Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1476:32001d93755b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 28 Jun 2024 20:55:38 +0900 |
| parents | 47d3cc596d68 |
| children | 0b30bb7c7501 |
| 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 |
| 1101 | 62 |
| 482 | 63 record Topology ( L : HOD ) : Set (suc n) where |
| 431 | 64 field |
| 65 OS : HOD | |
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66 OS⊆PL : OS ⊆ Power L |
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67 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) |
| 1161 | 68 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P |
| 1210 | 69 OS∋od∅ : OS ∋ od∅ -- OS ∋ Union od∅ |
| 1160 | 70 --- we may add |
| 71 -- OS∋L : OS ∋ L | |
| 1101 | 72 -- closed Set |
| 1476 | 73 open BAlgebra O HODAxiom ho< L ? |
| 1101 | 74 CS : HOD |
| 1119 | 75 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where |
| 76 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) | |
| 77 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) | |
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78 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L |
| 1476 | 79 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) ? xy ) |
| 1122 | 80 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L |
| 1476 | 81 cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) ? xy ) |
| 1122 | 82 CS∋L : CS ∋ L |
| 1476 | 83 CS∋L = ⟪ subst (λ k → k ⊆ L) ? (λ x → x) , subst (λ k → odef OS (& k)) (sym lem0) OS∋od∅ ⟫ where |
| 1123 | 84 lem0 : L ＼ * (& L) ≡ od∅ |
| 1476 | 85 lem0 = subst (λ k → L ＼ k ≡ od∅) ? ? -- L＼L=0 |
| 1154 | 86 CS⊆PL : CS ⊆ Power L |
| 1161 | 87 CS⊆PL {x} Cx y xy = proj1 Cx xy |
| 1160 | 88 P＼CS=OS : {cs : HOD} → CS ∋ cs → OS ∋ ( L ＼ cs ) |
| 1476 | 89 P＼CS=OS {cs} ⟪ cs⊆L , olcs ⟫ = subst (λ k → odef OS k) ? olcs |
| 1160 | 90 P＼OS=CS : {cs : HOD} → OS ∋ cs → CS ∋ ( L ＼ cs ) |
| 1476 | 91 P＼OS=CS {os} oos = ⟪ subst (λ k → k ⊆ L) ? proj1 |
| 92 , subst (λ k → odef OS k) (cong (&) (trans (sym ?) (cong (λ k → L ＼ k) ?) )) oos ⟫ | |
| 431 | 93 |
| 482 | 94 open Topology |
| 431 | 95 |
| 1163 | 96 -- Closure ( Intersection of Closed Set which include A ) |
| 97 | |
| 1162 | 98 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → HOD |
| 99 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 | 100 ; odmax = & L ; <odmax = odef∧< } |
| 1122 | 101 |
| 1476 | 102 ClL : {L : HOD} → (top : Topology L) → Cl top L =h= L |
| 103 ClL {L} top = record { eq→ = λ {x} ic | |
| 104 → subst (λ k → odef k x) ? ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) ? ( λ x → x))) | |
| 105 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } | |
| 1123 | 106 |
| 1163 | 107 -- Closure is Closed Set |
| 108 | |
| 1162 | 109 CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → CS top ∋ Cl top A |
| 1476 | 110 CS∋Cl {L} top A = subst (λ k → CS top ∋ k) ? (P＼OS=CS top UOCl-is-OS) where |
| 111 open BAlgebra O HODAxiom ho< L ? | |
| 1163 | 112 OCl : HOD -- set of open set which it not contains A |
| 1162 | 113 OCl = record { od = record { def = λ o → odef (OS top) o ∧ ( A ⊆ (L ＼ * o) ) } ; odmax = & (OS top) ; <odmax = odef∧< } |
| 114 OCl⊆OS : OCl ⊆ OS top | |
| 115 OCl⊆OS ox = proj1 ox | |
| 116 UOCl-is-OS : OS top ∋ Union OCl | |
| 117 UOCl-is-OS = o∪ top OCl⊆OS | |
| 118 cc00 : (L ＼ Union OCl) =h= Cl top A | |
| 119 cc00 = record { eq→ = cc01 ; eq← = cc03 } where | |
| 120 cc01 : {x : Ordinal} → odef (L ＼ Union OCl) x → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) | |
| 121 cc01 {x} ⟪ Lx , nul ⟫ = ⟪ Lx , ( λ c cc ac → cc02 c cc ac nul ) ⟫ where | |
| 122 cc02 : (c : Ordinal) → odef (CS top) c → A ⊆ * c → ¬ odef (Union OCl) x → odef (* c) x | |
| 1476 | 123 cc02 c cc ac nox with ODC.∋-p (* c) (* x) |
| 1162 | 124 ... | yes y = subst (λ k → odef (* c) k) &iso y |
| 1476 | 125 ... | no ncx = ⊥-elim ( nox record { owner = & ( L ＼ * c) ; ao = ⟪ proj2 cc , cc07 ⟫ ; ox = subst (λ k → odef k x) ? cc06 } ) where |
| 1162 | 126 cc06 : odef (L ＼ * c) x |
| 127 cc06 = ⟪ Lx , subst (λ k → ¬ odef (* c) k) &iso ncx ⟫ | |
| 128 cc08 : * c ⊆ L | |
| 129 cc08 = cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) cc ) | |
| 130 cc07 : A ⊆ (L ＼ * (& (L ＼ * c))) | |
| 131 cc07 {z} az = subst (λ k → odef k z ) ( | |
| 1476 | 132 begin * c ≡⟨ sym ? ⟩ |
| 133 L ＼ (L ＼ * c) ≡⟨ cong (λ k → L ＼ k ) ? ⟩ | |
| 1162 | 134 L ＼ * (& (L ＼ * c)) ∎ ) ( ac az ) where open ≡-Reasoning |
| 135 cc03 : {x : Ordinal} → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) → odef (L ＼ Union OCl) x | |
| 136 cc03 {x} ⟪ Lx , ccx ⟫ = ⟪ Lx , cc04 ⟫ where | |
| 1163 | 137 -- 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 | 138 cc04 : ¬ odef (Union OCl) x |
| 1476 | 139 cc04 record { owner = o ; ao = ⟪ oo , A⊆L-o ⟫ ; ox = ox } = proj2 ( subst (λ k → odef k x) ? cc05) ox where |
| 1162 | 140 cc05 : odef (* (& (L ＼ * o))) x |
| 1476 | 141 cc05 = ccx (& (L ＼ * o)) (P＼OS=CS top (subst (λ k → odef (OS top) k) (sym &iso) oo)) (subst (λ k → A ⊆ k) ? A⊆L-o) |
| 1161 | 142 |
| 1476 | 143 CS∋x→Clx=x : {L x : HOD} → (top : Topology L) → CS top ∋ x → Cl top x =h= x |
| 144 CS∋x→Clx=x {L} {x} top cx = record { eq→ = cc10 ; eq← = cc11 } where | |
| 1210 | 145 cc10 : {y : Ordinal} → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) → odef x y |
| 1476 | 146 cc10 {y} ⟪ Ly , cc ⟫ = subst (λ k → odef k y) ? ( cc (& x) cx (λ {z} xz → subst (λ k → odef k z) ? xz ) ) |
| 1210 | 147 cc11 : {y : Ordinal} → odef x y → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) |
| 148 cc11 {y} xy = ⟪ cs⊆L top cx xy , (λ c oc x⊆c → x⊆c xy ) ⟫ | |
| 1160 | 149 |
| 1119 | 150 -- Subbase P |
| 151 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 152 |
| 153 data Subbase (P : HOD) : Ordinal → Set n where | |
| 154 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 155 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 156 | |
| 1119 | 157 -- |
| 1150 | 158 -- if y is in a Subbase, some element of P contains it |
| 1119 | 159 |
| 1111 | 160 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 161 sbp P {x} (gi {y} px) = x | |
| 162 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 163 |
| 1111 | 164 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 165 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
| 1476 | 166 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) ? xy)) |
| 1107 | 167 |
| 1155 | 168 sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x |
| 169 sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) | |
| 170 sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) | |
| 171 | |
| 1119 | 172 -- An open set generate from a base |
| 173 -- | |
| 1161 | 174 -- OS = { U ⊆ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊆ U } |
| 1114 | 175 |
| 1115 | 176 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 177 field |
| 1150 | 178 b : Ordinal |
| 1161 | 179 u⊆L : * u ⊆ L |
| 1114 | 180 sb : Subbase P b |
| 181 b⊆u : * b ⊆ * u | |
| 182 bx : odef (* b) x | |
| 1150 | 183 x⊆L : odef L x |
| 1161 | 184 x⊆L = u⊆L (b⊆u bx) |
| 1114 | 185 |
| 1115 | 186 SO : (L P : HOD) → HOD |
| 1119 | 187 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 |
| 188 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 189 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 190 |
| 1111 | 191 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 192 field |
| 1122 | 193 P⊆PL : P ⊆ Power L |
| 1116 | 194 -- we may need these if OS ∋ L is necessary |
| 195 -- p : {x : HOD} → L ∋ x → HOD | |
| 1161 | 196 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx |
| 1116 | 197 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x |
| 1110 | 198 |
| 1152 | 199 InducedTopology : (L P : HOD) → IsSubBase L P → Topology L |
| 200 InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 | |
| 1122 | 201 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 202 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1476 | 203 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) ? (sym &iso) 0x )) |
| 1115 | 204 tp00 : SO L P ⊆ Power L |
| 205 tp00 {u} ou x ux with ou ux | |
| 1161 | 206 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊆L (b⊆u bx) |
| 1115 | 207 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1476 | 208 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊆L = subst (λ k → k ⊆ L) ? ul |
| 1116 | 209 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 210 px : odef (* (& p)) x |
| 1476 | 211 px = subst (λ k → odef k x ) ? ( proj1 (subst (λ k → odef k _ ) ? ux ) ) |
| 1115 | 212 qx : odef (* (& q)) x |
| 1476 | 213 qx = subst (λ k → odef k x ) ? ( proj2 (subst (λ k → odef k _ ) ? ux ) ) |
| 1115 | 214 b : Ordinal |
| 215 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 216 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1476 | 217 tp08 = subst₂ (λ j k → j ⊆ k ) ? ? (⊆∩-dist {(* (Base.b (op px)) ∩ * (Base.b (oq qx)))} {p} {q} tp09 tp10 ) where |
| 1116 | 218 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 219 tp11 = Base.b⊆u (op px) | |
| 220 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 221 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 222 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1476 | 223 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) ? tp11) |
| 1150 | 224 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1476 | 225 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) ? tp12) |
| 1116 | 226 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x |
| 1476 | 227 tp14 = subst (λ k → odef k x ) ? ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ |
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228 ul : (p ∩ q) ⊆ L |
| 1476 | 229 ul = subst (λ k → k ⊆ L ) ? (λ {z} pq → (Base.u⊆L (op px)) (pz pq) ) where |
| 1116 | 230 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 1476 | 231 pz {z} pq = subst (λ k → odef k z ) ? ( proj1 (subst (λ k → odef k _ ) ? pq ) ) |
| 1161 | 232 tp02 : { q : HOD} → q ⊆ SO L P → SO L P ∋ Union q |
| 1476 | 233 tp02 {q} q⊆O {x} = ? -- ux with subst (λ k → odef k x) ? ux |
| 234 -- . | record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 235 -- . | 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) ? tp04 | |
| 236 -- ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) ? tp06 ; bx = bx } where | |
| 237 -- tp05 : Union q ⊆ L | |
| 238 -- tp05 {z} record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 239 -- ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } | |
| 240 -- = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) | |
| 241 -- tp04 : Union q ⊆ L | |
| 242 -- tp04 = tp05 | |
| 243 -- tp06 : * b ⊆ Union q | |
| 244 -- tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } | |
| 1110 | 245 |
| 1142 | 246 -- Product Topology |
| 247 | |
| 248 open ZFProduct | |
| 249 | |
| 1150 | 250 -- Product Topology is not |
| 1142 | 251 -- ZFP (OS TP) (OS TQ) (box) |
| 252 | |
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253 -- Rectangle subset (zπ1 ⁻¹ p) |
| 1142 | 254 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where |
| 255 field | |
| 1172 | 256 p : Ordinal |
| 1142 | 257 op : odef (OS TP) p |
| 258 prod : x ≡ & (ZFP (* p) Q ) | |
| 259 | |
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260 -- Rectangle subset (zπ12⁻¹ q ) |
| 1142 | 261 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where |
| 262 field | |
| 1172 | 263 q : Ordinal |
| 1142 | 264 oq : odef (OS TQ) q |
| 265 prod : x ≡ & (ZFP P (* q )) | |
| 266 | |
| 267 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 | 268 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 | 269 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) |
| 1476 | 270 tp01 w wz = ? |
| 271 -- tp01 w wz with subst (λ k → odef k w ) ? wz | |
| 272 -- ... | 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 | |
| 273 -- tp03 : odef P a | |
| 274 -- tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 1172 | 275 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 | 276 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) |
| 1476 | 277 tp01 w wz = ? -- with subst (λ k → odef k w ) ? wz |
| 278 -- ... | 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 | |
| 279 -- tp03 : odef Q b | |
| 280 -- tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 1142 | 281 |
| 282 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 283 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 | |
| 284 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 285 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 286 |
| 287 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 1152 | 288 ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } |
| 1142 | 289 |
| 1152 | 290 -- covers ( q ⊆ Union P ) |
| 1101 | 291 |
| 1120 | 292 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 293 field |
| 1120 | 294 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 295 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 296 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 297 |
| 298 open _covers_ | |
| 431 | 299 |
| 300 -- Finite Intersection Property | |
| 301 | |
| 1120 | 302 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 303 field |
| 1150 | 304 limit : {X : Ordinal } → * X ⊆ CS top |
| 1187 | 305 → ( { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → Ordinal |
| 1150 | 306 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 307 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1143 | 308 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 309 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 310 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1150 | 311 → {x : Ordinal } → odef (* X) x |
| 1143 | 312 → odef L (limit CX fip) |
| 313 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 | 314 |
| 315 -- Compact | |
| 316 | |
| 1119 | 317 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1188 | 318 fin-e : Finite-∪ S o∅ |
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319 fin-i : {x : Ordinal } → odef S x → Finite-∪ S (& ( * x , * x )) -- an element of S |
| 1188 | 320 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 1198 | 321 -- Finite-∪ S y → Union y ⊆ S |
| 431 | 322 |
| 1120 | 323 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 324 field |
| 1120 | 325 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 326 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 327 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 328 |
| 329 -- FIP is Compact | |
| 330 | |
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331 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 332 FIP→Compact {L} top fip with trio< (& L) o∅ |
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333 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 334 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 335 -- L is empty | |
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336 fip02 : {x : Ordinal } → ¬ odef L x |
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337 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 338 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
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339 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 340 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1188 | 341 fip00 {X} xo xcp = fin-e |
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342 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 343 -- set of coset of X |
| 1476 | 344 open BAlgebra O HODAxiom ho< L ? |
| 1121 | 345 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal |
| 1293 | 346 CX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼ ) |
| 1150 | 347 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1476 | 348 CCX {X} os {x} ox = ? -- with subst (λ k → odef k x) ? ox |
| 349 -- ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 350 -- fip07 : z ≡ & (L ＼ * x) | |
| 351 -- 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 | |
| 352 -- fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 353 -- fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) ? not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ | |
| 354 -- ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) | |
| 355 -- fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 356 -- fip09 {w} zw = ⟪ os⊆L top (os (subst (λ k → odef (* X) k) (sym &iso) az)) zw , subst (λ k → ¬ odef k w) ? fip10 ⟫ where | |
| 357 -- fip10 : ¬ (odef (L ＼ * z) w) | |
| 358 -- fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 359 -- fip06 : odef (OS top) (& (L ＼ * x)) | |
| 360 -- fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 361 -- fip05 : * x ⊆ L | |
| 362 -- fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) ? ) xw ) | |
| 363 | |
| 1143 | 364 -- |
| 365 -- X covres L means Intersection of (CX X) contains nothing | |
| 1152 | 366 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) |
| 1143 | 367 -- it means there is a finite cover |
| 368 -- | |
| 1293 | 369 finCoverBase : {X : Ordinal } → * X ⊆ OS top → * X covers L → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅ |
| 1476 | 370 finCoverBase {X} ox oc with p∨¬p (Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅) |
| 1189 | 371 ... | case1 sb = sb |
| 1199 | 372 ... | case2 ¬sb = ⊥-elim (¬¬cover fip25 fip20) where |
| 373 ¬¬cover : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) | |
| 374 ¬¬cover {z} Lz nc = nc ( P∋cover oc Lz ) (isCover oc _ ) | |
| 375 -- ¬sb → we have finite intersection | |
| 1187 | 376 fip02 : {x : Ordinal} → Subbase (* (CX ox)) x → o∅ o< x |
| 377 fip02 {x} sc with trio< x o∅ | |
| 1148 | 378 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 379 ... | tri> ¬a ¬b c = c | |
| 1476 | 380 ... | tri≈ ¬a b ¬c = ⊥-elim (¬sb (subst₂ (λ j k → Subbase j k ) ? b sc )) |
| 1150 | 381 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) |
| 1476 | 382 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( x∋minimal L (0<P→ne 0<L) ) ))) |
| 383 fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( x∋minimal L (0<P→ne 0<L) ) )) )) ? | |
| 1150 | 384 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 | 385 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 386 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 387 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 388 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 389 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1476 | 390 fip22 = subst (λ k → odef k (& ( L ＼ * y ))) ? record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 391 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 1476 | 392 fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) ? ( FIP.is-limit fip (CCX ox) fip02 fip22 ) |
| 1199 | 393 -- create HOD from Subbase ( finite intersection ) |
| 1293 | 394 finCoverSet : {X : Ordinal } → (x : Ordinal) → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x → HOD |
| 1190 | 395 finCoverSet {X} x (gi rx) = ( L ＼ * x ) , ( L ＼ * x ) |
| 1189 | 396 finCoverSet {X} x∩y (g∩ {x} {y} sx sy) = finCoverSet {X} x sx ∪ finCoverSet {X} y sy |
| 1149 | 397 -- |
| 1121 | 398 -- this defines finite cover |
| 1120 | 399 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1189 | 400 finCover {X} ox oc = & ( finCoverSet o∅ (finCoverBase ox oc)) |
| 1199 | 401 -- create Finite-∪ from finCoverSet |
| 1120 | 402 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1189 | 403 isFinite {X} xo xcp = fip60 o∅ (finCoverBase xo xcp) where |
| 1293 | 404 fip60 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → Finite-∪ (* X) (& (finCoverSet {X} x sb)) |
| 1190 | 405 fip60 x (gi rx) = subst (λ k → Finite-∪ (* X) k) fip62 (fin-i (fip61 rx)) where |
| 406 fip62 : & (* (& (L ＼ * x)) , * (& (L ＼ * x))) ≡ & ((L ＼ * x) , (L ＼ * x)) | |
| 1476 | 407 fip62 = cong₂ (λ j k → & (j , k )) ? ? |
| 1293 | 408 fip61 : odef (Replace (* X) (_＼_ L) RC＼ ) x → odef (* X) ( & ((L ＼ * x ) )) |
| 1189 | 409 fip61 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
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410 fip34 : * z1 ⊆ L |
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411 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
| 1189 | 412 fip33 : z1 ≡ & (L ＼ * x) |
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413 fip33 = begin |
| 1152 | 414 z1 ≡⟨ sym &iso ⟩ |
| 1476 | 415 & (* z1) ≡⟨ cong (&) ? ⟩ |
| 416 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) ? ⟩ | |
| 1152 | 417 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ |
| 1189 | 418 & (L ＼ * x ) ∎ where open ≡-Reasoning |
| 419 fip60 x∩y (g∩ {x} {y} sx sy) = subst (λ k → Finite-∪ (* X) k) fip62 ( fin-∪ (fip60 x sx) (fip60 y sy) ) where | |
| 420 fip62 : & (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡ & (finCoverSet x sx ∪ finCoverSet y sy) | |
| 421 fip62 = cong (&) ( begin | |
| 1476 | 422 (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡⟨ cong₂ _∪_ ? ? ⟩ |
| 1189 | 423 finCoverSet x sx ∪ finCoverSet y sy ∎ ) where open ≡-Reasoning |
| 1120 | 424 -- is also a cover |
| 425 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
| 1476 | 426 isCover1 {X} xo xcp = subst₂ (λ j k → j covers k ) ? (subst (λ k → L ＼ k ≡ L) ? ? ) -- L＼0=L) |
| 1190 | 427 (fip70 o∅ (finCoverBase xo xcp)) where |
| 1293 | 428 fip70 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → (finCoverSet {X} x sb) covers (L ＼ * x) |
| 1199 | 429 fip70 x (gi rx) = fip73 where |
| 430 fip73 : ((L ＼ * x) , (L ＼ * x)) covers (L ＼ * x) -- obvious | |
| 431 fip73 = record { cover = λ _ → & (L ＼ * x) ; P∋cover = λ _ → case1 refl | |
| 1476 | 432 ; isCover = λ {x} lt → subst (λ k → odef k x) ? lt } |
| 1190 | 433 fip70 x∩y (g∩ {x} {y} sx sy) = subst (λ k → finCoverSet (& (* x ∩ * y)) (g∩ sx sy) covers |
| 1476 | 434 (L ＼ k)) ? ( fip43 {_} {L} {* x} {* y} (fip71 (fip70 x sx)) (fip72 (fip70 y sy)) ) where |
| 1194 | 435 fip71 : {a b c : HOD} → a covers c → (a ∪ b) covers c |
| 436 fip71 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case1 (P∋cover cov lt) | |
| 437 ; isCover = isCover cov } | |
| 438 fip72 : {a b c : HOD} → a covers c → (b ∪ a) covers c | |
| 439 fip72 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case2 (P∋cover cov lt) | |
| 440 ; isCover = isCover cov } | |
| 1190 | 441 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
| 1476 | 442 fip45 {L} {a} {b} {x} Lab with ∋-p b (* x) |
| 1190 | 443 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ |
| 444 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ | |
| 445 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) | |
| 446 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where | |
| 447 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal | |
| 448 fip44 {x} Lab with fip45 {L} {a} {b} Lab | |
| 449 ... | case1 La = cover ca La | |
| 450 ... | case2 Lb = cover cb Lb | |
| 451 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) | |
| 452 fip46 {x} Lab with fip45 {L} {a} {b} Lab | |
| 453 ... | case1 La = P∋cover ca La | |
| 454 ... | case2 Lb = P∋cover cb Lb | |
| 455 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x | |
| 456 fip47 {x} Lab with fip45 {L} {a} {b} Lab | |
| 457 ... | case1 La = isCover ca La | |
| 458 ... | case2 Lb = isCover cb Lb | |
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459 |
| 1158 | 460 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top |
| 1180 | 461 Compact→FIP {L} top compact with trio< (& L) o∅ |
| 1175 | 462 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1198 | 463 ... | 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 |
| 464 -- empty L case | |
| 465 -- if 0 < X then 0 < x ∧ L ∋ xfrom fip | |
| 466 -- if 0 ≡ X then ¬ odef X x | |
| 467 fip000 : {X x : Ordinal} (CX : * X ⊆ CS top) → ({y : Ordinal} → Subbase (* X) y → o∅ o< y) → ¬ odef (* X) x | |
| 468 fip000 {X} {x} CX fip xx with trio< o∅ X | |
| 1476 | 469 ... | tri< 0<X ¬b ¬c = ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans ? (cong (*) L=0)) ? ) (sym &iso) |
| 1198 | 470 ( cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) (CX xx)) Xe )) where |
| 471 0<x : o∅ o< x | |
| 472 0<x = fip (gi xx ) | |
| 473 e : HOD -- we have an element of x | |
| 1476 | 474 e = minimal (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) |
| 1198 | 475 Xe : odef (* x) (& e) |
| 1476 | 476 Xe = x∋minimal (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) |
| 1198 | 477 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin |
| 478 * X ≡⟨ cong (*) (sym 0=X) ⟩ | |
| 1476 | 479 * o∅ ≡⟨ ? ⟩ |
| 1198 | 480 od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning |
| 481 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1175 | 482 ... | tri> ¬a ¬b 0<L = record { limit = limit ; is-limit = fip00 } where |
| 483 -- set of coset of X | |
| 1476 | 484 open BAlgebra O HODAxiom ho< L ? |
| 1175 | 485 OX : {X : Ordinal} → * X ⊆ CS top → Ordinal |
| 1293 | 486 OX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼) |
| 1175 | 487 OOX : {X : Ordinal} → (cs : * X ⊆ CS top) → * (OX cs) ⊆ OS top |
| 1476 | 488 OOX {X} cs {x} ox = ? -- with subst (λ k → odef k x) ? ox |
| 489 -- ... | 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 | |
| 490 -- comp01 : odef (* X) (& (* z)) | |
| 491 -- comp01 = subst (λ k → odef (* X) k) (sym &iso) az | |
| 1183 | 492 -- if all finite intersection of X contains something, |
| 1175 | 493 -- there is no finite cover. From Compactness, (OX X) is not a cover of L ( contraposition of Compact) |
| 494 -- it means there is a limit | |
| 1199 | 495 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 |
| 496 field -- find an element x, which is not covered (which is a limit point) | |
| 497 x : Ordinal | |
| 498 yx : {y : Ordinal} (Xy : odef (* X) y) → odef (* y) x | |
| 1187 | 499 has-intersection : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1199 | 500 → (0<X : o∅ o< X ) → NC CX fip 0<X |
| 501 has-intersection {X} CX fip 0<X = intersection where | |
| 1198 | 502 e : HOD -- we have an element of X |
| 1476 | 503 e = minimal (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) |
| 1198 | 504 Xe : odef (* X) (& e) |
| 1476 | 505 Xe = x∋minimal (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) |
| 1183 | 506 no-cover : ¬ ( (* (OX CX)) covers L ) |
| 1198 | 507 no-cover cov = ⊥-elim (no-finite-cover (Compact.isCover compact (OOX CX) cov)) where |
| 508 -- construct Subase from Finite-∪ | |
| 1180 | 509 fp01 : Ordinal |
| 510 fp01 = Compact.finCover compact (OOX CX) cov | |
| 1194 | 511 record SB (t : Ordinal) : Set n where |
| 512 field | |
| 513 i : Ordinal | |
| 514 sb : Subbase (* X) (& (L ＼ * i)) | |
| 1198 | 515 t⊆i : (L ＼ * i) ⊆ (L ＼ Union ( * t ) ) |
| 1194 | 516 fp02 : (t : Ordinal) → Finite-∪ (* (OX CX)) t → SB t |
| 1198 | 517 fp02 t fin-e = record { i = & ( L ＼ e) ; sb = gi (subst (λ k → odef (* X) k) fp21 Xe) ; t⊆i = fp23 } where |
| 518 -- t ≡ o∅, no cover. Any subst of L is ok and we have e ⊆ L | |
| 519 fp22 : e ⊆ L | |
| 520 fp22 {x} lt = cs⊆L top (CX Xe) lt | |
| 521 fp21 : & e ≡ & (L ＼ * (& (L ＼ e))) | |
| 1476 | 522 fp21 = cong (&) (trans (sym ?) (cong (λ k → L ＼ k) ?)) |
| 1198 | 523 fp23 : (L ＼ * (& (L ＼ e))) ⊆ (L ＼ Union (* o∅)) |
| 1476 | 524 fp23 {x} ⟪ Lx , _ ⟫ = ⟪ Lx , ( λ lt → ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ? (sym &iso) (Own.ao lt )))) ⟫ |
| 1198 | 525 fp02 t (fin-i {x} tx ) = record { i = x ; sb = gi fp03 ; t⊆i = fp24 } where |
| 526 -- we have a single cover x, L ＼ * x is single finite intersection | |
| 1197 | 527 fp24 : (L ＼ * x) ⊆ (L ＼ Union (* (& (* x , * x)))) |
| 1476 | 528 fp24 {y} ⟪ Lx , not ⟫ = ⟪ Lx , subst (λ k → ¬ odef (Union k) y) ? fp25 ⟫ where |
| 1197 | 529 fp25 : ¬ odef (Union (* x , * x)) y |
| 1476 | 530 fp25 record { owner = .(& (* x)) ; ao = (case1 refl) ; ox = ox } = not (subst (λ k → odef k y) ? ox ) |
| 531 fp25 record { owner = .(& (* x)) ; ao = (case2 refl) ; ox = ox } = not (subst (λ k → odef k y) ? ox ) | |
| 1198 | 532 fp03 : odef (* X) (& (L ＼ * x)) -- becase x is an element of Replace (* X) (λ z → L ＼ z ) |
| 1476 | 533 fp03 = ? -- with subst (λ k → odef k x ) ? tx |
| 534 -- ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where | |
| 535 -- fip34 : * z1 ⊆ L | |
| 536 -- fip34 {w} wz1 = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX az1) ) wz1 | |
| 537 -- fip33 : z1 ≡ & (L ＼ * x) | |
| 538 -- fip33 = begin | |
| 539 -- z1 ≡⟨ sym &iso ⟩ | |
| 540 -- & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ | |
| 541 -- & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) ? ⟩ | |
| 542 -- & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ | |
| 543 -- & (L ＼ * x ) ∎ where open ≡-Reasoning | |
| 1198 | 544 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 |
| 545 fp35 : (L ＼ * (& (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))) ⊆ (L ＼ Union (* (& (* tx ∪ * ty)))) | |
| 1476 | 546 fp35 = subst₂ (λ j k → (L ＼ j ) ⊆ (L ＼ Union k)) ? ? fp36 where |
| 1197 | 547 fp40 : {z tz : Ordinal } → Finite-∪ (* (OX CX)) tz → odef (Union (* tz )) z → odef L z |
| 1199 | 548 fp40 {z} {.(Ordinals.o∅ O)} fin-e record { owner = owner ; ao = ao ; ox = ox } |
| 1476 | 549 = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ? (sym &iso) ao )) |
| 550 fp40 {z} {.(& (* _ , * _))} (fin-i {w} x) uz = fp41 x (subst (λ k → odef (Union k) z) ? uz) where | |
| 1197 | 551 fp41 : (x : odef (* (OX CX)) w) → (uz : odef (Union (* w , * w)) z ) → odef L z |
| 552 fp41 x record { owner = .(& (* w)) ; ao = (case1 refl) ; ox = ox } = | |
| 1476 | 553 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) ? ox ) |
| 1197 | 554 fp41 x record { owner = .(& (* w)) ; ao = (case2 refl) ; ox = ox } = |
| 1476 | 555 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) ? ox ) |
| 556 fp40 {z} {.(& (* _ ∪ * _))} (fin-∪ {x1} {y1} ftx fty) uz = ? -- with subst (λ k → odef (Union k) z ) ? uz | |
| 557 -- ... | record { owner = o ; ao = case1 x1o ; ox = oz } = fp40 ftx record { owner = o ; ao = x1o ; ox = oz } | |
| 558 -- ... | record { owner = o ; ao = case2 y1o ; ox = oz } = fp40 fty record { owner = o ; ao = y1o ; ox = oz } | |
| 1198 | 559 fp36 : (L ＼ (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* tx ∪ * ty)) |
| 1197 | 560 fp36 {z} ⟪ Lz , not ⟫ = ⟪ Lz , fp37 ⟫ where |
| 561 fp37 : ¬ odef (Union (* tx ∪ * ty)) z | |
| 562 fp37 record { owner = owner ; ao = (case1 ax) ; ox = ox } = not (case1 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where | |
| 1198 | 563 fp38 : (L ＼ (* (SB.i (fp02 tx ux)))) ⊆ (L ＼ Union (* tx)) |
| 564 fp38 = SB.t⊆i (fp02 tx ux) | |
| 565 fp39 : Union (* tx) ⊆ (* (SB.i (fp02 tx ux))) | |
| 1476 | 566 fp39 {w} txw = ? -- with ∨L＼X {L} {* (SB.i (fp02 tx ux))} (fp40 ux txw) |
| 567 -- ... | case1 sb = sb | |
| 568 -- ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) txw ) | |
| 1197 | 569 fp37 record { owner = owner ; ao = (case2 ax) ; ox = ox } = not (case2 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where |
| 1198 | 570 fp38 : (L ＼ (* (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* ty)) |
| 571 fp38 = SB.t⊆i (fp02 ty uy) | |
| 572 fp39 : Union (* ty) ⊆ (* (SB.i (fp02 ty uy))) | |
| 1476 | 573 fp39 {w} tyw = ? -- with ∨L＼X {L} {* (SB.i (fp02 ty uy))} (fp40 uy tyw) |
| 574 -- ... | case1 sb = sb | |
| 575 -- ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) tyw ) | |
| 1194 | 576 fp04 : {tx ty : Ordinal} → & (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) ≡ & (L ＼ * (& (* tx ∪ * ty))) |
| 1476 | 577 fp04 {tx} {ty} = ? where -- cong (&) ( ==→o≡ record { eq→ = fp05 ; eq← = fp09 } ) where |
| 1187 | 578 fp05 : {x : Ordinal} → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x → odef (L ＼ * (& (* tx ∪ * ty))) x |
| 1476 | 579 fp05 {x} lt = ? -- with subst₂ (λ j k → odef (j ∩ k) x ) ? ? lt |
| 580 -- ... | ⟪ ⟪ Lx , ¬tx ⟫ , ⟪ Ly , ¬ty ⟫ ⟫ = subst (λ k → odef (L ＼ k) x) ? ⟪ Lx , fp06 ⟫ where | |
| 581 -- fp06 : ¬ odef (* tx ∪ * ty) x | |
| 582 -- fp06 (case1 tx) = ¬tx tx | |
| 583 -- fp06 (case2 ty) = ¬ty ty | |
| 1187 | 584 fp09 : {x : Ordinal} → odef (L ＼ * (& (* tx ∪ * ty))) x → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x |
| 1476 | 585 fp09 {x} lt with subst (λ k → odef (L ＼ k) x) ? lt |
| 586 ... | ⟪ Lx , ¬tx∨ty ⟫ = subst₂ (λ j k → odef (j ∩ k) x ) ? ? | |
| 1187 | 587 ⟪ ⟪ Lx , ( λ tx → ¬tx∨ty (case1 tx)) ⟫ , ⟪ Lx , ( λ ty → ¬tx∨ty (case2 ty)) ⟫ ⟫ |
| 1198 | 588 fp11 : Subbase (* X) (& (L ＼ * (& ((* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))))) |
| 589 fp11 = subst (λ k → Subbase (* X) k ) fp04 ( g∩ (SB.sb (fp02 tx ux)) (SB.sb (fp02 ty uy )) ) | |
| 590 -- | |
| 591 -- becase of fip, finite cover cannot be a cover | |
| 592 -- | |
| 1195 | 593 fcov : Finite-∪ (* (OX CX)) (Compact.finCover compact (OOX CX) cov) |
| 594 fcov = Compact.isFinite compact (OOX CX) cov | |
| 1196 | 595 0<sb : {i : Ordinal } → (sb : Subbase (* X) (& (L ＼ * i))) → o∅ o< & (L ＼ * i) |
| 596 0<sb {i} sb = fip sb | |
| 597 sb : SB (Compact.finCover compact (OOX CX) cov) | |
| 598 sb = fp02 fp01 (Compact.isFinite compact (OOX CX) cov) | |
| 1198 | 599 no-finite-cover : ¬ ( (* (Compact.finCover compact (OOX CX) cov)) covers L ) |
| 1476 | 600 no-finite-cover fcovers = ? where -- ⊥-elim ( o<¬≡ (cong (&) (sym (==→o≡ f22))) f25 ) where |
| 1196 | 601 f23 : (L ＼ * (SB.i sb)) ⊆ ( L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) |
| 1198 | 602 f23 = SB.t⊆i sb |
| 1196 | 603 f22 : (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) =h= od∅ |
| 604 f22 = record { eq→ = λ lt → ⊥-elim ( f24 lt) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where | |
| 605 f24 : {x : Ordinal } → ¬ ( odef (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) x ) | |
| 606 f24 {x} ⟪ Lx , not ⟫ = not record { owner = cover fcovers Lx ; ao = P∋cover fcovers Lx ; ox = isCover fcovers Lx } | |
| 607 f25 : & od∅ o< (& (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ) | |
| 608 f25 = ordtrans<-≤ (subst (λ k → k o< & (L ＼ * (SB.i sb))) (sym ord-od∅) (0<sb (SB.sb sb) ) ) ( begin | |
| 609 & (L ＼ * (SB.i sb)) ≤⟨ ⊆→o≤ f23 ⟩ | |
| 610 & (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ∎ ) where open o≤-Reasoning O | |
| 1199 | 611 -- if we have no cover, we can consruct NC |
| 612 intersection : NC CX fip 0<X | |
| 1476 | 613 intersection with p∨¬p (NC CX fip 0<X) |
| 1184 | 614 ... | case1 nc = nc |
| 1185 | 615 ... | case2 ¬nc = ⊥-elim ( no-cover record { cover = λ Lx → & (L ＼ coverf Lx) ; P∋cover = fp22 ; isCover = fp23 } ) where |
| 616 coverSet : {x : Ordinal} → odef L x → HOD | |
| 1186 | 617 coverSet {x} Lx = record { od = record { def = λ y → odef (* X) y ∧ odef (L ＼ * y) x } ; odmax = X |
| 618 ; <odmax = λ {x} lt → subst (λ k → x o< k) &iso ( odef< (proj1 lt)) } | |
| 1185 | 619 fp17 : {x : Ordinal} → (Lx : odef L x ) → ¬ ( coverSet Lx =h= od∅ ) |
| 1187 | 620 fp17 {x} Lx eq = ⊥-elim (¬nc record { x = x ; yx = fp19 } ) where |
| 1185 | 621 fp19 : {y : Ordinal} → odef (* X) y → odef (* y) x |
| 1476 | 622 fp19 {y} Xy = ? -- with ∨L＼X {L} {* y} {x} Lx |
| 623 -- ... | case1 yx = yx | |
| 624 -- ... | case2 lyx = ⊥-elim ( ¬x<0 {y} ( eq→ eq fp20 )) where | |
| 625 -- fp20 : odef (* X) y ∧ odef (L ＼ * y) x | |
| 626 -- fp20 = ⟪ Xy , lyx ⟫ | |
| 1185 | 627 coverf : {x : Ordinal} → (Lx : odef L x ) → HOD |
| 1476 | 628 coverf Lx = minimal (coverSet Lx) (fp17 Lx) |
| 1186 | 629 fp22 : {x : Ordinal} (lt : odef L x) → odef (* (OX CX)) (& (L ＼ coverf lt)) |
| 1476 | 630 fp22 {x} Lx = subst (λ k → odef k (& (L ＼ coverf Lx ))) ? record { z = _ ; az = fp25 ; x=ψz = fp24 } where |
| 1186 | 631 fp24 : & (L ＼ coverf Lx) ≡ & (L ＼ * (& (coverf Lx))) |
| 1476 | 632 fp24 = cong (λ k → & ( L ＼ k )) ? |
| 1185 | 633 fp25 : odef (* X) (& (coverf Lx)) |
| 1476 | 634 fp25 = proj1 ( x∋minimal (coverSet Lx) (fp17 Lx) ) |
| 1186 | 635 fp23 : {x : Ordinal} (lt : odef L x) → odef (* (& (L ＼ coverf lt))) x |
| 1476 | 636 fp23 {x} Lx = subst (λ k → odef k x) ? ⟪ Lx , fp26 ⟫ where |
| 1186 | 637 fp26 : ¬ odef (coverf Lx) x |
| 1476 | 638 fp26 = subst (λ k → ¬ odef k x ) ? (proj2 (proj2 ( x∋minimal (coverSet Lx) (fp17 Lx) )) ) |
| 1199 | 639 limit : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) → Ordinal |
| 1180 | 640 limit {X} CX fip with trio< X o∅ |
| 641 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 642 ... | tri≈ ¬a b ¬c = o∅ | |
| 1199 | 643 ... | tri> ¬a ¬b c = NC.x ( has-intersection CX fip c) |
| 1175 | 644 fip00 : {X : Ordinal} (CX : * X ⊆ CS top) |
| 1187 | 645 (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1175 | 646 {x : Ordinal} → odef (* X) x → odef (* x) (limit CX fip ) |
| 1180 | 647 fip00 {X} CX fip {x} Xx with trio< X o∅ |
| 648 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 649 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) (subst (λ k → o∅ o< k) &iso (∈∅< Xx) ) ) | |
| 1199 | 650 ... | tri> ¬a ¬b c = NC.yx ( has-intersection CX fip c ) Xx |
| 431 | 651 |
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652 open Filter |
| 1102 | 653 |
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654 -- |
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655 -- {v | Neighbor lim v} set of u ⊆ v ⊆ P where u is an open set u ∋ x |
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656 -- |
| 1159 | 657 record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where |
| 658 field | |
| 659 u : Ordinal | |
| 660 ou : odef (OS TP) u | |
| 661 ux : odef (* u) x | |
| 662 v⊆P : * v ⊆ P | |
| 1170 | 663 u⊆v : * u ⊆ * v |
| 1102 | 664 |
| 1169 | 665 -- |
| 666 -- Neighbor on x is a Filter (on Power P) | |
| 667 -- | |
| 1170 | 668 NeighborF : {P : HOD} (TP : Topology P) (x : Ordinal ) → Filter {Power P} {P} (λ x → x) |
| 1169 | 669 NeighborF {P} TP x = record { filter = NF ; f⊆L = NF⊆PP ; filter1 = f1 ; filter2 = f2 } where |
| 1168 | 670 nf00 : {v : Ordinal } → Neighbor TP x v → odef (Power P) v |
| 671 nf00 {v} nei y vy = Neighbor.v⊆P nei vy | |
| 1167 | 672 NF : HOD |
| 1168 | 673 NF = record { od = record { def = λ v → Neighbor TP x v } ; odmax = & (Power P) ; <odmax = λ lt → odef< (nf00 lt) } |
| 1167 | 674 NF⊆PP : NF ⊆ Power P |
| 1168 | 675 NF⊆PP = nf00 |
| 676 f1 : {p q : HOD} → Power P ∋ q → NF ∋ p → p ⊆ q → NF ∋ q | |
| 1170 | 677 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 | 678 f11 : * (Neighbor.u Np) ⊆ * (& q) |
| 1476 | 679 f11 {x} ux = subst (λ k → odef k x ) ? ( p⊆q (subst (λ k → odef k x) ? (Neighbor.u⊆v Np ux)) ) |
| 1168 | 680 f2 : {p q : HOD} → NF ∋ p → NF ∋ q → Power P ∋ (p ∩ q) → NF ∋ (p ∩ q) |
| 1170 | 681 f2 {p} {q} Np Nq Ppq = record { u = upq ; ou = ou ; ux = ux ; v⊆P = Ppq _ ; u⊆v = f20 } where |
| 1168 | 682 upq : Ordinal |
| 683 upq = & ( * (Neighbor.u Np) ∩ * (Neighbor.u Nq) ) | |
| 684 ou : odef (OS TP) upq | |
| 685 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)) | |
| 686 ux : odef (* upq) x | |
| 1476 | 687 ux = subst ( λ k → odef k x ) ? ⟪ Neighbor.ux Np , Neighbor.ux Nq ⟫ |
| 1168 | 688 f20 : * upq ⊆ * (& (p ∩ q)) |
| 1476 | 689 f20 = subst₂ (λ j k → j ⊆ k ) ? ? ( λ {x} pq |
| 690 → ⟪ subst (λ k → odef k x) ? (Neighbor.u⊆v Np (proj1 pq)) , subst (λ k → odef k x) ? (Neighbor.u⊆v Nq (proj2 pq)) ⟫ ) | |
| 1153 | 691 |
| 1165 | 692 CAP : (P : HOD) {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) |
| 1476 | 693 CAP P {p} {q} Pp Pq x pqx with subst (λ k → odef k x ) ? pqx |
| 694 ... | ⟪ px , qx ⟫ = Pp _ (subst (λ k → odef k x) ? px ) | |
| 1165 | 695 |
| 1170 | 696 NEG : (P : HOD) {p : HOD } → Power P ∋ p → Power P ∋ (P ＼ p ) |
| 1476 | 697 NEG P {p} Pp x vx with subst (λ k → odef k x) ? vx |
| 1170 | 698 ... | ⟪ Px , npx ⟫ = Px |
| 1142 | 699 |