Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Tychonoff.agda @ 1180:8e04e3cad0b5
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Thu, 23 Feb 2023 18:44:47 +0900 |
parents | 7d2bae0ff36b |
children | d996fe8dd116 |
rev | line source |
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1175 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
431 | 2 open import Level |
3 open import Ordinals | |
1170 | 4 module Tychonoff {n : Level } (O : Ordinals {n}) where |
431 | 5 |
6 open import zf | |
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 | |
28 open Ordinals.IsNext isNext | |
29 open OrdUtil O | |
30 open ODUtil O | |
31 | |
32 import ODC | |
33 open ODC O | |
34 | |
1102 | 35 open import filter O |
1101 | 36 open import OPair O |
1170 | 37 open import Topology O |
38 open import maximum-filter O | |
431 | 39 |
1170 | 40 open Filter |
41 open Topology | |
1169 | 42 |
431 | 43 -- FIP is UFL |
44 | |
1159 | 45 -- filter Base |
46 record FBase (P : HOD )(X : Ordinal ) (u : Ordinal) : Set n where | |
1153 | 47 field |
1161 | 48 b x : Ordinal |
1155 | 49 b⊆X : * b ⊆ * X |
1161 | 50 sb : Subbase (* b) x |
1158 | 51 u⊆P : * u ⊆ P |
1154 | 52 x⊆u : * x ⊆ * u |
1155 | 53 |
1170 | 54 record UFLP {P : HOD} (TP : Topology P) (F : Filter {Power P} {P} (λ x → x) ) |
55 (ultra : ultra-filter F ) : Set (suc (suc n)) where | |
56 field | |
57 limit : Ordinal | |
58 P∋limit : odef P limit | |
59 is-limit : {v : Ordinal} → Neighbor TP limit v → (* v) ⊆ filter F | |
1165 | 60 |
1161 | 61 UFLP→FIP : {P : HOD} (TP : Topology P) → |
1169 | 62 ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP |
1163 | 63 UFLP→FIP {P} TP uflp with trio< (& P) o∅ |
64 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) | |
65 ... | tri≈ ¬a b ¬c = record { limit = ? ; is-limit = {!!} } where | |
66 -- P is empty | |
67 fip02 : {x : Ordinal } → ¬ odef P x | |
68 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym b) (∈∅< Px) ) | |
1170 | 69 ... | tri> ¬a ¬b 0<P = record { limit = limit ; is-limit = uf01 } where |
1143 | 70 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
1153 | 71 fip {X} CSX = {u x : Ordinal} → * u ⊆ * X → Subbase (* u) x → o∅ o< x |
1154 | 72 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
1161 | 73 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
1159 | 74 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } |
1158 | 75 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
1159 | 76 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
1165 | 77 nc : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → HOD |
78 nc = ? | |
79 N∋nc :{X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → odef (N CSX fp) (& (nc CSX fp) ) | |
80 N∋nc = ? | |
81 0<PP : o∅ o< & (Power P) | |
82 0<PP = ? | |
1174 | 83 -- |
84 -- FIP defines a filter | |
85 -- | |
1158 | 86 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
87 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
88 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
1161 | 89 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
90 record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = subst (λ k → k ⊆ P) (sym *iso) f10 ; x⊆u = λ {z} xp → | |
1158 | 91 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
92 f10 : q ⊆ P | |
93 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
94 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
95 f2 {p} {q} Np Nq Xpq = record { b = & Np+Nq ; x = & xp∧xq ; b⊆X = f20 ; sb = sbpq ; u⊆P = p∩q⊆p ; x⊆u = f22 } where | |
96 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
97 p∩q⊆p {x} pqx = subst (λ k → odef P k) &iso (power→ P _ Xpq (subst₂ (λ j k → odef j k ) *iso (sym &iso) pqx )) | |
1159 | 98 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
99 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
1155 | 100 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
1159 | 101 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
1155 | 102 f20 : * (& Np+Nq) ⊆ * X |
103 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
1159 | 104 ... | case1 np = FBase.b⊆X Np np |
105 ... | case2 nq = FBase.b⊆X Nq nq | |
1155 | 106 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
1161 | 107 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
1159 | 108 → ⟪ subst (λ k → odef k w) *iso ( FBase.x⊆u Np (proj1 xpq)) , subst (λ k → odef k w) *iso ( FBase.x⊆u Nq (proj2 xpq)) ⟫ ) |
1174 | 109 -- |
110 -- it contains no empty sets | |
111 -- | |
1155 | 112 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ¬ (N CSX fp ∋ od∅) |
113 proper = ? | |
1174 | 114 -- |
115 -- then we have maximum ultra filter | |
116 -- | |
1158 | 117 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
1165 | 118 maxf {X} CSX fp = F→Maximum {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp) |
1170 | 119 mf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) |
120 mf {X} CSX fp = MaximumFilter.mf (maxf CSX fp) | |
121 ultraf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf CSX fp) | |
1169 | 122 ultraf {X} CSX fp = F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc CSX fp) (proper CSX fp) |
1174 | 123 -- |
124 -- so i has a limit as a limit of UIP | |
125 -- | |
1170 | 126 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
127 limit {X} CSX fp = UFLP.limit ( uflp ( mf CSX fp ) (ultraf CSX fp)) | |
128 uf02 : {X v : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) | |
129 → Neighbor TP (limit CSX fp) v → * v ⊆ filter ( mf CSX fp ) | |
130 uf02 {X} {v} CSX fp nei {x} vx = UFLP.is-limit ( uflp ( mf CSX fp ) (ultraf CSX fp)) nei vx | |
1174 | 131 -- |
132 -- the limit is an element of entire elements of X | |
133 -- | |
1170 | 134 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) |
135 uf01 {X} CSX fp {x} xx with ODC.∋-p O (* x) (* (limit CSX fp)) | |
136 ... | yes y = subst (λ k → odef (* x) k) &iso y | |
137 ... | no nxl = ⊥-elim ( MaximumFilter.proper (maxf CSX fp) uf08 ) where | |
138 uf03 : OS TP ∋ ( P \ (* x)) | |
139 uf03 = ? | |
140 uf05 : odef ( P \ (* x)) (limit CSX fp) | |
141 uf05 = ⟪ ? , subst (λ k → ¬ odef (* x) k) ? nxl ⟫ | |
142 uf04 : Neighbor TP (limit CSX fp) (& ( P \ (* (limit CSX fp)))) | |
143 uf04 = record { u = & ( P \ (* x)) ; ou = ? ; ux = ? ; v⊆P = ? ; u⊆v = ? } | |
144 uf07 : odef (filter (mf CSX fp)) x | |
145 uf07 = ? | |
146 uf06 : odef (filter (mf CSX fp)) (& ( P \ (* x)) ) | |
147 uf06 = ? | |
148 uf08 : (filter (mf CSX fp)) ∋ od∅ | |
149 uf08 = ? | |
1169 | 150 |
1142 | 151 |
1158 | 152 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
1169 | 153 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF |
154 FIP→UFLP {P} TP fip F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 ; P∋limit = ? ; is-limit = ufl00 } where | |
1174 | 155 -- |
156 -- take closure of given filter elements | |
157 -- | |
1160 | 158 CF : HOD |
1162 | 159 CF = Replace' (filter F) (λ x fx → Cl TP x ) |
1160 | 160 CF⊆CS : CF ⊆ CS TP |
1162 | 161 CF⊆CS {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef (CS TP) k) (sym x=ψz) (CS∋Cl TP (* z)) |
1174 | 162 -- |
163 -- it is set of closed set and has FIP ( F is proper ) | |
164 -- | |
1162 | 165 ufl01 : {C x : Ordinal} → * C ⊆ * (& CF) → Subbase (* C) x → o∅ o< x |
166 ufl01 = ? | |
1174 | 167 -- |
168 -- so we have a limit | |
169 -- | |
1170 | 170 limit : Ordinal |
171 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 | |
172 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit | |
1169 | 173 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
1174 | 174 -- |
175 -- Neigbor of limit ⊆ Filter | |
176 -- | |
1171 | 177 ufl03 : {f v : Ordinal } → odef (filter F) f → Neighbor TP limit v → ¬ ( * f ∩ * v ) =h= od∅ -- because limit is in CF which is a closure |
1170 | 178 ufl03 {f} {v} ff nei fv=0 = ? |
179 pp : {v x : Ordinal} → Neighbor TP limit v → odef (* v) x → Power P ∋ (* x) | |
180 pp {v} {x} nei vx z pz = ? | |
181 ufl00 : {v : Ordinal} → Neighbor TP limit v → * v ⊆ filter F | |
182 ufl00 {v} nei {x} fx with ultra-filter.ultra UF (pp nei fx) (NEG P (pp nei fx)) | |
183 ... | case1 fv = subst (λ k → odef (filter F) k) &iso fv | |
1171 | 184 ... | case2 nfv = ? -- will contradicts ufl03 |
1163 | 185 |
1124 | 186 -- product topology of compact topology is compact |
431 | 187 |
1142 | 188 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
1158 | 189 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
1169 | 190 uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) |
191 → UFLP TP F UF | |
192 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF | |
193 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) | |
194 → UFLP TQ F UF | |
195 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF | |
1159 | 196 -- Product of UFL has limit point |
1169 | 197 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) |
198 → UFLP (ProductTopology TP TQ) F UF | |
199 uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where | |
200 FP : Filter {Power P} {P} (λ x → x) | |
1164 | 201 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
1169 | 202 ty00 : Proj1 (filter F) (Power P) (Power Q) ⊆ Power P |
203 ty00 {x} ⟪ PPx , ppf ⟫ = PPx | |
1161 | 204 UFP : ultra-filter FP |
1159 | 205 UFP = record { proper = ? ; ultra = ? } |
1169 | 206 uflp : UFLP TP FP UFP |
207 uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP | |
1154 | 208 |
1169 | 209 FQ : Filter {Power Q} {Q} (λ x → x) |
1166 | 210 FQ = record { filter = Proj2 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
1169 | 211 ty00 : Proj2 (filter F) (Power P) (Power Q) ⊆ Power Q |
212 ty00 {x} ⟪ QPx , ppf ⟫ = QPx | |
1166 | 213 UFQ : ultra-filter FQ |
214 UFQ = record { proper = ? ; ultra = ? } | |
1169 | 215 uflq : UFLP TQ FQ UFQ |
216 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ | |
1154 | 217 |
1166 | 218 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
219 Pf = ? | |
1171 | 220 pq⊆F : {p q : HOD} → Neighbor TP (& p) (UFLP.limit uflp) → Neighbor TP (& q) (UFLP.limit uflq) → ? ⊆ filter F |
1170 | 221 pq⊆F = ? |
222 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → * v ⊆ filter F | |
1173 | 223 isL {v} npq {x} fx = ? where |
1172 | 224 bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u npq) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
225 bpq = Neighbor.ou npq (Neighbor.ux npq) | |
226 pqb : Subbase (pbase TP TQ) (Base.b bpq ) | |
227 pqb = Base.sb bpq | |
1173 | 228 pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u npq ) |
229 pqb⊆opq = Base.b⊆u bpq | |
230 base⊆F : {b : Ordinal } → Subbase (pbase TP TQ) b → * b ⊆ * (Neighbor.u npq) → * b ⊆ filter F | |
231 base⊆F (gi (case1 px)) b⊆u {z} bz = fz where | |
232 -- F contains no od∅, because it projection contains no od∅ | |
233 -- x is an element of BaseP, which is a subset of Neighbor npq | |
234 -- x is also an elment of Proj1 F because Proj1 F has UFLP (uflp) | |
235 -- BaseP ∩ F is not empty | |
236 -- (Base P ∩ F) ⊆ F , (Base P ) ⊆ F , | |
237 il1 : odef (Power P) z ∧ ZProj1 (filter F) z | |
238 il1 = UFLP.is-limit uflp ? bz | |
239 nei1 : HOD | |
240 nei1 = Proj1 (* (Neighbor.u npq)) (Power P) (Power Q) | |
241 plimit : Ordinal | |
1174 | 242 plimit = UFLP.limit uflp |
1173 | 243 nproper : {b : Ordinal } → * b ⊆ nei1 → o∅ o< b |
244 nproper = ? | |
245 b∋z : odef nei1 z | |
246 b∋z = ? | |
247 bp : BaseP {P} TP Q z | |
248 bp = record { b = ? ; op = ? ; prod = ? } | |
249 neip : {p : Ordinal } → ( bp : BaseP TP Q p ) → * p ⊆ filter F | |
250 neip = ? | |
251 il2 : * z ⊆ ZFP (Power P) (Power Q) | |
252 il2 = ? | |
253 il3 : filter F ∋ ? | |
254 il3 = ? | |
255 fz : odef (filter F) z | |
256 fz = subst (λ k → odef (filter F) k) &iso (filter1 F ? ? ?) | |
257 base⊆F (gi (case2 qx)) b⊆u {z} bz = ? | |
258 base⊆F (g∩ b1 b2) b⊆u {z} bz = ? | |
1154 | 259 |
1170 | 260 |
261 | |
262 | |
263 | |
264 |