Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Topology.agda @ 1353:ddbc0726f9bb
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 18 Jun 2023 18:16:03 +0900 |
| parents | 47d3cc596d68 |
| children | 32001d93755b |
| rev | line source |
|---|---|
| 1170 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 431 | 3 open import Level |
| 4 open import Ordinals | |
| 5 module Topology {n : Level } (O : Ordinals {n}) where | |
| 6 | |
| 7 open import logic | |
| 8 open _∧_ | |
| 9 open _∨_ | |
| 10 open Bool | |
| 11 | |
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12 import OD |
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13 open import Relation.Nullary |
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14 open import Data.Empty |
| 431 | 15 open import Relation.Binary.Core |
| 1143 | 16 open import Relation.Binary.Definitions |
| 431 | 17 open import Relation.Binary.PropositionalEquality |
| 1124 | 18 import BAlgebra |
| 19 open BAlgebra O | |
| 431 | 20 open inOrdinal O |
| 21 open OD O | |
| 22 open OD.OD | |
| 23 open ODAxiom odAxiom | |
| 24 import OrdUtil | |
| 25 import ODUtil | |
| 26 open Ordinals.Ordinals O | |
| 27 open Ordinals.IsOrdinals isOrdinal | |
| 1300 | 28 -- open Ordinals.IsNext isNext |
| 431 | 29 open OrdUtil O |
| 30 open ODUtil O | |
| 31 | |
| 32 import ODC | |
| 33 open ODC O | |
| 34 | |
| 1102 | 35 open import filter O |
| 1218 | 36 open import ZProduct O |
| 1101 | 37 |
| 482 | 38 record Topology ( L : HOD ) : Set (suc n) where |
| 431 | 39 field |
| 40 OS : HOD | |
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41 OS⊆PL : OS ⊆ Power L |
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42 o∩ : { p q : HOD } → OS ∋ p → OS ∋ q → OS ∋ (p ∩ q) |
| 1161 | 43 o∪ : { P : HOD } → P ⊆ OS → OS ∋ Union P |
| 1210 | 44 OS∋od∅ : OS ∋ od∅ -- OS ∋ Union od∅ |
| 1160 | 45 --- we may add |
| 46 -- OS∋L : OS ∋ L | |
| 1101 | 47 -- closed Set |
| 48 CS : HOD | |
| 1119 | 49 CS = record { od = record { def = λ x → (* x ⊆ L) ∧ odef OS (& ( L ＼ (* x ))) } ; odmax = osuc (& L) ; <odmax = tp02 } where |
| 50 tp02 : {y : Ordinal } → (* y ⊆ L) ∧ odef OS (& (L ＼ * y)) → y o< osuc (& L) | |
| 51 tp02 {y} nop = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → proj1 nop yx )) | |
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52 os⊆L : {x : HOD} → OS ∋ x → x ⊆ L |
| 1108 | 53 os⊆L {x} Ox {y} xy = ( OS⊆PL Ox ) _ (subst (λ k → odef k y) (sym *iso) xy ) |
| 1122 | 54 cs⊆L : {x : HOD} → CS ∋ x → x ⊆ L |
| 55 cs⊆L {x} Cx {y} xy = proj1 Cx (subst (λ k → odef k y ) (sym *iso) xy ) | |
| 56 CS∋L : CS ∋ L | |
| 1123 | 57 CS∋L = ⟪ subst (λ k → k ⊆ L) (sym *iso) (λ x → x) , subst (λ k → odef OS (& k)) (sym lem0) OS∋od∅ ⟫ where |
| 58 lem0 : L ＼ * (& L) ≡ od∅ | |
| 59 lem0 = subst (λ k → L ＼ k ≡ od∅) (sym *iso) L＼L=0 | |
| 1154 | 60 CS⊆PL : CS ⊆ Power L |
| 1161 | 61 CS⊆PL {x} Cx y xy = proj1 Cx xy |
| 1160 | 62 P＼CS=OS : {cs : HOD} → CS ∋ cs → OS ∋ ( L ＼ cs ) |
| 63 P＼CS=OS {cs} ⟪ cs⊆L , olcs ⟫ = subst (λ k → odef OS k) (cong (λ k → & ( L ＼ k)) *iso) olcs | |
| 64 P＼OS=CS : {cs : HOD} → OS ∋ cs → CS ∋ ( L ＼ cs ) | |
| 1161 | 65 P＼OS=CS {os} oos = ⟪ subst (λ k → k ⊆ L) (sym *iso) proj1 |
| 1160 | 66 , subst (λ k → odef OS k) (cong (&) (trans (sym (L＼Lx=x (os⊆L oos))) (cong (λ k → L ＼ k) (sym *iso)) )) oos ⟫ |
| 431 | 67 |
| 482 | 68 open Topology |
| 431 | 69 |
| 1163 | 70 -- Closure ( Intersection of Closed Set which include A ) |
| 71 | |
| 1162 | 72 Cl : {L : HOD} → (top : Topology L) → (A : HOD) → HOD |
| 73 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 | 74 ; odmax = & L ; <odmax = odef∧< } |
| 1122 | 75 |
| 1162 | 76 ClL : {L : HOD} → (top : Topology L) → Cl top L ≡ L |
| 77 ClL {L} top = ==→o≡ ( record { eq→ = λ {x} ic | |
| 1142 | 78 → subst (λ k → odef k x) *iso ((proj2 ic) (& L) (CS∋L top) (subst (λ k → L ⊆ k) (sym *iso) ( λ x → x))) |
| 79 ; eq← = λ {x} lx → ⟪ lx , ( λ c cs l⊆c → l⊆c lx) ⟫ } ) | |
| 1123 | 80 |
| 1163 | 81 -- Closure is Closed Set |
| 82 | |
| 1162 | 83 CS∋Cl : {L : HOD} → (top : Topology L) → (A : HOD) → CS top ∋ Cl top A |
| 84 CS∋Cl {L} top A = subst (λ k → CS top ∋ k) (==→o≡ cc00) (P＼OS=CS top UOCl-is-OS) where | |
| 1163 | 85 OCl : HOD -- set of open set which it not contains A |
| 1162 | 86 OCl = record { od = record { def = λ o → odef (OS top) o ∧ ( A ⊆ (L ＼ * o) ) } ; odmax = & (OS top) ; <odmax = odef∧< } |
| 87 OCl⊆OS : OCl ⊆ OS top | |
| 88 OCl⊆OS ox = proj1 ox | |
| 89 UOCl-is-OS : OS top ∋ Union OCl | |
| 90 UOCl-is-OS = o∪ top OCl⊆OS | |
| 91 cc00 : (L ＼ Union OCl) =h= Cl top A | |
| 92 cc00 = record { eq→ = cc01 ; eq← = cc03 } where | |
| 93 cc01 : {x : Ordinal} → odef (L ＼ Union OCl) x → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) | |
| 94 cc01 {x} ⟪ Lx , nul ⟫ = ⟪ Lx , ( λ c cc ac → cc02 c cc ac nul ) ⟫ where | |
| 95 cc02 : (c : Ordinal) → odef (CS top) c → A ⊆ * c → ¬ odef (Union OCl) x → odef (* c) x | |
| 96 cc02 c cc ac nox with ODC.∋-p O (* c) (* x) | |
| 97 ... | yes y = subst (λ k → odef (* c) k) &iso y | |
| 98 ... | no ncx = ⊥-elim ( nox record { owner = & ( L ＼ * c) ; ao = ⟪ proj2 cc , cc07 ⟫ ; ox = subst (λ k → odef k x) (sym *iso) cc06 } ) where | |
| 99 cc06 : odef (L ＼ * c) x | |
| 100 cc06 = ⟪ Lx , subst (λ k → ¬ odef (* c) k) &iso ncx ⟫ | |
| 101 cc08 : * c ⊆ L | |
| 102 cc08 = cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) cc ) | |
| 103 cc07 : A ⊆ (L ＼ * (& (L ＼ * c))) | |
| 104 cc07 {z} az = subst (λ k → odef k z ) ( | |
| 105 begin * c ≡⟨ sym ( L＼Lx=x cc08 ) ⟩ | |
| 106 L ＼ (L ＼ * c) ≡⟨ cong (λ k → L ＼ k ) (sym *iso) ⟩ | |
| 107 L ＼ * (& (L ＼ * c)) ∎ ) ( ac az ) where open ≡-Reasoning | |
| 108 cc03 : {x : Ordinal} → odef L x ∧ ((c : Ordinal) → odef (CS top) c → A ⊆ * c → odef (* c) x) → odef (L ＼ Union OCl) x | |
| 109 cc03 {x} ⟪ Lx , ccx ⟫ = ⟪ Lx , cc04 ⟫ where | |
| 1163 | 110 -- 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 | 111 cc04 : ¬ odef (Union OCl) x |
| 112 cc04 record { owner = o ; ao = ⟪ oo , A⊆L-o ⟫ ; ox = ox } = proj2 ( subst (λ k → odef k x) *iso cc05) ox where | |
| 113 cc05 : odef (* (& (L ＼ * o))) x | |
| 114 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 | 115 |
| 1210 | 116 CS∋x→Clx=x : {L x : HOD} → (top : Topology L) → CS top ∋ x → Cl top x ≡ x |
| 117 CS∋x→Clx=x {L} {x} top cx = ==→o≡ record { eq→ = cc10 ; eq← = cc11 } where | |
| 118 cc10 : {y : Ordinal} → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) → odef x y | |
| 119 cc10 {y} ⟪ Ly , cc ⟫ = subst (λ k → odef k y) *iso ( cc (& x) cx (λ {z} xz → subst (λ k → odef k z) (sym *iso) xz ) ) | |
| 120 cc11 : {y : Ordinal} → odef x y → odef L y ∧ ((c : Ordinal) → odef (CS top) c → x ⊆ * c → odef (* c) y) | |
| 121 cc11 {y} xy = ⟪ cs⊆L top cx xy , (λ c oc x⊆c → x⊆c xy ) ⟫ | |
| 1160 | 122 |
| 1119 | 123 -- Subbase P |
| 124 -- A set of countable intersection of P will be a base (x ix an element of the base) | |
| 1107 | 125 |
| 126 data Subbase (P : HOD) : Ordinal → Set n where | |
| 127 gi : {x : Ordinal } → odef P x → Subbase P x | |
| 128 g∩ : {x y : Ordinal } → Subbase P x → Subbase P y → Subbase P (& (* x ∩ * y)) | |
| 129 | |
| 1119 | 130 -- |
| 1150 | 131 -- if y is in a Subbase, some element of P contains it |
| 1119 | 132 |
| 1111 | 133 sbp : (P : HOD) {x : Ordinal } → Subbase P x → Ordinal |
| 134 sbp P {x} (gi {y} px) = x | |
| 135 sbp P {.(& (* _ ∩ * _))} (g∩ sb sb₁) = sbp P sb | |
| 1107 | 136 |
| 1111 | 137 is-sbp : (P : HOD) {x y : Ordinal } → (px : Subbase P x) → odef (* x) y → odef P (sbp P px ) ∧ odef (* (sbp P px)) y |
| 138 is-sbp P {x} (gi px) xy = ⟪ px , xy ⟫ | |
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139 is-sbp P {.(& (* _ ∩ * _))} (g∩ {x} {y} px px₁) xy = is-sbp P px (proj1 (subst (λ k → odef k _ ) *iso xy)) |
| 1107 | 140 |
| 1155 | 141 sb⊆ : {P Q : HOD} {x : Ordinal } → P ⊆ Q → Subbase P x → Subbase Q x |
| 142 sb⊆ {P} {Q} P⊆Q (gi px) = gi (P⊆Q px) | |
| 143 sb⊆ {P} {Q} P⊆Q (g∩ px qx) = g∩ (sb⊆ P⊆Q px) (sb⊆ P⊆Q qx) | |
| 144 | |
| 1119 | 145 -- An open set generate from a base |
| 146 -- | |
| 1161 | 147 -- OS = { U ⊆ L | ∀ x ∈ U → ∃ b ∈ P → x ∈ b ⊆ U } |
| 1114 | 148 |
| 1115 | 149 record Base (L P : HOD) (u x : Ordinal) : Set n where |
| 1114 | 150 field |
| 1150 | 151 b : Ordinal |
| 1161 | 152 u⊆L : * u ⊆ L |
| 1114 | 153 sb : Subbase P b |
| 154 b⊆u : * b ⊆ * u | |
| 155 bx : odef (* b) x | |
| 1150 | 156 x⊆L : odef L x |
| 1161 | 157 x⊆L = u⊆L (b⊆u bx) |
| 1114 | 158 |
| 1115 | 159 SO : (L P : HOD) → HOD |
| 1119 | 160 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 |
| 161 tp00 : {y : Ordinal} → ({x : Ordinal} → odef (* y) x → Base L P y x) → y o< osuc (& L) | |
| 1150 | 162 tp00 {y} op = subst (λ k → k o≤ & L ) &iso ( ⊆→o≤ (λ {x} yx → Base.x⊆L (op yx) )) |
| 1114 | 163 |
| 1111 | 164 record IsSubBase (L P : HOD) : Set (suc n) where |
| 1110 | 165 field |
| 1122 | 166 P⊆PL : P ⊆ Power L |
| 1116 | 167 -- we may need these if OS ∋ L is necessary |
| 168 -- p : {x : HOD} → L ∋ x → HOD | |
| 1161 | 169 -- Pp : {x : HOD} → {lx : L ∋ x } → P ∋ p lx |
| 1116 | 170 -- px : {x : HOD} → {lx : L ∋ x } → p lx ∋ x |
| 1110 | 171 |
| 1152 | 172 InducedTopology : (L P : HOD) → IsSubBase L P → Topology L |
| 173 InducedTopology L P isb = record { OS = SO L P ; OS⊆PL = tp00 | |
| 1122 | 174 ; o∪ = tp02 ; o∩ = tp01 ; OS∋od∅ = tp03 } where |
| 175 tp03 : {x : Ordinal } → odef (* (& od∅)) x → Base L P (& od∅) x | |
| 1150 | 176 tp03 {x} 0x = ⊥-elim ( empty (* x) ( subst₂ (λ j k → odef j k ) *iso (sym &iso) 0x )) |
| 1115 | 177 tp00 : SO L P ⊆ Power L |
| 178 tp00 {u} ou x ux with ou ux | |
| 1161 | 179 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } = u⊆L (b⊆u bx) |
| 1115 | 180 tp01 : {p q : HOD} → SO L P ∋ p → SO L P ∋ q → SO L P ∋ (p ∩ q) |
| 1161 | 181 tp01 {p} {q} op oq {x} ux = record { b = b ; u⊆L = subst (λ k → k ⊆ L) (sym *iso) ul |
| 1116 | 182 ; sb = g∩ (Base.sb (op px)) (Base.sb (oq qx)) ; b⊆u = tp08 ; bx = tp14 } where |
| 1115 | 183 px : odef (* (& p)) x |
| 184 px = subst (λ k → odef k x ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 185 qx : odef (* (& q)) x | |
| 186 qx = subst (λ k → odef k x ) (sym *iso) ( proj2 (subst (λ k → odef k _ ) *iso ux ) ) | |
| 187 b : Ordinal | |
| 188 b = & (* (Base.b (op px)) ∩ * (Base.b (oq qx))) | |
| 1116 | 189 tp08 : * b ⊆ * (& (p ∩ q) ) |
| 1150 | 190 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 | 191 tp11 : * (Base.b (op px)) ⊆ * (& p ) |
| 192 tp11 = Base.b⊆u (op px) | |
| 193 tp12 : * (Base.b (oq qx)) ⊆ * (& q ) | |
| 194 tp12 = Base.b⊆u (oq qx) | |
| 1150 | 195 tp09 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ p |
| 1116 | 196 tp09 = ⊆∩-incl-1 {* (Base.b (op px))} {* (Base.b (oq qx))} {p} (subst (λ k → (* (Base.b (op px))) ⊆ k ) *iso tp11) |
| 1150 | 197 tp10 : (* (Base.b (op px)) ∩ * (Base.b (oq qx))) ⊆ q |
| 1116 | 198 tp10 = ⊆∩-incl-2 {* (Base.b (oq qx))} {* (Base.b (op px))} {q} (subst (λ k → (* (Base.b (oq qx))) ⊆ k ) *iso tp12) |
| 199 tp14 : odef (* (& (* (Base.b (op px)) ∩ * (Base.b (oq qx))))) x | |
| 200 tp14 = subst (λ k → odef k x ) (sym *iso) ⟪ Base.bx (op px) , Base.bx (oq qx) ⟫ | |
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201 ul : (p ∩ q) ⊆ L |
| 1161 | 202 ul = subst (λ k → k ⊆ L ) *iso (λ {z} pq → (Base.u⊆L (op px)) (pz pq) ) where |
| 1116 | 203 pz : {z : Ordinal } → odef (* (& (p ∩ q))) z → odef (* (& p)) z |
| 204 pz {z} pq = subst (λ k → odef k z ) (sym *iso) ( proj1 (subst (λ k → odef k _ ) *iso pq ) ) | |
| 1161 | 205 tp02 : { q : HOD} → q ⊆ SO L P → SO L P ∋ Union q |
| 206 tp02 {q} q⊆O {x} ux with subst (λ k → odef k x) *iso ux | |
| 207 ... | record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx | |
| 208 ... | 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 | 209 ; sb = sb ; b⊆u = subst ( λ k → * b ⊆ k ) (sym *iso) tp06 ; bx = bx } where |
| 210 tp05 : Union q ⊆ L | |
| 1161 | 211 tp05 {z} record { owner = y ; ao = qy ; ox = yx } with q⊆O qy yx |
| 212 ... | record { b = b ; u⊆L = u⊆L ; sb = sb ; b⊆u = b⊆u ; bx = bx } | |
| 1116 | 213 = IsSubBase.P⊆PL isb (proj1 (is-sbp P sb bx )) _ (proj2 (is-sbp P sb bx )) |
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214 tp04 : Union q ⊆ L |
| 1150 | 215 tp04 = tp05 |
| 1116 | 216 tp06 : * b ⊆ Union q |
| 1150 | 217 tp06 {z} bz = record { owner = y ; ao = qy ; ox = b⊆u bz } |
| 1110 | 218 |
| 1142 | 219 -- Product Topology |
| 220 | |
| 221 open ZFProduct | |
| 222 | |
| 1150 | 223 -- Product Topology is not |
| 1142 | 224 -- ZFP (OS TP) (OS TQ) (box) |
| 225 | |
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226 -- Rectangle subset (zπ1 ⁻¹ p) |
| 1142 | 227 record BaseP {P : HOD} (TP : Topology P ) (Q : HOD) (x : Ordinal) : Set n where |
| 228 field | |
| 1172 | 229 p : Ordinal |
| 1142 | 230 op : odef (OS TP) p |
| 231 prod : x ≡ & (ZFP (* p) Q ) | |
| 232 | |
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233 -- Rectangle subset (zπ12⁻¹ q ) |
| 1142 | 234 record BaseQ (P : HOD) {Q : HOD} (TQ : Topology Q ) (x : Ordinal) : Set n where |
| 235 field | |
| 1172 | 236 q : Ordinal |
| 1142 | 237 oq : odef (OS TQ) q |
| 238 prod : x ≡ & (ZFP P (* q )) | |
| 239 | |
| 240 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 | 241 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 | 242 tp01 : odef (Power (ZFP P Q)) (& (ZFP (* p) Q)) |
| 243 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 244 ... | 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 | |
| 245 tp03 : odef P a | |
| 246 tp03 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) op) pa | |
| 1172 | 247 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 | 248 tp01 : odef (Power (ZFP P Q)) (& (ZFP P (* q) )) |
| 249 tp01 w wz with subst (λ k → odef k w ) *iso wz | |
| 250 ... | 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 | |
| 251 tp03 : odef Q b | |
| 252 tp03 = os⊆L TQ (subst (λ k → odef (OS TQ) k) (sym &iso) oq) qb | |
| 253 | |
| 254 pbase : {P Q : HOD} → Topology P → Topology Q → HOD | |
| 255 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 | |
| 256 tp00 : {y : Ordinal} → BaseP TP Q y ∨ BaseQ P TQ y → y o< & (Power (ZFP P Q)) | |
| 1150 | 257 tp00 {y} bpq = odef< ( pbase⊆PL TP TQ bpq ) |
| 1142 | 258 |
| 259 ProductTopology : {P Q : HOD} → Topology P → Topology Q → Topology (ZFP P Q) | |
| 1152 | 260 ProductTopology {P} {Q} TP TQ = InducedTopology (ZFP P Q) (pbase TP TQ) record { P⊆PL = pbase⊆PL TP TQ } |
| 1142 | 261 |
| 1152 | 262 -- covers ( q ⊆ Union P ) |
| 1101 | 263 |
| 1120 | 264 record _covers_ ( P q : HOD ) : Set n where |
| 431 | 265 field |
| 1120 | 266 cover : {x : Ordinal } → odef q x → Ordinal |
| 1145 | 267 P∋cover : {x : Ordinal } → (lt : odef q x) → odef P (cover lt) |
| 268 isCover : {x : Ordinal } → (lt : odef q x) → odef (* (cover lt)) x | |
| 1120 | 269 |
| 270 open _covers_ | |
| 431 | 271 |
| 272 -- Finite Intersection Property | |
| 273 | |
| 1120 | 274 record FIP {L : HOD} (top : Topology L) : Set n where |
| 431 | 275 field |
| 1150 | 276 limit : {X : Ordinal } → * X ⊆ CS top |
| 1187 | 277 → ( { x : Ordinal } → Subbase (* X) x → o∅ o< x ) → Ordinal |
| 1150 | 278 is-limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 279 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1143 | 280 → {x : Ordinal } → odef (* X) x → odef (* x) (limit CX fip) |
| 1150 | 281 L∋limit : {X : Ordinal } → (CX : * X ⊆ CS top ) |
| 1187 | 282 → ( fip : { x : Ordinal } → Subbase (* X) x → o∅ o< x ) |
| 1150 | 283 → {x : Ordinal } → odef (* X) x |
| 1143 | 284 → odef L (limit CX fip) |
| 285 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 | 286 |
| 287 -- Compact | |
| 288 | |
| 1119 | 289 data Finite-∪ (S : HOD) : Ordinal → Set n where |
| 1188 | 290 fin-e : Finite-∪ S o∅ |
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291 fin-i : {x : Ordinal } → odef S x → Finite-∪ S (& ( * x , * x )) -- an element of S |
| 1188 | 292 fin-∪ : {x y : Ordinal } → Finite-∪ S x → Finite-∪ S y → Finite-∪ S (& (* x ∪ * y)) |
| 1198 | 293 -- Finite-∪ S y → Union y ⊆ S |
| 431 | 294 |
| 1120 | 295 record Compact {L : HOD} (top : Topology L) : Set n where |
| 431 | 296 field |
| 1120 | 297 finCover : {X : Ordinal } → (* X) ⊆ OS top → (* X) covers L → Ordinal |
| 298 isCover : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → (* (finCover xo xcp )) covers L | |
| 1150 | 299 isFinite : {X : Ordinal } → (xo : (* X) ⊆ OS top) → (xcp : (* X) covers L ) → Finite-∪ (* X) (finCover xo xcp ) |
| 431 | 300 |
| 301 -- FIP is Compact | |
| 302 | |
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303 FIP→Compact : {L : HOD} → (top : Topology L ) → FIP top → Compact top |
| 1150 | 304 FIP→Compact {L} top fip with trio< (& L) o∅ |
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305 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1148 | 306 ... | tri≈ ¬a b ¬c = record { finCover = λ _ _ → o∅ ; isCover = λ {X} _ xcp → fip01 xcp ; isFinite = fip00 } where |
| 307 -- L is empty | |
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308 fip02 : {x : Ordinal } → ¬ odef L x |
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309 fip02 {x} Lx = ⊥-elim ( o<¬≡ (sym b) (∈∅< Lx) ) |
| 1148 | 310 fip01 : {X : Ordinal } → (xcp : * X covers L) → (* o∅) covers L |
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311 fip01 xcp = record { cover = λ Lx → ⊥-elim (fip02 Lx) ; P∋cover = λ Lx → ⊥-elim (fip02 Lx) ; isCover = λ Lx → ⊥-elim (fip02 Lx) } |
| 1148 | 312 fip00 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) o∅ |
| 1188 | 313 fip00 {X} xo xcp = fin-e |
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314 ... | tri> ¬a ¬b 0<L = record { finCover = finCover ; isCover = isCover1 ; isFinite = isFinite } where |
| 1121 | 315 -- set of coset of X |
| 316 CX : {X : Ordinal} → * X ⊆ OS top → Ordinal | |
| 1293 | 317 CX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼ ) |
| 1150 | 318 CCX : {X : Ordinal} → (os : * X ⊆ OS top) → * (CX os) ⊆ CS top |
| 1143 | 319 CCX {X} os {x} ox with subst (λ k → odef k x) *iso ox |
| 320 ... | record { z = z ; az = az ; x=ψz = x=ψz } = ⟪ fip05 , fip06 ⟫ where -- x ≡ & (L ＼ * z) | |
| 321 fip07 : z ≡ & (L ＼ * x) | |
| 322 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 | |
| 323 fip08 : {x : Ordinal} → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) → odef (* z) x | |
| 1150 | 324 fip08 {x} ⟪ Lx , not ⟫ with subst (λ k → (¬ odef k x)) *iso not -- ( odef L x ∧ odef (* z) x → ⊥) → ⊥ |
| 1143 | 325 ... | Lx∧¬zx = ODC.double-neg-elim O ( λ nz → Lx∧¬zx ⟪ Lx , nz ⟫ ) |
| 326 fip09 : {x : Ordinal} → odef (* z) x → odef L x ∧ (¬ odef (* (& (L ＼ * z))) x) | |
| 327 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 | |
| 328 fip10 : ¬ (odef (L ＼ * z) w) | |
| 329 fip10 ⟪ Lw , nzw ⟫ = nzw zw | |
| 330 fip06 : odef (OS top) (& (L ＼ * x)) | |
| 331 fip06 = os ( subst (λ k → odef (* X) k ) fip07 az ) | |
| 332 fip05 : * x ⊆ L | |
| 333 fip05 {w} xw = proj1 ( subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso ) xw ) | |
| 334 -- | |
| 335 -- X covres L means Intersection of (CX X) contains nothing | |
| 1152 | 336 -- then some finite Intersection of (CX X) contains nothing ( contraposition of FIP .i.e. CFIP) |
| 1143 | 337 -- it means there is a finite cover |
| 338 -- | |
| 1293 | 339 finCoverBase : {X : Ordinal } → * X ⊆ OS top → * X covers L → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅ |
| 340 finCoverBase {X} ox oc with ODC.p∨¬p O (Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) o∅) | |
| 1189 | 341 ... | case1 sb = sb |
| 1199 | 342 ... | case2 ¬sb = ⊥-elim (¬¬cover fip25 fip20) where |
| 343 ¬¬cover : {z : Ordinal } → odef L z → ¬ ( {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) z )) | |
| 344 ¬¬cover {z} Lz nc = nc ( P∋cover oc Lz ) (isCover oc _ ) | |
| 345 -- ¬sb → we have finite intersection | |
| 1187 | 346 fip02 : {x : Ordinal} → Subbase (* (CX ox)) x → o∅ o< x |
| 347 fip02 {x} sc with trio< x o∅ | |
| 1148 | 348 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 349 ... | tri> ¬a ¬b c = c | |
| 1189 | 350 ... | tri≈ ¬a b ¬c = ⊥-elim (¬sb (subst₂ (λ j k → Subbase j k ) *iso b sc )) |
| 1150 | 351 -- we have some intersection because L is not empty (if we have an element of L, we don't need choice) |
| 1148 | 352 fip26 : odef (* (CX ox)) (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) ))) |
| 1150 | 353 fip26 = subst (λ k → odef k (& (L ＼ * ( cover oc ( ODC.x∋minimal O L (0<P→ne 0<L) ) )) )) (sym *iso) |
| 354 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 | 355 fip25 : odef L( FIP.limit fip (CCX ox) fip02 ) |
| 356 fip25 = FIP.L∋limit fip (CCX ox) fip02 fip26 | |
| 357 fip20 : {y : Ordinal } → (Xy : odef (* X) y) → ¬ ( odef (* y) ( FIP.limit fip (CCX ox) fip02 )) | |
| 358 fip20 {y} Xy yl = proj2 fip21 yl where | |
| 359 fip22 : odef (* (CX ox)) (& ( L ＼ * y )) | |
| 1150 | 360 fip22 = subst (λ k → odef k (& ( L ＼ * y ))) (sym *iso) record { z = y ; az = Xy ; x=ψz = refl } |
| 1148 | 361 fip21 : odef (L ＼ * y) ( FIP.limit fip (CCX ox) fip02 ) |
| 362 fip21 = subst (λ k → odef k ( FIP.limit fip (CCX ox) fip02 ) ) *iso ( FIP.is-limit fip (CCX ox) fip02 fip22 ) | |
| 1199 | 363 -- create HOD from Subbase ( finite intersection ) |
| 1293 | 364 finCoverSet : {X : Ordinal } → (x : Ordinal) → Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x → HOD |
| 1190 | 365 finCoverSet {X} x (gi rx) = ( L ＼ * x ) , ( L ＼ * x ) |
| 1189 | 366 finCoverSet {X} x∩y (g∩ {x} {y} sx sy) = finCoverSet {X} x sx ∪ finCoverSet {X} y sy |
| 1149 | 367 -- |
| 1121 | 368 -- this defines finite cover |
| 1120 | 369 finCover : {X : Ordinal} → * X ⊆ OS top → * X covers L → Ordinal |
| 1189 | 370 finCover {X} ox oc = & ( finCoverSet o∅ (finCoverBase ox oc)) |
| 1199 | 371 -- create Finite-∪ from finCoverSet |
| 1120 | 372 isFinite : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → Finite-∪ (* X) (finCover xo xcp) |
| 1189 | 373 isFinite {X} xo xcp = fip60 o∅ (finCoverBase xo xcp) where |
| 1293 | 374 fip60 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → Finite-∪ (* X) (& (finCoverSet {X} x sb)) |
| 1190 | 375 fip60 x (gi rx) = subst (λ k → Finite-∪ (* X) k) fip62 (fin-i (fip61 rx)) where |
| 376 fip62 : & (* (& (L ＼ * x)) , * (& (L ＼ * x))) ≡ & ((L ＼ * x) , (L ＼ * x)) | |
| 377 fip62 = cong₂ (λ j k → & (j , k )) *iso *iso | |
| 1293 | 378 fip61 : odef (Replace (* X) (_＼_ L) RC＼ ) x → odef (* X) ( & ((L ＼ * x ) )) |
| 1189 | 379 fip61 record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where |
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380 fip34 : * z1 ⊆ L |
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381 fip34 {w} wz1 = os⊆L top (subst (λ k → odef (OS top) k) (sym &iso) (xo az1)) wz1 |
| 1189 | 382 fip33 : z1 ≡ & (L ＼ * x) |
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383 fip33 = begin |
| 1152 | 384 z1 ≡⟨ sym &iso ⟩ |
| 385 & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ | |
| 386 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ | |
| 387 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ | |
| 1189 | 388 & (L ＼ * x ) ∎ where open ≡-Reasoning |
| 389 fip60 x∩y (g∩ {x} {y} sx sy) = subst (λ k → Finite-∪ (* X) k) fip62 ( fin-∪ (fip60 x sx) (fip60 y sy) ) where | |
| 390 fip62 : & (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡ & (finCoverSet x sx ∪ finCoverSet y sy) | |
| 391 fip62 = cong (&) ( begin | |
| 392 (* (& (finCoverSet x sx)) ∪ * (& (finCoverSet y sy))) ≡⟨ cong₂ _∪_ *iso *iso ⟩ | |
| 393 finCoverSet x sx ∪ finCoverSet y sy ∎ ) where open ≡-Reasoning | |
| 1120 | 394 -- is also a cover |
| 395 isCover1 : {X : Ordinal} (xo : * X ⊆ OS top) (xcp : * X covers L) → * (finCover xo xcp) covers L | |
| 1190 | 396 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) |
| 397 (fip70 o∅ (finCoverBase xo xcp)) where | |
| 1293 | 398 fip70 : (x : Ordinal) → (sb : Subbase (Replace (* X) (λ z → L ＼ z) RC＼ ) x ) → (finCoverSet {X} x sb) covers (L ＼ * x) |
| 1199 | 399 fip70 x (gi rx) = fip73 where |
| 400 fip73 : ((L ＼ * x) , (L ＼ * x)) covers (L ＼ * x) -- obvious | |
| 401 fip73 = record { cover = λ _ → & (L ＼ * x) ; P∋cover = λ _ → case1 refl | |
| 402 ; isCover = λ {x} lt → subst (λ k → odef k x) (sym *iso) lt } | |
| 1190 | 403 fip70 x∩y (g∩ {x} {y} sx sy) = subst (λ k → finCoverSet (& (* x ∩ * y)) (g∩ sx sy) covers |
| 1194 | 404 (L ＼ k)) (sym *iso) ( fip43 {_} {L} {* x} {* y} (fip71 (fip70 x sx)) (fip72 (fip70 y sy)) ) where |
| 405 fip71 : {a b c : HOD} → a covers c → (a ∪ b) covers c | |
| 406 fip71 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case1 (P∋cover cov lt) | |
| 407 ; isCover = isCover cov } | |
| 408 fip72 : {a b c : HOD} → a covers c → (b ∪ a) covers c | |
| 409 fip72 {a} {b} {c} cov = record { cover = cover cov ; P∋cover = λ lt → case2 (P∋cover cov lt) | |
| 410 ; isCover = isCover cov } | |
| 1190 | 411 fip45 : {L a b : HOD} → (L ＼ (a ∩ b)) ⊆ ( (L ＼ a) ∪ (L ＼ b)) |
| 412 fip45 {L} {a} {b} {x} Lab with ODC.∋-p O b (* x) | |
| 413 ... | yes bx = case1 ⟪ proj1 Lab , (λ ax → proj2 Lab ⟪ ax , subst (λ k → odef b k) &iso bx ⟫ ) ⟫ | |
| 414 ... | no ¬bx = case2 ⟪ proj1 Lab , subst (λ k → ¬ ( odef b k)) &iso ¬bx ⟫ | |
| 415 fip43 : {A L a b : HOD } → A covers (L ＼ a) → A covers (L ＼ b ) → A covers ( L ＼ ( a ∩ b ) ) | |
| 416 fip43 {A} {L} {a} {b} ca cb = record { cover = fip44 ; P∋cover = fip46 ; isCover = fip47 } where | |
| 417 fip44 : {x : Ordinal} → odef (L ＼ (a ∩ b)) x → Ordinal | |
| 418 fip44 {x} Lab with fip45 {L} {a} {b} Lab | |
| 419 ... | case1 La = cover ca La | |
| 420 ... | case2 Lb = cover cb Lb | |
| 421 fip46 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef A (fip44 lt) | |
| 422 fip46 {x} Lab with fip45 {L} {a} {b} Lab | |
| 423 ... | case1 La = P∋cover ca La | |
| 424 ... | case2 Lb = P∋cover cb Lb | |
| 425 fip47 : {x : Ordinal} (lt : odef (L ＼ (a ∩ b)) x) → odef (* (fip44 lt)) x | |
| 426 fip47 {x} Lab with fip45 {L} {a} {b} Lab | |
| 427 ... | case1 La = isCover ca La | |
| 428 ... | case2 Lb = isCover cb Lb | |
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429 |
| 1180 | 430 open _==_ |
| 431 | |
| 1158 | 432 Compact→FIP : {L : HOD} → (top : Topology L ) → Compact top → FIP top |
| 1180 | 433 Compact→FIP {L} top compact with trio< (& L) o∅ |
| 1175 | 434 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) |
| 1198 | 435 ... | 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 |
| 436 -- empty L case | |
| 437 -- if 0 < X then 0 < x ∧ L ∋ xfrom fip | |
| 438 -- if 0 ≡ X then ¬ odef X x | |
| 439 fip000 : {X x : Ordinal} (CX : * X ⊆ CS top) → ({y : Ordinal} → Subbase (* X) y → o∅ o< y) → ¬ odef (* X) x | |
| 440 fip000 {X} {x} CX fip xx with trio< o∅ X | |
| 441 ... | tri< 0<X ¬b ¬c = ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) L=0)) o∅≡od∅ ) (sym &iso) | |
| 442 ( cs⊆L top (subst (λ k → odef (CS top) k ) (sym &iso) (CX xx)) Xe )) where | |
| 443 0<x : o∅ o< x | |
| 444 0<x = fip (gi xx ) | |
| 445 e : HOD -- we have an element of x | |
| 446 e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 447 Xe : odef (* x) (& e) | |
| 448 Xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 449 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin | |
| 450 * X ≡⟨ cong (*) (sym 0=X) ⟩ | |
| 451 * o∅ ≡⟨ o∅≡od∅ ⟩ | |
| 452 od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning | |
| 453 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1175 | 454 ... | tri> ¬a ¬b 0<L = record { limit = limit ; is-limit = fip00 } where |
| 455 -- set of coset of X | |
| 456 OX : {X : Ordinal} → * X ⊆ CS top → Ordinal | |
| 1293 | 457 OX {X} ox = & ( Replace (* X) (λ z → L ＼ z ) RC＼) |
| 1175 | 458 OOX : {X : Ordinal} → (cs : * X ⊆ CS top) → * (OX cs) ⊆ OS top |
| 459 OOX {X} cs {x} ox with subst (λ k → odef k x) *iso ox | |
| 460 ... | 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 | |
| 461 comp01 : odef (* X) (& (* z)) | |
| 462 comp01 = subst (λ k → odef (* X) k) (sym &iso) az | |
| 1183 | 463 -- if all finite intersection of X contains something, |
| 1175 | 464 -- there is no finite cover. From Compactness, (OX X) is not a cover of L ( contraposition of Compact) |
| 465 -- it means there is a limit | |
| 1199 | 466 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 |
| 467 field -- find an element x, which is not covered (which is a limit point) | |
| 468 x : Ordinal | |
| 469 yx : {y : Ordinal} (Xy : odef (* X) y) → odef (* y) x | |
| 1187 | 470 has-intersection : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1199 | 471 → (0<X : o∅ o< X ) → NC CX fip 0<X |
| 472 has-intersection {X} CX fip 0<X = intersection where | |
| 1198 | 473 e : HOD -- we have an element of X |
| 474 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 475 Xe : odef (* X) (& e) | |
| 476 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 1183 | 477 no-cover : ¬ ( (* (OX CX)) covers L ) |
| 1198 | 478 no-cover cov = ⊥-elim (no-finite-cover (Compact.isCover compact (OOX CX) cov)) where |
| 479 -- construct Subase from Finite-∪ | |
| 1180 | 480 fp01 : Ordinal |
| 481 fp01 = Compact.finCover compact (OOX CX) cov | |
| 1194 | 482 record SB (t : Ordinal) : Set n where |
| 483 field | |
| 484 i : Ordinal | |
| 485 sb : Subbase (* X) (& (L ＼ * i)) | |
| 1198 | 486 t⊆i : (L ＼ * i) ⊆ (L ＼ Union ( * t ) ) |
| 1194 | 487 fp02 : (t : Ordinal) → Finite-∪ (* (OX CX)) t → SB t |
| 1198 | 488 fp02 t fin-e = record { i = & ( L ＼ e) ; sb = gi (subst (λ k → odef (* X) k) fp21 Xe) ; t⊆i = fp23 } where |
| 489 -- t ≡ o∅, no cover. Any subst of L is ok and we have e ⊆ L | |
| 490 fp22 : e ⊆ L | |
| 491 fp22 {x} lt = cs⊆L top (CX Xe) lt | |
| 492 fp21 : & e ≡ & (L ＼ * (& (L ＼ e))) | |
| 1194 | 493 fp21 = cong (&) (trans (sym (L＼Lx=x fp22)) (cong (λ k → L ＼ k) (sym *iso))) |
| 1198 | 494 fp23 : (L ＼ * (& (L ＼ e))) ⊆ (L ＼ Union (* o∅)) |
| 1197 | 495 fp23 {x} ⟪ Lx , _ ⟫ = ⟪ Lx , ( λ lt → ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) o∅≡od∅ (sym &iso) (Own.ao lt )))) ⟫ |
| 1198 | 496 fp02 t (fin-i {x} tx ) = record { i = x ; sb = gi fp03 ; t⊆i = fp24 } where |
| 497 -- we have a single cover x, L ＼ * x is single finite intersection | |
| 1197 | 498 fp24 : (L ＼ * x) ⊆ (L ＼ Union (* (& (* x , * x)))) |
| 499 fp24 {y} ⟪ Lx , not ⟫ = ⟪ Lx , subst (λ k → ¬ odef (Union k) y) (sym *iso) fp25 ⟫ where | |
| 500 fp25 : ¬ odef (Union (* x , * x)) y | |
| 501 fp25 record { owner = .(& (* x)) ; ao = (case1 refl) ; ox = ox } = not (subst (λ k → odef k y) *iso ox ) | |
| 502 fp25 record { owner = .(& (* x)) ; ao = (case2 refl) ; ox = ox } = not (subst (λ k → odef k y) *iso ox ) | |
| 1198 | 503 fp03 : odef (* X) (& (L ＼ * x)) -- becase x is an element of Replace (* X) (λ z → L ＼ z ) |
| 1194 | 504 fp03 with subst (λ k → odef k x ) *iso tx |
| 505 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef (* X) k) fip33 az1 where | |
| 506 fip34 : * z1 ⊆ L | |
| 507 fip34 {w} wz1 = cs⊆L top (subst (λ k → odef (CS top) k) (sym &iso) (CX az1) ) wz1 | |
| 508 fip33 : z1 ≡ & (L ＼ * x) | |
| 509 fip33 = begin | |
| 510 z1 ≡⟨ sym &iso ⟩ | |
| 511 & (* z1) ≡⟨ cong (&) (sym (L＼Lx=x fip34 )) ⟩ | |
| 512 & (L ＼ ( L ＼ * z1)) ≡⟨ cong (λ k → & ( L ＼ k )) (sym *iso) ⟩ | |
| 513 & (L ＼ * (& ( L ＼ * z1))) ≡⟨ cong (λ k → & ( L ＼ * k )) (sym x=ψz1) ⟩ | |
| 514 & (L ＼ * x ) ∎ where open ≡-Reasoning | |
| 1198 | 515 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 |
| 516 fp35 : (L ＼ * (& (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))) ⊆ (L ＼ Union (* (& (* tx ∪ * ty)))) | |
| 1197 | 517 fp35 = subst₂ (λ j k → (L ＼ j ) ⊆ (L ＼ Union k)) (sym *iso) (sym *iso) fp36 where |
| 518 fp40 : {z tz : Ordinal } → Finite-∪ (* (OX CX)) tz → odef (Union (* tz )) z → odef L z | |
| 1199 | 519 fp40 {z} {.(Ordinals.o∅ O)} fin-e record { owner = owner ; ao = ao ; ox = ox } |
| 520 = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) o∅≡od∅ (sym &iso) ao )) | |
| 1197 | 521 fp40 {z} {.(& (* _ , * _))} (fin-i {w} x) uz = fp41 x (subst (λ k → odef (Union k) z) *iso uz) where |
| 522 fp41 : (x : odef (* (OX CX)) w) → (uz : odef (Union (* w , * w)) z ) → odef L z | |
| 523 fp41 x record { owner = .(& (* w)) ; ao = (case1 refl) ; ox = ox } = | |
| 524 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) *iso ox ) | |
| 525 fp41 x record { owner = .(& (* w)) ; ao = (case2 refl) ; ox = ox } = | |
| 526 os⊆L top (OOX CX (subst (λ k → odef (* (OX CX)) k) (sym &iso) x )) (subst (λ k → odef k z) *iso ox ) | |
| 527 fp40 {z} {.(& (* _ ∪ * _))} (fin-∪ {x1} {y1} ftx fty) uz with subst (λ k → odef (Union k) z ) *iso uz | |
| 528 ... | record { owner = o ; ao = case1 x1o ; ox = oz } = fp40 ftx record { owner = o ; ao = x1o ; ox = oz } | |
| 529 ... | record { owner = o ; ao = case2 y1o ; ox = oz } = fp40 fty record { owner = o ; ao = y1o ; ox = oz } | |
| 1198 | 530 fp36 : (L ＼ (* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* tx ∪ * ty)) |
| 1197 | 531 fp36 {z} ⟪ Lz , not ⟫ = ⟪ Lz , fp37 ⟫ where |
| 532 fp37 : ¬ odef (Union (* tx ∪ * ty)) z | |
| 533 fp37 record { owner = owner ; ao = (case1 ax) ; ox = ox } = not (case1 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where | |
| 1198 | 534 fp38 : (L ＼ (* (SB.i (fp02 tx ux)))) ⊆ (L ＼ Union (* tx)) |
| 535 fp38 = SB.t⊆i (fp02 tx ux) | |
| 536 fp39 : Union (* tx) ⊆ (* (SB.i (fp02 tx ux))) | |
| 537 fp39 {w} txw with ∨L＼X {L} {* (SB.i (fp02 tx ux))} (fp40 ux txw) | |
| 1197 | 538 ... | case1 sb = sb |
| 539 ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) txw ) | |
| 540 fp37 record { owner = owner ; ao = (case2 ax) ; ox = ox } = not (case2 (fp39 record { owner = _ ; ao = ax ; ox = ox }) ) where | |
| 1198 | 541 fp38 : (L ＼ (* (SB.i (fp02 ty uy)))) ⊆ (L ＼ Union (* ty)) |
| 542 fp38 = SB.t⊆i (fp02 ty uy) | |
| 543 fp39 : Union (* ty) ⊆ (* (SB.i (fp02 ty uy))) | |
| 544 fp39 {w} tyw with ∨L＼X {L} {* (SB.i (fp02 ty uy))} (fp40 uy tyw) | |
| 1197 | 545 ... | case1 sb = sb |
| 546 ... | case2 lsb = ⊥-elim ( proj2 (fp38 lsb) tyw ) | |
| 1194 | 547 fp04 : {tx ty : Ordinal} → & (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) ≡ & (L ＼ * (& (* tx ∪ * ty))) |
| 548 fp04 {tx} {ty} = cong (&) ( ==→o≡ record { eq→ = fp05 ; eq← = fp09 } ) where | |
| 1187 | 549 fp05 : {x : Ordinal} → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x → odef (L ＼ * (& (* tx ∪ * ty))) x |
| 550 fp05 {x} lt with subst₂ (λ j k → odef (j ∩ k) x ) *iso *iso lt | |
| 551 ... | ⟪ ⟪ Lx , ¬tx ⟫ , ⟪ Ly , ¬ty ⟫ ⟫ = subst (λ k → odef (L ＼ k) x) (sym *iso) ⟪ Lx , fp06 ⟫ where | |
| 552 fp06 : ¬ odef (* tx ∪ * ty) x | |
| 553 fp06 (case1 tx) = ¬tx tx | |
| 554 fp06 (case2 ty) = ¬ty ty | |
| 555 fp09 : {x : Ordinal} → odef (L ＼ * (& (* tx ∪ * ty))) x → odef (* (& (L ＼ * tx)) ∩ * (& (L ＼ * ty))) x | |
| 556 fp09 {x} lt with subst (λ k → odef (L ＼ k) x) (*iso) lt | |
| 557 ... | ⟪ Lx , ¬tx∨ty ⟫ = subst₂ (λ j k → odef (j ∩ k) x ) (sym *iso) (sym *iso) | |
| 558 ⟪ ⟪ Lx , ( λ tx → ¬tx∨ty (case1 tx)) ⟫ , ⟪ Lx , ( λ ty → ¬tx∨ty (case2 ty)) ⟫ ⟫ | |
| 1198 | 559 fp11 : Subbase (* X) (& (L ＼ * (& ((* (SB.i (fp02 tx ux)) ∪ * (SB.i (fp02 ty uy))))))) |
| 560 fp11 = subst (λ k → Subbase (* X) k ) fp04 ( g∩ (SB.sb (fp02 tx ux)) (SB.sb (fp02 ty uy )) ) | |
| 561 -- | |
| 562 -- becase of fip, finite cover cannot be a cover | |
| 563 -- | |
| 1195 | 564 fcov : Finite-∪ (* (OX CX)) (Compact.finCover compact (OOX CX) cov) |
| 565 fcov = Compact.isFinite compact (OOX CX) cov | |
| 1196 | 566 0<sb : {i : Ordinal } → (sb : Subbase (* X) (& (L ＼ * i))) → o∅ o< & (L ＼ * i) |
| 567 0<sb {i} sb = fip sb | |
| 568 sb : SB (Compact.finCover compact (OOX CX) cov) | |
| 569 sb = fp02 fp01 (Compact.isFinite compact (OOX CX) cov) | |
| 1198 | 570 no-finite-cover : ¬ ( (* (Compact.finCover compact (OOX CX) cov)) covers L ) |
| 571 no-finite-cover fcovers = ⊥-elim ( o<¬≡ (cong (&) (sym (==→o≡ f22))) f25 ) where | |
| 1196 | 572 f23 : (L ＼ * (SB.i sb)) ⊆ ( L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) |
| 1198 | 573 f23 = SB.t⊆i sb |
| 1196 | 574 f22 : (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) =h= od∅ |
| 575 f22 = record { eq→ = λ lt → ⊥-elim ( f24 lt) ; eq← = λ lt → ⊥-elim (¬x<0 lt) } where | |
| 576 f24 : {x : Ordinal } → ¬ ( odef (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) x ) | |
| 577 f24 {x} ⟪ Lx , not ⟫ = not record { owner = cover fcovers Lx ; ao = P∋cover fcovers Lx ; ox = isCover fcovers Lx } | |
| 578 f25 : & od∅ o< (& (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ) | |
| 579 f25 = ordtrans<-≤ (subst (λ k → k o< & (L ＼ * (SB.i sb))) (sym ord-od∅) (0<sb (SB.sb sb) ) ) ( begin | |
| 580 & (L ＼ * (SB.i sb)) ≤⟨ ⊆→o≤ f23 ⟩ | |
| 581 & (L ＼ Union (* (Compact.finCover compact (OOX CX) cov))) ∎ ) where open o≤-Reasoning O | |
| 1199 | 582 -- if we have no cover, we can consruct NC |
| 583 intersection : NC CX fip 0<X | |
| 584 intersection with ODC.p∨¬p O (NC CX fip 0<X) | |
| 1184 | 585 ... | case1 nc = nc |
| 1185 | 586 ... | case2 ¬nc = ⊥-elim ( no-cover record { cover = λ Lx → & (L ＼ coverf Lx) ; P∋cover = fp22 ; isCover = fp23 } ) where |
| 587 coverSet : {x : Ordinal} → odef L x → HOD | |
| 1186 | 588 coverSet {x} Lx = record { od = record { def = λ y → odef (* X) y ∧ odef (L ＼ * y) x } ; odmax = X |
| 589 ; <odmax = λ {x} lt → subst (λ k → x o< k) &iso ( odef< (proj1 lt)) } | |
| 1185 | 590 fp17 : {x : Ordinal} → (Lx : odef L x ) → ¬ ( coverSet Lx =h= od∅ ) |
| 1187 | 591 fp17 {x} Lx eq = ⊥-elim (¬nc record { x = x ; yx = fp19 } ) where |
| 1185 | 592 fp19 : {y : Ordinal} → odef (* X) y → odef (* y) x |
| 593 fp19 {y} Xy with ∨L＼X {L} {* y} {x} Lx | |
| 594 ... | case1 yx = yx | |
| 595 ... | case2 lyx = ⊥-elim ( ¬x<0 {y} ( eq→ eq fp20 )) where | |
| 596 fp20 : odef (* X) y ∧ odef (L ＼ * y) x | |
| 597 fp20 = ⟪ Xy , lyx ⟫ | |
| 598 coverf : {x : Ordinal} → (Lx : odef L x ) → HOD | |
| 599 coverf Lx = ODC.minimal O (coverSet Lx) (fp17 Lx) | |
| 1186 | 600 fp22 : {x : Ordinal} (lt : odef L x) → odef (* (OX CX)) (& (L ＼ coverf lt)) |
| 601 fp22 {x} Lx = subst (λ k → odef k (& (L ＼ coverf Lx ))) (sym *iso) record { z = _ ; az = fp25 ; x=ψz = fp24 } where | |
| 602 fp24 : & (L ＼ coverf Lx) ≡ & (L ＼ * (& (coverf Lx))) | |
| 603 fp24 = cong (λ k → & ( L ＼ k )) (sym *iso) | |
| 1185 | 604 fp25 : odef (* X) (& (coverf Lx)) |
| 605 fp25 = proj1 ( ODC.x∋minimal O (coverSet Lx) (fp17 Lx) ) | |
| 1186 | 606 fp23 : {x : Ordinal} (lt : odef L x) → odef (* (& (L ＼ coverf lt))) x |
| 607 fp23 {x} Lx = subst (λ k → odef k x) (sym *iso) ⟪ Lx , fp26 ⟫ where | |
| 608 fp26 : ¬ odef (coverf Lx) x | |
| 609 fp26 = subst (λ k → ¬ odef k x ) *iso (proj2 (proj2 ( ODC.x∋minimal O (coverSet Lx) (fp17 Lx) )) ) | |
| 1199 | 610 limit : {X : Ordinal} (CX : * X ⊆ CS top) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) → Ordinal |
| 1180 | 611 limit {X} CX fip with trio< X o∅ |
| 612 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 613 ... | tri≈ ¬a b ¬c = o∅ | |
| 1199 | 614 ... | tri> ¬a ¬b c = NC.x ( has-intersection CX fip c) |
| 1175 | 615 fip00 : {X : Ordinal} (CX : * X ⊆ CS top) |
| 1187 | 616 (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) |
| 1175 | 617 {x : Ordinal} → odef (* X) x → odef (* x) (limit CX fip ) |
| 1180 | 618 fip00 {X} CX fip {x} Xx with trio< X o∅ |
| 619 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
| 620 ... | tri≈ ¬a b ¬c = ⊥-elim ( o<¬≡ (sym b) (subst (λ k → o∅ o< k) &iso (∈∅< Xx) ) ) | |
| 1199 | 621 ... | tri> ¬a ¬b c = NC.yx ( has-intersection CX fip c ) Xx |
| 431 | 622 |
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623 open Filter |
| 1102 | 624 |
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625 -- |
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626 -- {v | Neighbor lim v} set of u ⊆ v ⊆ P where u is an open set u ∋ x |
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627 -- |
| 1159 | 628 record Neighbor {P : HOD} (TP : Topology P) (x v : Ordinal) : Set n where |
| 629 field | |
| 630 u : Ordinal | |
| 631 ou : odef (OS TP) u | |
| 632 ux : odef (* u) x | |
| 633 v⊆P : * v ⊆ P | |
| 1170 | 634 u⊆v : * u ⊆ * v |
| 1102 | 635 |
| 1169 | 636 -- |
| 637 -- Neighbor on x is a Filter (on Power P) | |
| 638 -- | |
| 1170 | 639 NeighborF : {P : HOD} (TP : Topology P) (x : Ordinal ) → Filter {Power P} {P} (λ x → x) |
| 1169 | 640 NeighborF {P} TP x = record { filter = NF ; f⊆L = NF⊆PP ; filter1 = f1 ; filter2 = f2 } where |
| 1168 | 641 nf00 : {v : Ordinal } → Neighbor TP x v → odef (Power P) v |
| 642 nf00 {v} nei y vy = Neighbor.v⊆P nei vy | |
| 1167 | 643 NF : HOD |
| 1168 | 644 NF = record { od = record { def = λ v → Neighbor TP x v } ; odmax = & (Power P) ; <odmax = λ lt → odef< (nf00 lt) } |
| 1167 | 645 NF⊆PP : NF ⊆ Power P |
| 1168 | 646 NF⊆PP = nf00 |
| 647 f1 : {p q : HOD} → Power P ∋ q → NF ∋ p → p ⊆ q → NF ∋ q | |
| 1170 | 648 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 | 649 f11 : * (Neighbor.u Np) ⊆ * (& q) |
| 1170 | 650 f11 {x} ux = subst (λ k → odef k x ) (sym *iso) ( p⊆q (subst (λ k → odef k x) *iso (Neighbor.u⊆v Np ux)) ) |
| 1168 | 651 f2 : {p q : HOD} → NF ∋ p → NF ∋ q → Power P ∋ (p ∩ q) → NF ∋ (p ∩ q) |
| 1170 | 652 f2 {p} {q} Np Nq Ppq = record { u = upq ; ou = ou ; ux = ux ; v⊆P = Ppq _ ; u⊆v = f20 } where |
| 1168 | 653 upq : Ordinal |
| 654 upq = & ( * (Neighbor.u Np) ∩ * (Neighbor.u Nq) ) | |
| 655 ou : odef (OS TP) upq | |
| 656 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)) | |
| 657 ux : odef (* upq) x | |
| 1170 | 658 ux = subst ( λ k → odef k x ) (sym *iso) ⟪ Neighbor.ux Np , Neighbor.ux Nq ⟫ |
| 1168 | 659 f20 : * upq ⊆ * (& (p ∩ q)) |
| 660 f20 = subst₂ (λ j k → j ⊆ k ) (sym *iso) (sym *iso) ( λ {x} pq | |
| 1170 | 661 → ⟪ subst (λ k → odef k x) *iso (Neighbor.u⊆v Np (proj1 pq)) , subst (λ k → odef k x) *iso (Neighbor.u⊆v Nq (proj2 pq)) ⟫ ) |
| 1153 | 662 |
| 1165 | 663 CAP : (P : HOD) {p q : HOD } → Power P ∋ p → Power P ∋ q → Power P ∋ (p ∩ q) |
| 664 CAP P {p} {q} Pp Pq x pqx with subst (λ k → odef k x ) *iso pqx | |
| 665 ... | ⟪ px , qx ⟫ = Pp _ (subst (λ k → odef k x) (sym *iso) px ) | |
| 666 | |
| 1170 | 667 NEG : (P : HOD) {p : HOD } → Power P ∋ p → Power P ∋ (P ＼ p ) |
| 668 NEG P {p} Pp x vx with subst (λ k → odef k x) *iso vx | |
| 669 ... | ⟪ Px , npx ⟫ = Px | |
| 1142 | 670 |