00 : {X : Ordinal } → * X ⊆ OS top → * X covers L → ¬ ( Cex X =h= od∅ ) |
| 337 fip00 {X} ox oc cex=0 = ⊥-elim (fip09 fip25 fip20) where | |
| 1148 | 338 -- CX is finite intersection |
| 339 fip02 : {C x : Ordinal} → * C ⊆ * (CX ox) → Subbase (* C) x → o∅ o< x | |
| 1150 | 340 fip02 {C} {x} C<CX sc with trio< x o∅ |
| 1148 | 341 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 342 ... | tri> ¬a ¬b c = c | |
| 343 ... | 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 | 344 fip10 : * C ⊆ Replace' (* X) (λ z xz → L ＼ z) |
| 345 fip10 {w} cw = subst (λ k → odef k w) *iso ( C<CX cw ) | |
| 346 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) | |
| 1148 | 347 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) ))) |
| 1150 | 348 fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) )) )) (sym *iso) |
| 349 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 | 350 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 351 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 352 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 353 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 354 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1150 | 355 fip22 = subst (λ k → odef k (& ( L ＼ * y ))) (sym *iso) record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 356 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 357 fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) | |
| 1150 | 358 fip09 : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) |
| 1148 | 359 fip09 {z} Lz nc = nc ( P∋cover oc Lz ) (subst (λ k → odef (* (cover oc Lz)) k) refl (isCover oc _ )) |
| 1121 | 360 cex : {X : Ordinal } → * X ⊆ OS top → * X covers L → Ordinal |
| 1152 | 361 cex {X} ox oc = & ( ODC.minimal O (Cex X) (fip00 ox oc)) -- this will be the finite cover |
| 1150 | 362 CXfip : {X : Ordinal } → (ox : * X ⊆ OS top) → (oc : * X covers L) → CFIP X (cex ox oc) |
| 363 CXfip {X} ox oc = ODC.x∋minimal O (Cex X) (fip00 ox oc) | |
| 1149 | 364 -- |
| 1121 | 365 -- this defines finite cover |
| 1120 | 366 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1121 | 367 finCover {X} ox oc = & ( Replace' (* (cex ox oc)) (λ z xz → L ＼ z )) |
| 1150 | 368 -- create Finite-∪ from cex |
| 1120 | 369 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1150 | 370 isFinite {X} xo xcp = fip30 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp)) where |
| 371 fip30 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y → Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z ))) | |
| 372 fip30 x y x⊆cs (gi sb) = fip31 where | |
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373 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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374 fip32 {w} record { z = z ; az = xz ; x=ψz = x=ψz } with x⊆cs xz |
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375 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
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376 fip34 : * z1 ⊆ L |
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377 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
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378 fip33 : z1 ≡ w |
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379 fip33 = begin |
| 1152 | 380 z1 ≡⟨ sym &iso ⟩ |
| 381 & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ | |
| 382 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ | |
| 383 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ | |
| 384 & (L ＼ * z) ≡⟨ sym x=ψz ⟩ | |
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385 w ∎ where open ≡-Reasoning |
| 1150 | 386 fip31 : Finite-∪ (* X) (& (Replace' (* x) (λ z xz → L ＼ z))) |
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387 fip31 = fin-e (subst (λ k → k ⊆ * X ) (sym *iso) fip32 ) |
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388 fip30 x yz x⊆cs (g∩ {y} {z} sy sz) = fip35 where |
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389 fip35 : Finite-∪ (* X) (& (Replace' (* x) (λ z₁ xz → L ＼ z₁))) |
| 1152 | 390 fip35 = subst (λ k → Finite-∪ (* X) k) |
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391 (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 | 392 -- is also a cover |
| 393 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
| 1152 | 394 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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395 ( fip40 (cex xo xcp) o∅ (CFIP.is-CS (CXfip xo xcp)) (CFIP.sx (CXfip xo xcp))) where |
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396 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
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397 fip45 {L} {a} {b} {x} Lab with ODC.∋-p O b (* x) |
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398 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ |
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399 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ |
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400 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) |
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401 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where |
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402 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal |
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403 fip44 {x} Lab with fip45 {L} {a} {b} Lab |
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404 ... | case1 La = cover ca La |
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405 ... | case2 Lb = cover cb Lb |
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406 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) |
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407 fip46 {x} Lab with fip45 {L} {a} {b} Lab |
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408 ... | case1 La = P∋cover ca La |
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409 ... | case2 Lb = P∋cover cb Lb |
| 1152 | 410 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x |
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411 fip47 {x} Lab with fip45 {L} {a} {b} Lab |
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412 ... | case1 La = isCover ca La |
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413 ... | case2 Lb = isCover cb Lb |
| 1150 | 414 fip40 : ( x y : Ordinal ) → * x ⊆ Replace' (* X) (λ z xz → L ＼ z) → Subbase (* x) y |
| 415 → (Replace' (* x) (λ z xz → L ＼ z )) covers (L ＼ * y ) | |
| 1152 | 416 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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417 ( fip43 {_} {L} {* a} {* b} fip41 fip42 ) where |
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418 fip41 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * a) |
| 1152 | 419 fip41 = fip40 x a x⊆r sa |
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420 fip42 : Replace' (* x) (λ z xz → L ＼ z) covers (L ＼ * b) |
| 1152 | 421 fip42 = fip40 x b x⊆r sb |
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422 fip40 x y x⊆r (gi sb) with x⊆r sb |
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423 ... | record { z = z ; az = az ; x=ψz = x=ψz } = record { cover = fip51 ; P∋cover = fip53 ; isCover = fip50 }where |
| 1152 | 424 fip51 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → Ordinal |
| 425 fip51 {w} Lyw = z | |
| 426 fip52 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* X) z | |
| 427 fip52 {w} Lyw = az | |
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428 fip55 : * z ⊆ L |
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429 fip55 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az)) wz1 |
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430 fip56 : * z ≡ L ＼ * y |
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431 fip56 = begin |
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432 * z ≡⟨ sym (L＼Lx=x fip55 ) ⟩ |
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433 L ＼ ( L ＼ * z ) ≡⟨ cong (λ k → L ＼ k) (sym *iso) ⟩ |
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434 L ＼ * ( & ( L ＼ * z )) ≡⟨ cong (λ k → L ＼ * k) (sym x=ψz) ⟩ |
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435 L ＼ * y ∎ where open ≡-Reasoning |
| 1152 | 436 fip53 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (Replace' (* x) (λ z₁ xz → L ＼ z₁)) z |
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437 fip53 {w} Lyw = record { z = _ ; az = sb ; x=ψz = fip54 } where |
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438 fip54 : z ≡ & ( L ＼ * y ) |
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439 fip54 = begin |
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440 z ≡⟨ sym &iso ⟩ |
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441 & (* z) ≡⟨ cong (&) fip56 ⟩ |
| 1152 | 442 & (L ＼ * y ) |
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443 ∎ where open ≡-Reasoning |
| 1152 | 444 fip50 : {w : Ordinal} (Lyw : odef (L ＼ * y) w) → odef (* z) w |
| 445 fip50 {w} Lyw = subst (λ k → odef k w ) (sym fip56) Lyw | |
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446 |
| 1158 | 447 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top |
| 448 Compact→FIP = ? | |
| 431 | 449 |
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450 -- existence of Ultra Filter |
| 431 | 451 |
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452 open Filter |
| 1102 | 453 |
| 431 | 454 -- Ultra Filter has limit point |
| 455 | |
| 1159 | 456 record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where |
| 457 field | |
| 458 u : Ordinal | |
| 459 ou : odef (OS TP) u | |
| 460 ux : odef (* u) x | |
| 461 v⊆P : * v ⊆ P | |
| 462 o⊆u : * u ⊆ * v | |
| 463 | |
| 1150 | 464 record UFLP {P : HOD} (TP : Topology P) {L : HOD} (LP : L ⊆ Power P ) (F : Filter {L} {P} LP ) |
| 1155 | 465 (ultra : ultra-filter F ) : Set (suc (suc n)) where |
| 1102 | 466 field |
| 467 limit : Ordinal | |
| 468 P∋limit : odef P limit | |
| 1159 | 469 is-limit : {v : Ordinal} → Neighbor TP limit v → filter F ⊆ (* v) |
| 1102 | 470 |
| 1167 | 471 Neighbor→F : {P : HOD} (TP : Topology P) (x : Ordinal ) → Filter {Power P} {P} (λ x → x) |
| 472 Neighbor→F {P} TP x = record { filter = NF ; f⊆L = NF⊆PP ; filter1 = ? ; filter2 = ? } where | |
| 473 NF : HOD | |
| 474 NF = record { od = record { def = λ v → Neighbor TP x v } ; odmax = ? ; <odmax = ? } | |
| 475 NF⊆PP : NF ⊆ Power P | |
| 476 NF⊆PP = nf00 where | |
| 477 nf00 : {v : Ordinal } → Neighbor TP x v → odef (Power P) v | |
| 478 nf00 {v} nei y vy = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou nei)) ? | |
| 479 | |
| 431 | 480 -- FIP is UFL |
| 481 | |
| 1159 | 482 -- filter Base |
| 483 record FBase (P : HOD )(X : Ordinal ) (u : Ordinal) : Set n where | |
| 1153 | 484 field |
| 1161 | 485 b x : Ordinal |
| 1155 | 486 b⊆X : * b ⊆ * X |
| 1161 | 487 sb : Subbase (* b) x |
| 1158 | 488 u⊆P : * u ⊆ P |
| 1154 | 489 x⊆u : * x ⊆ * u |
| 1155 | 490 |
| 491 open import maximum-filter O | |
| 1153 | 492 |
| 1165 | 493 CAP : (P : HOD) {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) |
| 494 CAP P {p} {q} Pp Pq x pqx with subst (λ k → odef k x ) *iso pqx | |
| 495 ... | ⟪ px , qx ⟫ = Pp _ (subst (λ k → odef k x) (sym *iso) px ) | |
| 496 | |
| 1161 | 497 UFLP→FIP : {P : HOD} (TP : Topology P) → |
| 1155 | 498 ( {L : HOD} (LP : L ⊆ Power P ) → (F : Filter {L} {P} LP ) (UF : ultra-filter F ) → UFLP TP LP F UF ) → FIP TP |
| 1163 | 499 UFLP→FIP {P} TP uflp with trio< (& P) o∅ |
| 500 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 501 ... | tri≈ ¬a b ¬c = record { limit = ? ; is-limit = {!!} } where | |
| 502 -- P is empty | |
| 503 fip02 : {x : Ordinal } → ¬ odef P x | |
| 504 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym b) (∈∅< Px) ) | |
| 1165 | 505 ... | tri> ¬a ¬b 0<P = record { limit = uf00 ; is-limit = uf01 } where |
| 1143 | 506 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 1153 | 507 fip {X} CSX = {u x : Ordinal} → * u ⊆ * X → Subbase (* u) x → o∅ o< x |
| 1154 | 508 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
| 1161 | 509 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
| 1159 | 510 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } |
| 1158 | 511 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
| 1159 | 512 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
| 1165 | 513 nc : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → HOD |
| 514 nc = ? | |
| 515 N∋nc :{X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → odef (N CSX fp) (& (nc CSX fp) ) | |
| 516 N∋nc = ? | |
| 517 0<PP : o∅ o< & (Power P) | |
| 518 0<PP = ? | |
| 1158 | 519 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 520 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 521 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 1161 | 522 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
| 523 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 | 524 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
| 525 f10 : q ⊆ P | |
| 526 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 527 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 528 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 | |
| 529 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 530 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 | 531 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 532 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 533 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 534 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 535 f20 : * (& Np+Nq) ⊆ * X |
| 536 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 537 ... | case1 np = FBase.b⊆X Np np |
| 538 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 539 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 1161 | 540 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
| 1159 | 541 → ⟪ 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 | 542 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ¬ (N CSX fp ∋ od∅) |
| 543 proper = ? | |
| 1158 | 544 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
| 1165 | 545 maxf {X} CSX fp = F→Maximum {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp) |
| 1143 | 546 uf00 : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1161 | 547 uf00 {X} CSX fp = UFLP.limit ( uflp (λ x → x) |
| 1155 | 548 ( MaximumFilter.mf (maxf CSX fp) ) |
| 1165 | 549 (F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp))) |
| 1166 | 550 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fip : {C x : Ordinal} → * C ⊆ * X → Subbase (* C) x → o∅ o< x) |
| 551 {x : Ordinal} → odef (* X) x → odef (* x) (uf00 CSX fip) | |
| 1167 | 552 uf01 {X} CSX fp {x} xx = UFLP.is-limit ( uflp (λ x → x) |
| 553 ( MaximumFilter.mf (maxf CSX fp) ) | |
| 554 (F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp))) | |
| 555 record { u = ? ; ou = ? ; ux = ? ; v⊆P = ? ; o⊆u = ? } ? | |
| 1142 | 556 |
| 1158 | 557 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 558 → {L : HOD} (LP : L ⊆ Power P ) (F : Filter LP ) (UF : ultra-filter F ) → UFLP {P} TP {L} LP F UF | |
| 1162 | 559 FIP→UFLP {P} TP fip {L} LP F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 ; P∋limit = ? ; is-limit = ufl00 } where |
| 1160 | 560 CF : HOD |
| 1162 | 561 CF = Replace' (filter F) (λ x fx → Cl TP x ) |
| 1160 | 562 CF⊆CS : CF ⊆ CS TP |
| 1162 | 563 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)) |
| 564 ufl01 : {C x : Ordinal} → * C ⊆ * (& CF) → Subbase (* C) x → o∅ o< x | |
| 565 ufl01 = ? | |
| 1165 | 566 ufl00 : {v : Ordinal} → Neighbor TP (FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 ) v → filter F ⊆ * v |
| 567 ufl00 {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; o⊆u = o⊆u } {x} fx = ? | |
| 1163 | 568 |
| 1124 | 569 -- product topology of compact topology is compact |
| 431 | 570 |
| 1142 | 571 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 572 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1159 | 573 uflP : {L : HOD} → (LP : L ⊆ Power P) (F : Filter {L} LP) (UF : ultra-filter F) |
| 1158 | 574 → UFLP TP LP F UF |
| 1159 | 575 uflP {L} LP F UF = FIP→UFLP TP (Compact→FIP TP CP) LP F UF |
| 576 uflQ : {L : HOD} → (LP : L ⊆ Power Q) (F : Filter {L} LP) (UF : ultra-filter F) | |
| 1158 | 577 → UFLP TQ LP F UF |
| 1159 | 578 uflQ {L} LP F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) LP F UF |
| 579 -- Product of UFL has limit point | |
| 580 uflPQ : {L : HOD} → (LPQ : L ⊆ Power (ZFP P Q )) (F : Filter {L} LPQ) (UF : ultra-filter F) | |
| 1158 | 581 → UFLP (ProductTopology TP TQ) LPQ F UF |
| 1166 | 582 uflPQ {L} LPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where |
| 1159 | 583 LP : HOD -- proj1 of LPQ |
| 1164 | 584 LP = Proj1 L (Power P) (Power Q) |
| 1159 | 585 LPP : LP ⊆ Power P |
| 586 LPP {x} ⟪ Px , p1 ⟫ = Px | |
| 587 FP : Filter {LP} {P} LPP | |
| 1164 | 588 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
| 589 ty00 : Proj1 (filter F) (Power P) (Power Q) ⊆ LP | |
| 590 ty00 {x} ⟪ PPx , ppf ⟫ = ⟪ PPx , ( λ y → record { pq = ZProj1.pq (ppf y) | |
| 591 ; opq = ZProj1.opq (ppf y) ; Lpq = f⊆L F (ZProj1.Lpq (ppf y)) ; x=pi1 = ZProj1.x=pi1 (ppf y) } ) ⟫ | |
| 1161 | 592 UFP : ultra-filter FP |
| 1159 | 593 UFP = record { proper = ? ; ultra = ? } |
| 1158 | 594 uflp : UFLP TP LPP FP UFP |
| 1159 | 595 uflp = FIP→UFLP TP (Compact→FIP TP CP) LPP FP UFP |
| 1154 | 596 |
| 1166 | 597 LQ : HOD -- proj1 of LPQ |
| 598 LQ = Proj2 L (Power P) (Power Q) | |
| 599 LQP : LQ ⊆ Power Q | |
| 600 LQP {x} ⟪ Qx , q1 ⟫ = Qx | |
| 601 FQ : Filter {LQ} {Q} LQP | |
| 602 FQ = record { filter = Proj2 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where | |
| 603 ty00 : Proj2 (filter F) (Power P) (Power Q) ⊆ LQ | |
| 604 ty00 {x} ⟪ QPx , ppf ⟫ = ⟪ QPx , ( λ y → record { pq = ZProj2.pq (ppf y) | |
| 605 ; opq = ZProj2.opq (ppf y) ; Lpq = f⊆L F (ZProj2.Lpq (ppf y)) ; x=pi2 = ZProj2.x=pi2 (ppf y) } ) ⟫ | |
| 606 UFQ : ultra-filter FQ | |
| 607 UFQ = record { proper = ? ; ultra = ? } | |
| 608 uflq : UFLP TQ LQP FQ UFQ | |
| 609 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) LQP FQ UFQ | |
| 1154 | 610 |
| 1166 | 611 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 612 Pf = ? | |
| 613 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → filter F ⊆ * v | |
| 614 isL {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; o⊆u = o⊆u } {x} fx = ? | |
| 1154 | 615 |