Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1486:49c3ef1e9b4f
Topology fixed
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 01 Jul 2024 16:07:57 +0900 |
| parents | 5dacb669f13b |
| children |
| rev | line source |
|---|---|
| 1476 | 1 {-# OPTIONS --cubical-compatible --safe #-} |
| 2 open import Level | |
| 3 open import Ordinals | |
| 4 open import logic | |
| 5 open import Relation.Nullary | |
| 1170 | 6 |
| 431 | 7 open import Level |
| 8 open import Ordinals | |
| 1476 | 9 import HODBase |
| 10 import OD | |
| 11 open import Relation.Nullary | |
| 12 module Topology {n : Level } (O : Ordinals {n} ) (HODAxiom : HODBase.ODAxiom O) (ho< : OD.ODAxiom-ho< O HODAxiom ) | |
| 13 (AC : OD.AxiomOfChoice O HODAxiom ) where | |
| 14 | |
| 15 | |
| 16 open import Relation.Binary.PropositionalEquality hiding ( [_] ) | |
| 17 open import Relation.Binary.Definitions | |
| 18 | |
| 19 open import Data.Empty | |
| 20 | |
| 21 import OrdUtil | |
| 22 | |
| 23 open Ordinals.Ordinals O | |
| 24 open Ordinals.IsOrdinals isOrdinal | |
| 25 import ODUtil | |
| 431 | 26 |
| 27 open import logic | |
| 1476 | 28 open import nat |
| 29 | |
| 30 open OrdUtil O | |
| 31 open ODUtil O HODAxiom ho< | |
| 32 | |
| 431 | 33 open _∧_ |
| 34 open _∨_ | |
| 35 open Bool | |
| 36 | |
| 1476 | 37 open HODBase._==_ |
| 38 | |
| 39 open HODBase.ODAxiom HODAxiom | |
| 40 open OD O HODAxiom | |
| 41 | |
| 42 open HODBase.HOD | |
| 43 | |
| 44 open AxiomOfChoice AC | |
| 45 open import ODC O HODAxiom AC as ODC | |
| 46 | |
| 47 open import Level | |
| 48 open import Ordinals | |
| 49 | |
| 50 import filter | |
| 51 | |
| 52 open import Relation.Nullary | |
| 53 -- open import Relation.Binary hiding ( _⇔_ ) | |
| 54 open import Data.Empty | |
| 431 | 55 open import Relation.Binary.PropositionalEquality |
| 1476 | 56 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 57 import BAlgebra | |
| 431 | 58 |
| 1476 | 59 open import ZProduct O HODAxiom ho< |
| 60 open import filter O HODAxiom ho< AC | |
| 431 | 61 |
| 1484 | 62 import Relation.Binary.Reasoning.Setoid as EqHOD |
| 63 | |
| 64 LDec : (L : HOD) (P : HOD) → P ⊆ L → (x : HOD) → Dec (x ∈ P) | |
| 65 LDec L P _ x = ∋-p P x | |
| 1101 | 66 |
| 482 | 67 record Topology ( L : HOD ) : Set (suc n) where |
| 431 | 68 field |
| 69 OS : HOD | |
|
1113
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
|
70 OS⊆PL : OS ⊆ Power L |
|
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
|
71 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) |
| 1161 | 72 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P |
| 1210 | 73 OS∋od∅ : OS ∋ od∅ -- OS ∋ Union od∅ |
| 1160 | 74 --- we may add |
| 75 -- OS∋L : OS ∋ L | |
| 1101 | 76 -- closed Set |
| 1484 | 77 open BAlgebra O HODAxiom ho< L (LDec L) |
| 1101 | 78 CS : HOD |
| 1119 | 79 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where |
| 80 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) | |
| 81 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) | |
|
1113
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
|
82 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L |
| 1484 | 83 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (eq← *iso xy) |
| 1122 | 84 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L |
| 1484 | 85 cs⊆L {x} Cx {y} xy = proj1 Cx (eq← *iso xy) |
| 1122 | 86 CS∋L : CS ∋ L |
| 1484 | 87 CS∋L = ⟪ (λ lt → eq→ *iso lt) , subst (λ k → odef OS k) (sym lem0) OS∋od∅ ⟫ where |
| 88 lem0 : & (L ＼ * (& L)) ≡ & od∅ | |
| 1485 | 89 lem0 = ==→o≡ ( ==-trans (＼-cong L L (* (& L)) L ==-refl *iso ) L＼L=0 ) |
| 1154 | 90 CS⊆PL : CS ⊆ Power L |
| 1161 | 91 CS⊆PL {x} Cx y xy = proj1 Cx xy |
| 1160 | 92 P＼CS=OS : {cs : HOD} → CS ∋ cs → OS ∋ ( L ＼ cs ) |
| 1485 | 93 P＼CS=OS {cs} ⟪ cs⊆L , olcs ⟫ = subst (λ k → odef OS k) (==→o≡ (＼-cong L L (* (& cs)) cs ==-refl *iso)) olcs |
| 1160 | 94 P＼OS=CS : {cs : HOD} → OS ∋ cs → CS ∋ ( L ＼ cs ) |
| 1484 | 95 P＼OS=CS {os} oos = ⟪ (λ lt → proj1 (eq→ *iso lt)) |
| 96 , subst (λ k → odef OS k) (==→o≡ (==-sym (==-trans ( | |
| 1485 | 97 ＼-cong L L (* ( & (L ＼ os))) (L ＼ os) ==-refl *iso ) (L＼Lx=x {os} (os⊆L oos)) ))) oos ⟫ |
| 1484 | 98 |
| 431 | 99 |
| 482 | 100 open Topology |
| 431 | 101 |
| 1163 | 102 -- Closure ( Intersection of Closed Set which include A ) |
| 103 | |
| 1162 | 104 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → HOD |
| 105 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 | 106 ; odmax = & L ; <odmax = odef∧< } |
| 1122 | 107 |
| 1476 | 108 ClL : {L : HOD} → (top : Topology L) → Cl top L =h= L |
| 109 ClL {L} top = record { eq→ = λ {x} ic | |
| 1484 | 110 → eq→ *iso ((proj2 ic) (& L) (CS∋L top) (λ lt → eq← *iso lt)) |
| 1476 | 111 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } |
| 1123 | 112 |
| 1163 | 113 -- Closure is Closed Set |
| 114 | |
| 1484 | 115 |
| 1162 | 116 CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → CS top ∋ Cl top A |
| 1484 | 117 CS∋Cl {L} top A = subst (λ k → odef (CS top) k) (==→o≡ cc00) (P＼OS=CS top UOCl-is-OS) where |
| 118 open BAlgebra O HODAxiom ho< L (LDec L) | |
| 1163 | 119 OCl : HOD -- set of open set which it not contains A |
| 1162 | 120 OCl = record { od = record { def = λ o → odef (OS top) o ∧ ( A ⊆ (L ＼ * o) ) } ; odmax = & (OS top) ; <odmax = odef∧< } |
| 121 OCl⊆OS : OCl ⊆ OS top | |
| 122 OCl⊆OS ox = proj1 ox | |
| 123 UOCl-is-OS : OS top ∋ Union OCl | |
| 124 UOCl-is-OS = o∪ top OCl⊆OS | |
| 125 cc00 : (L ＼ Union OCl) =h= Cl top A | |
| 126 cc00 = record { eq→ = cc01 ; eq← = cc03 } where | |
| 127 cc01 : {x : Ordinal} → odef (L ＼ Union OCl) x → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) | |
| 128 cc01 {x} ⟪ Lx , nul ⟫ = ⟪ Lx , ( λ c cc ac → cc02 c cc ac nul ) ⟫ where | |
| 129 cc02 : (c : Ordinal) → odef (CS top) c → A ⊆ * c → ¬ odef (Union OCl) x → odef (* c) x | |
| 1476 | 130 cc02 c cc ac nox with ODC.∋-p (* c) (* x) |
| 1162 | 131 ... | yes y = subst (λ k → odef (* c) k) &iso y |
| 1484 | 132 ... | no ncx = ⊥-elim ( nox record { owner = & ( L ＼ * c) ; ao = ⟪ proj2 cc , cc07 ⟫ ; ox = eq← *iso cc06 } ) where |
| 1162 | 133 cc06 : odef (L ＼ * c) x |
| 134 cc06 = ⟪ Lx , subst (λ k → ¬ odef (* c) k) &iso ncx ⟫ | |
| 135 cc08 : * c ⊆ L | |
| 136 cc08 = cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) cc ) | |
| 1484 | 137 cc09 : (L ＼ * (& (L ＼ * c))) =h= (* c) |
| 138 cc09 = begin | |
| 1485 | 139 L ＼ * (& (L ＼ * c)) ≈⟨ ＼-cong L L (* (& (L ＼ * c))) (L ＼ * c) ==-refl *iso ⟩ |
| 1484 | 140 L ＼ (L ＼ * c) ≈⟨ L＼Lx=x {* c} cc08 ⟩ |
| 141 * c ∎ | |
| 142 where open EqHOD ==-Setoid | |
| 1162 | 143 cc07 : A ⊆ (L ＼ * (& (L ＼ * c))) |
| 1484 | 144 cc07 {z} az = eq← cc09 ( ac az ) |
| 1162 | 145 cc03 : {x : Ordinal} → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) → odef (L ＼ Union OCl) x |
| 146 cc03 {x} ⟪ Lx , ccx ⟫ = ⟪ Lx , cc04 ⟫ where | |
| 1163 | 147 -- 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 | 148 cc04 : ¬ odef (Union OCl) x |
| 1484 | 149 cc04 record { owner = o ; ao = ⟪ oo , A⊆L-o ⟫ ; ox = ox } = proj2 ( eq→ *iso cc05) ox where |
| 1162 | 150 cc05 : odef (* (& (L ＼ * o))) x |
| 1484 | 151 cc05 = ccx (& (L ＼ * o)) (P＼OS=CS top (subst (λ k → odef (OS top) k) (sym &iso) oo)) (λ lt → eq← *iso (A⊆L-o lt)) |
| 1161 | 152 |
| 1476 | 153 CS∋x→Clx=x : {L x : HOD} → (top : Topology L) → CS top ∋ x → Cl top x =h= x |
| 154 CS∋x→Clx=x {L} {x} top cx = record { eq→ = cc10 ; eq← = cc11 } where | |
| 1210 | 155 cc10 : {y : Ordinal} → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) → odef x y |
| 1484 | 156 cc10 {y} ⟪ Ly , cc ⟫ = eq→ *iso ( cc (& x) cx (λ {z} xz → eq← *iso xz ) ) |
| 1210 | 157 cc11 : {y : Ordinal} → odef x y → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) |
| 158 cc11 {y} xy = ⟪ cs⊆L top cx xy , (λ c oc x⊆c → x⊆c xy ) ⟫ | |
| 1160 | 159 |
| 1119 | 160 -- Subbase P |
| 161 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 162 |
| 163 data Subbase (P : HOD) : Ordinal → Set n where | |
| 164 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 165 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 166 | |
| 1119 | 167 -- |
| 1150 | 168 -- if y is in a Subbase, some element of P contains it |
| 1119 | 169 |
| 1111 | 170 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 171 sbp P {x} (gi {y} px) = x | |
| 172 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 173 |
| 1111 | 174 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 175 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
| 1484 | 176 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (eq→ *iso xy)) |
| 1107 | 177 |
| 1155 | 178 sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x |
| 179 sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) | |
| 180 sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) | |
| 181 | |
| 1119 | 182 -- An open set generate from a base |
| 183 -- | |
| 1161 | 184 -- OS = { U ⊆ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊆ U } |
| 1114 | 185 |
| 1115 | 186 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 187 field |
| 1150 | 188 b : Ordinal |
| 1161 | 189 u⊆L : * u ⊆ L |
| 1114 | 190 sb : Subbase P b |
| 191 b⊆u : * b ⊆ * u | |
| 192 bx : odef (* b) x | |
| 1150 | 193 x⊆L : odef L x |
| 1161 | 194 x⊆L = u⊆L (b⊆u bx) |
| 1114 | 195 |
| 1115 | 196 SO : (L P : HOD) → HOD |
| 1119 | 197 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 |
| 198 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 199 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 200 |
| 1111 | 201 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 202 field |
| 1122 | 203 P⊆PL : P ⊆ Power L |
| 1116 | 204 -- we may need these if OS ∋ L is necessary |
| 205 -- p : {x : HOD} → L ∋ x → HOD | |
| 1161 | 206 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx |
| 1116 | 207 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x |
| 1110 | 208 |
| 1152 | 209 InducedTopology : (L P : HOD) → IsSubBase L P → Topology L |
| 210 InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 | |
| 1122 | 211 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 212 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1484 | 213 tp03 {x} 0x = ⊥-elim ( empty (* x) (eq→ *iso ( subst (λ k → odef (* (& od∅)) k ) (sym &iso) 0x ))) |
| 1115 | 214 tp00 : SO L P ⊆ Power L |
| 215 tp00 {u} ou x ux with ou ux | |
| 1161 | 216 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊆L (b⊆u bx) |
| 1115 | 217 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1484 | 218 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊆L = λ lt → ul (eq→ *iso lt) |
| 1116 | 219 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 220 px : odef (* (& p)) x |
| 1484 | 221 px = eq← *iso ( proj1 (eq→ *iso ux) ) |
| 1115 | 222 qx : odef (* (& q)) x |
| 1484 | 223 qx = eq← *iso ( proj2 (eq→ *iso ux) ) |
| 1115 | 224 b : Ordinal |
| 225 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 226 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1484 | 227 tp08 = λ lt → ( eq← *iso ((⊆∩-dist {(* (Base.b (op px)) ∩ * (Base.b (oq qx)))} {p} {q} tp09 tp10 ) (eq→ *iso lt))) where |
| 1116 | 228 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 229 tp11 = Base.b⊆u (op px) | |
| 230 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 231 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 232 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1484 | 233 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (λ lt → eq→ *iso (tp11 lt)) |
| 1150 | 234 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1484 | 235 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (λ lt → eq→ *iso (tp12 lt)) |
| 1116 | 236 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x |
| 1484 | 237 tp14 = eq← *iso ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ |
|
1117
53ca3c609f0e
generated topology from subbase done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1116
diff
changeset
|
238 ul : (p ∩ q) ⊆ L |
| 1484 | 239 ul = λ pq → (Base.u⊆L (op px)) (pz (eq← *iso pq)) where |
| 1116 | 240 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 1484 | 241 pz {z} pq = eq← *iso ( proj1 (eq→ *iso pq ) ) |
| 1161 | 242 tp02 : { q : HOD} → q ⊆ SO L P → SO L P ∋ Union q |
| 1484 | 243 tp02 {q} q⊆O {x} ux with eq→ *iso ux |
| 244 ... | record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 245 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = record { b = b ; u⊆L = λ lt → tp04 (eq→ *iso lt) | |
| 246 ; sb = sb ; b⊆u = λ lt → eq← *iso (tp06 lt) ; bx = bx } where | |
| 247 tp05 : Union q ⊆ L | |
| 248 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 249 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } | |
| 250 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) | |
| 251 tp04 : Union q ⊆ L | |
| 252 tp04 = tp05 | |
| 253 tp06 : * b ⊆ Union q | |
| 254 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } | |
| 1110 | 255 |
| 1142 | 256 -- Product Topology |
| 257 | |
| 258 open ZFProduct | |
| 259 | |
| 1150 | 260 -- Product Topology is not |
| 1142 | 261 -- ZFP (OS TP) (OS TQ) (box) |
| 262 | |
|
1228
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
263 -- Rectangle subset (zπ1 ⁻¹ p) |
| 1142 | 264 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where |
| 265 field | |
| 1172 | 266 p : Ordinal |
| 1142 | 267 op : odef (OS TP) p |
| 268 prod : x ≡ & (ZFP (* p) Q ) | |
| 269 | |
|
1228
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
270 -- Rectangle subset (zπ12⁻¹ q ) |
| 1142 | 271 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where |
| 272 field | |
| 1172 | 273 q : Ordinal |
| 1142 | 274 oq : odef (OS TQ) q |
| 275 prod : x ≡ & (ZFP P (* q )) | |
| 276 | |
| 277 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 | |
| 1172 | 278 pbase⊆PL {P} {Q} TP TQ {z} (case1 record { p = p ; op = op ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where |
| 1142 | 279 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) |
| 1485 | 280 tp01 w wz with eq→ *iso wz |
| 281 ... | 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 | |
| 282 tp03 : odef P a | |
| 283 tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 1172 | 284 pbase⊆PL {P} {Q} TP TQ {z} (case2 record { q = q ; oq = oq ; prod = prod }) = subst (λ k → odef (Power (ZFP P Q)) k ) (sym prod) tp01 where |
| 1142 | 285 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) |
| 1485 | 286 tp01 w wz with eq→ *iso wz |
| 287 ... | 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 | |
| 288 tp03 : odef Q b | |
| 289 tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 1142 | 290 |
| 291 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 292 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 | |
| 293 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 294 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 295 |
| 296 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 1152 | 297 ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } |
| 1142 | 298 |
| 1152 | 299 -- covers ( q ⊆ Union P ) |
| 1101 | 300 |
| 1120 | 301 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 302 field |
| 1120 | 303 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 304 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 305 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 306 |
| 307 open _covers_ | |
| 431 | 308 |
| 1485 | 309 cong-covers : (P Q R S : HOD) → P =h= Q → R =h= S → P covers R → Q covers S |
| 310 cong-covers P Q R S P=Q R=S record { cover = cover ; P∋cover = P∋cover ; isCover = isCover } | |
| 311 = record { cover = λ {x} px → cover (eq← R=S px) ; P∋cover = λ {x} px → eq→ P=Q (P∋cover (eq← R=S px)) | |
| 312 ; isCover = λ {x} px → isCover (eq← R=S px) } | |
| 313 | |
| 314 | |
| 431 | 315 -- Finite Intersection Property |
| 316 | |
| 1120 | 317 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 318 field |
| 1150 | 319 limit : {X : Ordinal } → * X ⊆ CS top |
| 1187 | 320 → ( { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → Ordinal |
| 1150 | 321 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 322 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1143 | 323 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 324 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 325 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1150 | 326 → {x : Ordinal } → odef (* X) x |
| 1143 | 327 → odef L (limit CX fip) |
| 328 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 | 329 |
| 330 -- Compact | |
| 331 | |
| 1119 | 332 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1188 | 333 fin-e : Finite-∪ S o∅ |
|
1228
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
334 fin-i : {x : Ordinal } → odef S x → Finite-∪ S (& ( * x , * x )) -- an element of S |
| 1188 | 335 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 1198 | 336 -- Finite-∪ S y → Union y ⊆ S |
| 431 | 337 |
| 1120 | 338 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 339 field |
| 1120 | 340 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 341 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 342 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 343 |
| 344 -- FIP is Compact | |
| 345 | |
|
1113
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
|
346 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 347 FIP→Compact {L} top fip with trio< (& L) o∅ |
|
1146
1966127fc14f
wrong cover definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1145
diff
changeset
|
348 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 349 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 350 -- L is empty | |
|
1146
1966127fc14f
wrong cover definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1145
diff
changeset
|
351 fip02 : {x : Ordinal } → ¬ odef L x |
|
1966127fc14f
wrong cover definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1145
diff
changeset
|
352 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 353 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
|
1146
1966127fc14f
wrong cover definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1145
diff
changeset
|
354 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 355 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1188 | 356 fip00 {X} xo xcp = fin-e |
|
1146
1966127fc14f
wrong cover definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1145
diff
changeset
|
357 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 358 -- set of coset of X |
| 1484 | 359 open BAlgebra O HODAxiom ho< L (LDec L) |
| 1121 | 360 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal |
| 1293 | 361 CX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼ ) |
| 1150 | 362 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1485 | 363 CCX {X} os {x} ox with eq→ *iso ox |
| 364 ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 365 fip07 : z ≡ & (L ＼ * x) | |
| 366 fip07 = subst₂ (λ j k → j ≡ k) &iso (cong (λ k → & ( L ＼ k )) (cong (*) (sym x=ψz))) ( ==→o≡ record { eq→ = fip09 ; eq← = fip08 } ) where | |
| 367 fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 368 fip08 {x} ⟪ Lx , not ⟫ = double-neg-elim ( λ nz → not (eq← *iso ⟪ Lx , nz ⟫) ) -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ | |
| 369 fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 370 fip09 {w} zw = ⟪ os⊆L top (os (subst (λ k → odef (* X) k) (sym &iso) az)) zw , (λ lt → fip10 (eq→ *iso lt)) ⟫ where | |
| 371 fip10 : ¬ (odef (L ＼ * z) w) | |
| 372 fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 373 fip06 : odef (OS top) (& (L ＼ * x)) | |
| 374 fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 375 fip05 : * x ⊆ L | |
| 376 fip05 {w} xw = proj1 ( eq→ (begin | |
| 377 * x ≈⟨ o≡→== x=ψz ⟩ | |
| 378 * (& ( L ＼ * z )) ≈⟨ *iso ⟩ | |
| 379 L ＼ * z ∎ ) xw ) where | |
| 380 open EqHOD ==-Setoid | |
| 1143 | 381 -- |
| 382 -- X covres L means Intersection of (CX X) contains nothing | |
| 1152 | 383 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) |
| 1143 | 384 -- it means there is a finite cover |
| 385 -- | |
| 1293 | 386 finCoverBase : {X : Ordinal } → * X ⊆ OS top → * X covers L → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅ |
| 1476 | 387 finCoverBase {X} ox oc with p∨¬p (Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅) |
| 1189 | 388 ... | case1 sb = sb |
| 1199 | 389 ... | case2 ¬sb = ⊥-elim (¬¬cover fip25 fip20) where |
| 390 ¬¬cover : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) | |
| 391 ¬¬cover {z} Lz nc = nc ( P∋cover oc Lz ) (isCover oc _ ) | |
| 392 -- ¬sb → we have finite intersection | |
| 1187 | 393 fip02 : {x : Ordinal} → Subbase (* (CX ox)) x → o∅ o< x |
| 394 fip02 {x} sc with trio< x o∅ | |
| 1148 | 395 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 396 ... | tri> ¬a ¬b c = c | |
| 1485 | 397 ... | tri≈ ¬a b ¬c = ⊥-elim (¬sb (sb⊆ (eq→ *iso) (subst (λ k → Subbase (* (CX ox)) k ) b sc ))) |
| 1150 | 398 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) |
| 1476 | 399 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( x∋minimal L (0<P→ne 0<L) ) ))) |
| 1485 | 400 fip26 = eq← *iso 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 | 401 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 402 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 403 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 404 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 405 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1485 | 406 fip22 = eq← *iso record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 407 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 1485 | 408 fip21 = eq→ *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) |
| 1199 | 409 -- create HOD from Subbase ( finite intersection ) |
| 1293 | 410 finCoverSet : {X : Ordinal } → (x : Ordinal) → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x → HOD |
| 1190 | 411 finCoverSet {X} x (gi rx) = ( L ＼ * x ) , ( L ＼ * x ) |
| 1189 | 412 finCoverSet {X} x∩y (g∩ {x} {y} sx sy) = finCoverSet {X} x sx ∪ finCoverSet {X} y sy |
| 1149 | 413 -- |
| 1121 | 414 -- this defines finite cover |
| 1120 | 415 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1189 | 416 finCover {X} ox oc = & ( finCoverSet o∅ (finCoverBase ox oc)) |
| 1199 | 417 -- create Finite-∪ from finCoverSet |
| 1120 | 418 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1189 | 419 isFinite {X} xo xcp = fip60 o∅ (finCoverBase xo xcp) where |
| 1293 | 420 fip60 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → Finite-∪ (* X) (& (finCoverSet {X} x sb)) |
| 1190 | 421 fip60 x (gi rx) = subst (λ k → Finite-∪ (* X) k) fip62 (fin-i (fip61 rx)) where |
| 422 fip62 : & (* (& (L ＼ * x)) , * (& (L ＼ * x))) ≡ & ((L ＼ * x) , (L ＼ * x)) | |
| 1485 | 423 fip62 = ==→o≡ (pair-subst2 *iso *iso ) |
| 1293 | 424 fip61 : odef (Replace (* X) (_＼_ L) RC＼ ) x → odef (* X) ( & ((L ＼ * x ) )) |
| 1189 | 425 fip61 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
|
1151
8a071bf52407
Finite intersection property to Compact done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1150
diff
changeset
|
426 fip34 : * z1 ⊆ L |
|
8a071bf52407
Finite intersection property to Compact done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1150
diff
changeset
|
427 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
| 1189 | 428 fip33 : z1 ≡ & (L ＼ * x) |
|
1151
8a071bf52407
Finite intersection property to Compact done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1150
diff
changeset
|
429 fip33 = begin |
| 1152 | 430 z1 ≡⟨ sym &iso ⟩ |
| 1485 | 431 & (* z1) ≡⟨ sym ( ==→o≡ (L＼Lx=x {* z1} fip34)) ⟩ |
| 432 & (L ＼ ( L ＼ * z1)) ≡⟨ ==→o≡ ( ＼-cong L L ( L ＼ * z1) (* (& ( L ＼ * z1))) ==-refl (==-sym *iso) ) ⟩ | |
| 1152 | 433 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ |
| 1189 | 434 & (L ＼ * x ) ∎ where open ≡-Reasoning |
| 435 fip60 x∩y (g∩ {x} {y} sx sy) = subst (λ k → Finite-∪ (* X) k) fip62 ( fin-∪ (fip60 x sx) (fip60 y sy) ) where | |
| 436 fip62 : & (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡ & (finCoverSet x sx ∪ finCoverSet y sy) | |
| 1485 | 437 fip62 = ==→o≡ (begin |
| 438 (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) | |
| 439 ≈⟨ ∪-cong {* (& (finCoverSet x sx))} {finCoverSet x sx} {* (& (finCoverSet y sy))} {finCoverSet y sy} *iso *iso ⟩ | |
| 440 finCoverSet x sx ∪ finCoverSet y sy ∎ ) | |
| 441 where open EqHOD ==-Setoid | |
| 1120 | 442 -- is also a cover |
| 443 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
| 1485 | 444 isCover1 {X} xo xcp = cong-covers _ _ _ _ (==-sym *iso) (==-trans (＼-cong L L (* o∅) od∅ ==-refl o∅==od∅) L＼0=L) |
| 445 -- subst₂ (λ j k → j covers k ) ? (subst (λ k → L ＼ k ≡ L) ? ? ) -- L＼0=L) | |
| 1190 | 446 (fip70 o∅ (finCoverBase xo xcp)) where |
| 1293 | 447 fip70 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → (finCoverSet {X} x sb) covers (L ＼ * x) |
| 1199 | 448 fip70 x (gi rx) = fip73 where |
| 449 fip73 : ((L ＼ * x) , (L ＼ * x)) covers (L ＼ * x) -- obvious | |
| 450 fip73 = record { cover = λ _ → & (L ＼ * x) ; P∋cover = λ _ → case1 refl | |
| 1485 | 451 ; isCover = λ {x} lt → eq← *iso lt } |
| 452 fip70 x∩y (g∩ {x} {y} sx sy) = cong-covers _ _ _ _ ==-refl (＼-cong L L (* x ∩ * y) (* (& (* x ∩ * y))) ==-refl (==-sym *iso)) | |
| 453 ( fip43 {_} {L} {* x} {* y} (fip71 (fip70 x sx)) (fip72 (fip70 y sy)) ) where | |
| 1194 | 454 fip71 : {a b c : HOD} → a covers c → (a ∪ b) covers c |
| 455 fip71 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case1 (P∋cover cov lt) | |
| 456 ; isCover = isCover cov } | |
| 457 fip72 : {a b c : HOD} → a covers c → (b ∪ a) covers c | |
| 458 fip72 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case2 (P∋cover cov lt) | |
| 459 ; isCover = isCover cov } | |
| 1190 | 460 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
| 1476 | 461 fip45 {L} {a} {b} {x} Lab with ∋-p b (* x) |
| 1190 | 462 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ |
| 463 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ | |
| 464 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) | |
| 465 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where | |
| 466 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal | |
| 467 fip44 {x} Lab with fip45 {L} {a} {b} Lab | |
| 468 ... | case1 La = cover ca La | |
| 469 ... | case2 Lb = cover cb Lb | |
| 470 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) | |
| 471 fip46 {x} Lab with fip45 {L} {a} {b} Lab | |
| 472 ... | case1 La = P∋cover ca La | |
| 473 ... | case2 Lb = P∋cover cb Lb | |
| 474 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x | |
| 475 fip47 {x} Lab with fip45 {L} {a} {b} Lab | |
| 476 ... | case1 La = isCover ca La | |
| 477 ... | case2 Lb = isCover cb Lb | |
|
1151
8a071bf52407
Finite intersection property to Compact done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1150
diff
changeset
|
478 |
| 1485 | 479 |
| 480 | |
| 1158 | 481 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top |
| 1180 | 482 Compact→FIP {L} top compact with trio< (& L) o∅ |
| 1175 | 483 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1198 | 484 ... | tri≈ ¬a L=0 ¬c = record { limit = λ {X} CX fip → o∅ ; is-limit = λ {X} CX fip xx → ⊥-elim (fip000 CX fip xx) } where |
| 485 -- empty L case | |
| 486 -- if 0 < X then 0 < x ∧ L ∋ xfrom fip | |
| 487 -- if 0 ≡ X then ¬ odef X x | |
| 488 fip000 : {X x : Ordinal} (CX : * X ⊆ CS top) → ({y : Ordinal} → Subbase (* X) y → o∅ o< y) → ¬ odef (* X) x | |
| 489 fip000 {X} {x} CX fip xx with trio< o∅ X | |
| 1485 | 490 ... | tri< 0<X ¬b ¬c = ¬∅∋ (eq→ (begin |
| 491 L ≈⟨ ==-sym *iso ⟩ | |
| 492 * (& L) ≈⟨ o≡→== L=0 ⟩ | |
| 493 * o∅ ≈⟨ o∅==od∅ ⟩ | |
| 494 od∅ ∎ ) (subst (λ k → odef L k ) (sym &iso) | |
| 495 ( cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) (CX xx)) Xe ))) where | |
| 496 open EqHOD ==-Setoid | |
| 1198 | 497 0<x : o∅ o< x |
| 498 0<x = fip (gi xx ) | |
| 499 e : HOD -- we have an element of x | |
| 1476 | 500 e = minimal (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) |
| 1198 | 501 Xe : odef (* x) (& e) |
| 1476 | 502 Xe = x∋minimal (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) |
| 1485 | 503 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (eq→ ( begin |
| 504 * X ≈⟨ o≡→== (sym 0=X) ⟩ | |
| 505 * o∅ ≈⟨ o∅==od∅ ⟩ | |
| 506 od∅ ∎ ) (subst (λ k → odef (* X) k ) (sym &iso) xx ) )) where open EqHOD ==-Setoid | |
| 1198 | 507 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1175 | 508 ... | tri> ¬a ¬b 0<L = record { limit = limit ; is-limit = fip00 } where |
| 509 -- set of coset of X | |
| 1484 | 510 open BAlgebra O HODAxiom ho< L (LDec L) |
| 1175 | 511 OX : {X : Ordinal} → * X ⊆ CS top → Ordinal |
| 1293 | 512 OX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼) |
| 1175 | 513 OOX : {X : Ordinal} → (cs : * X ⊆ CS top) → * (OX cs) ⊆ OS top |
| 1485 | 514 OOX {X} cs {x} ox with eq→ *iso ox |
| 515 ... | record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef (OS top) k) (sym x=ψz) ( P＼CS=OS top (cs comp01)) where | |
| 516 comp01 : odef (* X) (& (* z)) | |
| 517 comp01 = subst (λ k → odef (* X) k) (sym &iso) az | |
| 518 -- if all finite intersection of X contains something, | |
| 519 -- there is no finite cover. From Compactness, (OX X) is not a cover of L ( contraposition of Compact) | |
| 520 -- it means there is a limit | |
| 1199 | 521 record NC {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) (0<X : o∅ o< X) : Set n where |
| 522 field -- find an element x, which is not covered (which is a limit point) | |
| 523 x : Ordinal | |
| 524 yx : {y : Ordinal} (Xy : odef (* X) y) → odef (* y) x | |
| 1187 | 525 has-intersection : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1199 | 526 → (0<X : o∅ o< X ) → NC CX fip 0<X |
| 527 has-intersection {X} CX fip 0<X = intersection where | |
| 1198 | 528 e : HOD -- we have an element of X |
| 1476 | 529 e = minimal (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) |
| 1198 | 530 Xe : odef (* X) (& e) |
| 1476 | 531 Xe = x∋minimal (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) |
| 1183 | 532 no-cover : ¬ ( (* (OX CX)) covers L ) |
| 1198 | 533 no-cover cov = ⊥-elim (no-finite-cover (Compact.isCover compact (OOX CX) cov)) where |
| 534 -- construct Subase from Finite-∪ | |
| 1180 | 535 fp01 : Ordinal |
| 536 fp01 = Compact.finCover compact (OOX CX) cov | |
| 1194 | 537 record SB (t : Ordinal) : Set n where |
| 538 field | |
| 539 i : Ordinal | |
| 540 sb : Subbase (* X) (& (L ＼ * i)) | |
| 1198 | 541 t⊆i : (L ＼ * i) ⊆ (L ＼ Union ( * t ) ) |
| 1194 | 542 fp02 : (t : Ordinal) → Finite-∪ (* (OX CX)) t → SB t |
| 1198 | 543 fp02 t fin-e = record { i = & ( L ＼ e) ; sb = gi (subst (λ k → odef (* X) k) fp21 Xe) ; t⊆i = fp23 } where |
| 544 -- t ≡ o∅, no cover. Any subst of L is ok and we have e ⊆ L | |
| 545 fp22 : e ⊆ L | |
| 546 fp22 {x} lt = cs⊆L top (CX Xe) lt | |
| 547 fp21 : & e ≡ & (L ＼ * (& (L ＼ e))) | |
| 1485 | 548 fp21 = ==→o≡ ( ==-trans (==-sym (L＼Lx=x {e} fp22 )) (＼-cong L L (L ＼ e) (* (& (L ＼ e))) ==-refl (==-sym *iso ))) |
| 1198 | 549 fp23 : (L ＼ * (& (L ＼ e))) ⊆ (L ＼ Union (* o∅)) |
| 1485 | 550 fp23 {x} ⟪ Lx , _ ⟫ = ⟪ Lx , ( λ lt → ⊥-elim ( ¬∅∋ (eq→ (begin |
| 551 (* o∅) ≈⟨ o∅==od∅ ⟩ | |
| 552 od∅ ∎ ) (subst (λ k → odef (* o∅) k ) (sym &iso) (Own.ao lt ))))) ⟫ | |
| 553 where open EqHOD ==-Setoid | |
| 1198 | 554 fp02 t (fin-i {x} tx ) = record { i = x ; sb = gi fp03 ; t⊆i = fp24 } where |
| 555 -- we have a single cover x, L ＼ * x is single finite intersection | |
| 1197 | 556 fp24 : (L ＼ * x) ⊆ (L ＼ Union (* (& (* x , * x)))) |
| 1485 | 557 fp24 {y} ⟪ Lx , not ⟫ = ⟪ Lx , fp26 ⟫ where |
| 558 fp26 : ¬ odef (Union (* (& (* x , * x)))) y | |
| 559 fp26 record { owner = owner ; ao = ao ; ox = ox } with eq→ *iso ao | |
| 560 ... | case1 refl = not (eq→ *iso ox ) | |
| 561 ... | case2 refl = not (eq→ *iso ox ) | |
| 1198 | 562 fp03 : odef (* X) (& (L ＼ * x)) -- becase x is an element of Replace (* X) (λ z → L ＼ z ) |
| 1485 | 563 fp03 with eq→ *iso tx |
| 564 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where | |
| 565 fip34 : * z1 ⊆ L | |
| 566 fip34 {w} wz1 = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX az1) ) wz1 | |
| 567 fip33 : z1 ≡ & (L ＼ * x) | |
| 568 fip33 = trans (sym &iso ) (==→o≡ ( begin | |
| 569 (* z1) ≈⟨ ==-sym (L＼Lx=x {* z1} fip34 ) ⟩ | |
| 570 (L ＼ ( L ＼ * z1)) ≈⟨ ＼-cong L L ( L ＼ * z1) (* x) ==-refl (==-trans (==-sym *iso) (o≡→== (sym x=ψz1)) ) ⟩ | |
| 571 (L ＼ * x ) ∎ ) ) where open EqHOD ==-Setoid | |
| 1198 | 572 fp02 t (fin-∪ {tx} {ty} ux uy ) = record { i = & (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))) ; sb = fp11 ; t⊆i = fp35 } where |
| 573 fp35 : (L ＼ * (& (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))) ⊆ (L ＼ Union (* (& (* tx ∪ * ty)))) | |
| 1485 | 574 fp35 = λ lt → eq← fp42 ( fp36 (eq→ ( ＼-cong L L (* (& (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))) |
| 575 (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))) ==-refl *iso ) lt ) ) where | |
| 576 fp43 : (P : HOD ) → Union (* (& P)) =h= Union P | |
| 577 eq← (fp43 P) record { owner = owner ; ao = ao ; ox = ox } = record { owner = owner ; ao = eq← *iso ao ; ox = ox } | |
| 578 eq→ (fp43 P) record { owner = owner ; ao = ao ; ox = ox } = record { owner = owner ; ao = eq→ *iso ao ; ox = ox } | |
| 579 fp42 : ( L ＼ Union (* (& (* tx ∪ * ty)))) =h= (L ＼ Union (* tx ∪ * ty)) | |
| 580 eq→ fp42 {x} ⟪ lx , nux ⟫ = ⟪ lx , (λ lt → nux (eq← (fp43 _) lt) ) ⟫ | |
| 581 eq← fp42 {x} ⟪ lx , nux ⟫ = ⟪ lx , (λ lt → nux (eq→ (fp43 _) lt) ) ⟫ | |
| 1197 | 582 fp40 : {z tz : Ordinal } → Finite-∪ (* (OX CX)) tz → odef (Union (* tz )) z → odef L z |
| 1199 | 583 fp40 {z} {.(Ordinals.o∅ O)} fin-e record { owner = owner ; ao = ao ; ox = ox } |
| 1485 | 584 = ⊥-elim ( ¬∅∋ (eq→ o∅==od∅ (subst (λ k → odef (* o∅) k ) (sym &iso) ao ))) |
| 585 fp40 {z} {.(& (* _ , * _))} (fin-i {w} x) uz = fp41 x (eq→ (fp43 _) uz) where | |
| 1197 | 586 fp41 : (x : odef (* (OX CX)) w) → (uz : odef (Union (* w , * w)) z ) → odef L z |
| 587 fp41 x record { owner = .(& (* w)) ; ao = (case1 refl) ; ox = ox } = | |
| 1485 | 588 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (eq→ *iso ox ) |
| 1197 | 589 fp41 x record { owner = .(& (* w)) ; ao = (case2 refl) ; ox = ox } = |
| 1485 | 590 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (eq→ *iso ox ) |
| 591 fp40 {z} {.(& (* _ ∪ * _))} (fin-∪ {x1} {y1} ftx fty) uz with eq→ (fp43 _ ) uz | |
| 592 ... | record { owner = o ; ao = case1 x1o ; ox = oz } = fp40 ftx record { owner = o ; ao = x1o ; ox = oz } | |
| 593 ... | record { owner = o ; ao = case2 y1o ; ox = oz } = fp40 fty record { owner = o ; ao = y1o ; ox = oz } | |
| 1198 | 594 fp36 : (L ＼ (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* tx ∪ * ty)) |
| 1197 | 595 fp36 {z} ⟪ Lz , not ⟫ = ⟪ Lz , fp37 ⟫ where |
| 596 fp37 : ¬ odef (Union (* tx ∪ * ty)) z | |
| 597 fp37 record { owner = owner ; ao = (case1 ax) ; ox = ox } = not (case1 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where | |
| 1198 | 598 fp38 : (L ＼ (* (SB.i (fp02 tx ux)))) ⊆ (L ＼ Union (* tx)) |
| 599 fp38 = SB.t⊆i (fp02 tx ux) | |
| 600 fp39 : Union (* tx) ⊆ (* (SB.i (fp02 tx ux))) | |
| 1486 | 601 fp39 {w} txw with ∨L＼X {* (SB.i (fp02 tx ux))} (fp40 ux txw) |
| 602 ... | case1 sb = sb | |
| 603 ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) txw ) | |
| 1197 | 604 fp37 record { owner = owner ; ao = (case2 ax) ; ox = ox } = not (case2 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where |
| 1198 | 605 fp38 : (L ＼ (* (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* ty)) |
| 606 fp38 = SB.t⊆i (fp02 ty uy) | |
| 607 fp39 : Union (* ty) ⊆ (* (SB.i (fp02 ty uy))) | |
| 1486 | 608 fp39 {w} tyw with ∨L＼X {* (SB.i (fp02 ty uy))} (fp40 uy tyw) |
| 609 ... | case1 sb = sb | |
| 610 ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) tyw ) | |
| 1194 | 611 fp04 : {tx ty : Ordinal} → & (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) ≡ & (L ＼ * (& (* tx ∪ * ty))) |
| 1485 | 612 fp04 {tx} {ty} = ==→o≡ record { eq→ = fp05 ; eq← = fp09 } where |
| 1187 | 613 fp05 : {x : Ordinal} → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x → odef (L ＼ * (& (* tx ∪ * ty))) x |
| 1485 | 614 fp05 {x} lt with eq← *iso∩ lt |
| 615 ... | ⟪ ⟪ Lx , ¬tx ⟫ , ⟪ Ly , ¬ty ⟫ ⟫ = eq→ (＼-cong L L (* tx ∪ * ty) (* (& (* tx ∪ * ty))) ==-refl (==-sym *iso) ) ⟪ Lx , fp06 ⟫ where | |
| 616 fp06 : ¬ odef (* tx ∪ * ty) x | |
| 617 fp06 (case1 tx) = ¬tx tx | |
| 618 fp06 (case2 ty) = ¬ty ty | |
| 1187 | 619 fp09 : {x : Ordinal} → odef (L ＼ * (& (* tx ∪ * ty))) x → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x |
| 1485 | 620 fp09 {x} lt with eq→ (＼-cong L L (* (& (* tx ∪ * ty))) (* tx ∪ * ty) ==-refl *iso ) lt |
| 621 ... | ⟪ Lx , ¬tx∨ty ⟫ = eq→ *iso∩ | |
| 1187 | 622 ⟪ ⟪ Lx , ( λ tx → ¬tx∨ty (case1 tx)) ⟫ , ⟪ Lx , ( λ ty → ¬tx∨ty (case2 ty)) ⟫ ⟫ |
| 1198 | 623 fp11 : Subbase (* X) (& (L ＼ * (& ((* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))))) |
| 624 fp11 = subst (λ k → Subbase (* X) k ) fp04 ( g∩ (SB.sb (fp02 tx ux)) (SB.sb (fp02 ty uy )) ) | |
| 625 -- | |
| 626 -- becase of fip, finite cover cannot be a cover | |
| 627 -- | |
| 1195 | 628 fcov : Finite-∪ (* (OX CX)) (Compact.finCover compact (OOX CX) cov) |
| 629 fcov = Compact.isFinite compact (OOX CX) cov | |
| 1196 | 630 0<sb : {i : Ordinal } → (sb : Subbase (* X) (& (L ＼ * i))) → o∅ o< & (L ＼ * i) |
| 631 0<sb {i} sb = fip sb | |
| 632 sb : SB (Compact.finCover compact (OOX CX) cov) | |
| 633 sb = fp02 fp01 (Compact.isFinite compact (OOX CX) cov) | |
| 1198 | 634 no-finite-cover : ¬ ( (* (Compact.finCover compact (OOX CX) cov)) covers L ) |
| 1486 | 635 no-finite-cover fcovers = ⊥-elim ( o<¬≡ (sym (==→o≡ f22)) f25 ) where |
| 1196 | 636 f23 : (L ＼ * (SB.i sb)) ⊆ ( L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) |
| 1198 | 637 f23 = SB.t⊆i sb |
| 1196 | 638 f22 : (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) =h= od∅ |
| 639 f22 = record { eq→ = λ lt → ⊥-elim ( f24 lt) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where | |
| 640 f24 : {x : Ordinal } → ¬ ( odef (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) x ) | |
| 641 f24 {x} ⟪ Lx , not ⟫ = not record { owner = cover fcovers Lx ; ao = P∋cover fcovers Lx ; ox = isCover fcovers Lx } | |
| 642 f25 : & od∅ o< (& (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ) | |
| 643 f25 = ordtrans<-≤ (subst (λ k → k o< & (L ＼ * (SB.i sb))) (sym ord-od∅) (0<sb (SB.sb sb) ) ) ( begin | |
| 644 & (L ＼ * (SB.i sb)) ≤⟨ ⊆→o≤ f23 ⟩ | |
| 645 & (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ∎ ) where open o≤-Reasoning O | |
| 1199 | 646 -- if we have no cover, we can consruct NC |
| 647 intersection : NC CX fip 0<X | |
| 1476 | 648 intersection with p∨¬p (NC CX fip 0<X) |
| 1184 | 649 ... | case1 nc = nc |
| 1185 | 650 ... | case2 ¬nc = ⊥-elim ( no-cover record { cover = λ Lx → & (L ＼ coverf Lx) ; P∋cover = fp22 ; isCover = fp23 } ) where |
| 651 coverSet : {x : Ordinal} → odef L x → HOD | |
| 1186 | 652 coverSet {x} Lx = record { od = record { def = λ y → odef (* X) y ∧ odef (L ＼ * y) x } ; odmax = X |
| 653 ; <odmax = λ {x} lt → subst (λ k → x o< k) &iso ( odef< (proj1 lt)) } | |
| 1185 | 654 fp17 : {x : Ordinal} → (Lx : odef L x ) → ¬ ( coverSet Lx =h= od∅ ) |
| 1187 | 655 fp17 {x} Lx eq = ⊥-elim (¬nc record { x = x ; yx = fp19 } ) where |
| 1185 | 656 fp19 : {y : Ordinal} → odef (* X) y → odef (* y) x |
| 1486 | 657 fp19 {y} Xy with ∨L＼X {* y} {x} Lx |
| 658 ... | case1 yx = yx | |
| 659 ... | case2 lyx = ⊥-elim ( ¬x<0 {y} ( eq→ eq fp20 )) where | |
| 660 fp20 : odef (* X) y ∧ odef (L ＼ * y) x | |
| 661 fp20 = ⟪ Xy , lyx ⟫ | |
| 1185 | 662 coverf : {x : Ordinal} → (Lx : odef L x ) → HOD |
| 1476 | 663 coverf Lx = minimal (coverSet Lx) (fp17 Lx) |
| 1186 | 664 fp22 : {x : Ordinal} (lt : odef L x) → odef (* (OX CX)) (& (L ＼ coverf lt)) |
| 1486 | 665 fp22 {x} Lx = eq← *iso record { z = _ ; az = fp25 ; x=ψz = fp24 } where |
| 1186 | 666 fp24 : & (L ＼ coverf Lx) ≡ & (L ＼ * (& (coverf Lx))) |
| 1486 | 667 fp24 = ==→o≡ (＼-cong L L (coverf Lx) (* (& (coverf Lx))) ==-refl (==-sym *iso) ) |
| 1185 | 668 fp25 : odef (* X) (& (coverf Lx)) |
| 1476 | 669 fp25 = proj1 ( x∋minimal (coverSet Lx) (fp17 Lx) ) |
| 1186 | 670 fp23 : {x : Ordinal} (lt : odef L x) → odef (* (& (L ＼ coverf lt))) x |
| 1486 | 671 fp23 {x} Lx = eq← *iso ⟪ Lx , fp26 ⟫ where |
| 1186 | 672 fp26 : ¬ odef (coverf Lx) x |
| 1486 | 673 fp26 llx = proj2 (proj2 ( x∋minimal (coverSet Lx) (fp17 Lx) )) (eq← *iso llx) |
| 1199 | 674 limit : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) → Ordinal |
| 1180 | 675 limit {X} CX fip with trio< X o∅ |
| 676 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 677 ... | tri≈ ¬a b ¬c = o∅ | |
| 1199 | 678 ... | tri> ¬a ¬b c = NC.x ( has-intersection CX fip c) |
| 1175 | 679 fip00 : {X : Ordinal} (CX : * X ⊆ CS top) |
| 1187 | 680 (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1175 | 681 {x : Ordinal} → odef (* X) x → odef (* x) (limit CX fip ) |
| 1180 | 682 fip00 {X} CX fip {x} Xx with trio< X o∅ |
| 683 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 684 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) (subst (λ k → o∅ o< k) &iso (∈∅< Xx) ) ) | |
| 1199 | 685 ... | tri> ¬a ¬b c = NC.yx ( has-intersection CX fip c ) Xx |
| 431 | 686 |
|
1113
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
|
687 open Filter |
| 1102 | 688 |
|
1228
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
689 -- |
|
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
690 -- {v | Neighbor lim v} set of u ⊆ v ⊆ P where u is an open set u ∋ x |
|
e3f20bc4fac9
last part of Tychonoff
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1218
diff
changeset
|
691 -- |
| 1159 | 692 record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where |
| 693 field | |
| 694 u : Ordinal | |
| 695 ou : odef (OS TP) u | |
| 696 ux : odef (* u) x | |
| 697 v⊆P : * v ⊆ P | |
| 1170 | 698 u⊆v : * u ⊆ * v |
| 1102 | 699 |
| 1169 | 700 -- |
| 701 -- Neighbor on x is a Filter (on Power P) | |
| 702 -- | |
| 1170 | 703 NeighborF : {P : HOD} (TP : Topology P) (x : Ordinal ) → Filter {Power P} {P} (λ x → x) |
| 1169 | 704 NeighborF {P} TP x = record { filter = NF ; f⊆L = NF⊆PP ; filter1 = f1 ; filter2 = f2 } where |
| 1168 | 705 nf00 : {v : Ordinal } → Neighbor TP x v → odef (Power P) v |
| 706 nf00 {v} nei y vy = Neighbor.v⊆P nei vy | |
| 1167 | 707 NF : HOD |
| 1168 | 708 NF = record { od = record { def = λ v → Neighbor TP x v } ; odmax = & (Power P) ; <odmax = λ lt → odef< (nf00 lt) } |
| 1167 | 709 NF⊆PP : NF ⊆ Power P |
| 1168 | 710 NF⊆PP = nf00 |
| 711 f1 : {p q : HOD} → Power P ∋ q → NF ∋ p → p ⊆ q → NF ∋ q | |
| 1170 | 712 f1 {p} {q} Pq Np p⊆q = record { u = Neighbor.u Np ; ou = Neighbor.ou Np ; ux = Neighbor.ux Np ; v⊆P = Pq _ ; u⊆v = f11 } where |
| 1168 | 713 f11 : * (Neighbor.u Np) ⊆ * (& q) |
| 1486 | 714 f11 {x} ux = eq← *iso ( p⊆q (eq→ *iso (Neighbor.u⊆v Np ux)) ) |
| 1168 | 715 f2 : {p q : HOD} → NF ∋ p → NF ∋ q → Power P ∋ (p ∩ q) → NF ∋ (p ∩ q) |
| 1170 | 716 f2 {p} {q} Np Nq Ppq = record { u = upq ; ou = ou ; ux = ux ; v⊆P = Ppq _ ; u⊆v = f20 } where |
| 1168 | 717 upq : Ordinal |
| 718 upq = & ( * (Neighbor.u Np) ∩ * (Neighbor.u Nq) ) | |
| 719 ou : odef (OS TP) upq | |
| 720 ou = o∩ TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou Np)) (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou Nq)) | |
| 721 ux : odef (* upq) x | |
| 1486 | 722 ux = eq← *iso ⟪ Neighbor.ux Np , Neighbor.ux Nq ⟫ |
| 1168 | 723 f20 : * upq ⊆ * (& (p ∩ q)) |
| 1486 | 724 f20 {x} lt = eq← *iso ⟪ eq→ *iso (Neighbor.u⊆v Np (proj1 pq)) , eq→ *iso (Neighbor.u⊆v Nq (proj2 pq)) ⟫ where |
| 725 pq : odef (* (Neighbor.u Np) ∩ * (Neighbor.u Nq)) x | |
| 726 pq = eq→ *iso lt | |
| 1153 | 727 |
| 1165 | 728 CAP : (P : HOD) {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) |
| 1486 | 729 CAP P {p} {q} Pp Pq x pqx with eq→ *iso pqx |
| 730 ... | ⟪ px , qx ⟫ = Pp _ (eq← *iso px ) | |
| 1165 | 731 |
| 1170 | 732 NEG : (P : HOD) {p : HOD } → Power P ∋ p → Power P ∋ (P ＼ p ) |
| 1486 | 733 NEG P {p} Pp x vx with eq→ *iso vx |
| 734 ... | ⟪ Px , npx ⟫ = Px | |
| 1142 | 735 |
| 1484 | 736 |
| 737 |