Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Tychonoff.agda @ 1204:4d894c166762
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| author | kono |
|---|---|
| date | Fri, 03 Mar 2023 08:37:02 +0800 |
| parents | 2f09ce1dd2a1 |
| children | 83ac320583f8 |
| rev | line source |
|---|---|
| 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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1113
384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
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changeset
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12 import OD |
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384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
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13 open import Relation.Nullary |
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384ba5a3c019
fix Topology definition
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
1112
diff
changeset
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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 |
| 1201 | 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 ) | |
| 1201 | 65 ... | tri≈ ¬a P=0 ¬c = record { limit = λ CX fip → o∅ ; is-limit = fip03 } where |
| 1163 | 66 -- P is empty |
| 67 fip02 : {x : Ordinal } → ¬ odef P x | |
| 1201 | 68 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym P=0) (∈∅< Px) ) |
| 69 fip03 : {X : Ordinal} (CX : * X ⊆ CS TP) (fip : {x : Ordinal} → Subbase (* X) x → o∅ o< x) {x : Ordinal} → | |
| 70 odef (* X) x → odef (* x) o∅ | |
| 71 -- empty P case | |
| 72 -- if 0 < X then 0 < x ∧ P ∋ xfrom fip | |
| 73 -- if 0 ≡ X then ¬ odef X x | |
| 74 fip03 {X} CX fip {x} xx with trio< o∅ X | |
| 75 ... | tri< 0<X ¬b ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) (trans (trans (sym *iso) (cong (*) P=0)) o∅≡od∅ ) (sym &iso) | |
| 76 ( cs⊆L TP (subst (λ k → odef (CS TP) k ) (sym &iso) (CX xx)) xe ))) where | |
| 77 0<x : o∅ o< x | |
| 78 0<x = fip (gi xx ) | |
| 79 e : HOD -- we have an element of x | |
| 80 e = ODC.minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 81 xe : odef (* x) (& e) | |
| 82 xe = ODC.x∋minimal O (* x) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<x) ) | |
| 83 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬∅∋ (subst₂ (λ j k → odef j k ) ( begin | |
| 84 * X ≡⟨ cong (*) (sym 0=X) ⟩ | |
| 85 * o∅ ≡⟨ o∅≡od∅ ⟩ | |
| 86 od∅ ∎ ) (sym &iso) xx ) ) where open ≡-Reasoning | |
| 87 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1204 | 88 ... | tri> ¬a ¬b 0<P = record { limit = λ CSX fip → limit CSX fip ; is-limit = uf01 } where |
| 1143 | 89 fip : {X : Ordinal} → * X ⊆ CS TP → Set n |
| 1187 | 90 fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x |
| 1154 | 91 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD |
| 1161 | 92 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) |
| 1159 | 93 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } |
| 1158 | 94 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P |
| 1159 | 95 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb |
| 1202 | 96 -- filter base is not empty |
| 1204 | 97 nc : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD |
| 98 nc {X} 0<X CSX fip = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) where | |
| 1201 | 99 e : HOD -- we have an element of X |
| 100 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 101 Xe : odef (* X) (& e) | |
| 102 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 1204 | 103 N∋nc :{X : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) |
| 104 → (fip : fip CSX) → odef (N CSX fip) (& (nc 0<X CSX fip) ) | |
| 105 N∋nc {X} 0<X CSX fip = record { b = ? ; x = ? ; b⊆X = ? ; sb = ? ; u⊆P = ? ; x⊆u = ? } where | |
| 1201 | 106 e : HOD -- we have an element of X |
| 107 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 108 Xe : odef (* X) (& e) | |
| 109 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 110 nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) | |
| 1165 | 111 0<PP : o∅ o< & (Power P) |
| 1201 | 112 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where |
| 113 nn00 : odef (Power P) o∅ | |
| 114 nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt | |
| 115 ... | x<0 = ⊥-elim ( ¬x<0 x<0) | |
| 1174 | 116 -- |
| 117 -- FIP defines a filter | |
| 118 -- | |
| 1158 | 119 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 120 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 121 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 1161 | 122 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
| 123 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 | 124 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
| 125 f10 : q ⊆ P | |
| 126 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 127 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 128 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 | |
| 129 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 130 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 | 131 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 132 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 133 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 134 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 135 f20 : * (& Np+Nq) ⊆ * X |
| 136 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 137 ... | case1 np = FBase.b⊆X Np np |
| 138 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 139 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 1161 | 140 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
| 1159 | 141 → ⟪ 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 | 142 -- |
| 143 -- it contains no empty sets | |
| 144 -- | |
| 1155 | 145 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ¬ (N CSX fp ∋ od∅) |
| 146 proper = ? | |
| 1174 | 147 -- |
| 148 -- then we have maximum ultra filter | |
| 149 -- | |
| 1158 | 150 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
| 1201 | 151 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 | 152 mf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) |
| 153 mf {X} CSX fp = MaximumFilter.mf (maxf CSX fp) | |
| 1204 | 154 ultraf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf CSX fp) |
| 155 ultraf {X} 0<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 | 156 -- |
| 157 -- so i has a limit as a limit of UIP | |
| 158 -- | |
| 1170 | 159 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1203 | 160 limit {X} CSX fp with trio< o∅ X |
| 1204 | 161 ... | tri< 0<X ¬b ¬c = UFLP.limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp)) |
| 162 ... | tri≈ ¬a 0=X ¬c = o∅ | |
| 1203 | 163 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1204 | 164 uf02 : {X v : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) |
| 1170 | 165 → Neighbor TP (limit CSX fp) v → * v ⊆ filter ( mf CSX fp ) |
| 1204 | 166 uf02 {X} {v} 0<X CSX fp nei {x} vx with trio< o∅ X |
| 167 ... | tri< 0<X ¬b ¬c = UFLP.is-limit ( uflp ( mf CSX fp ) (ultraf 0<X CSX fp)) nei vx | |
| 168 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬a 0<X ) | |
| 1203 | 169 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1174 | 170 -- |
| 1201 | 171 -- the limit is an limit of entire elements of X |
| 1174 | 172 -- |
| 1170 | 173 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) |
| 1204 | 174 uf01 {X} CSX fp {x} xx with trio< o∅ X |
| 175 ... | tri< 0<X ¬b ¬c = ? | |
| 176 ... | tri≈ ¬a 0=X ¬c = ? | |
| 177 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) | |
| 1142 | 178 |
| 1158 | 179 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 1169 | 180 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF |
| 1201 | 181 FIP→UFLP {P} TP fip F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ? ; P∋limit = ? ; is-limit = ufl00 } where |
| 1174 | 182 -- |
| 183 -- take closure of given filter elements | |
| 184 -- | |
| 1160 | 185 CF : HOD |
| 1188 | 186 CF = Replace (filter F) (λ x → Cl TP x ) |
| 1160 | 187 CF⊆CS : CF ⊆ CS TP |
| 1162 | 188 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 | 189 -- |
| 190 -- it is set of closed set and has FIP ( F is proper ) | |
| 191 -- | |
| 1187 | 192 ufl01 : {x : Ordinal} → Subbase (* (& CF)) x → o∅ o< x |
| 1162 | 193 ufl01 = ? |
| 1174 | 194 -- |
| 195 -- so we have a limit | |
| 196 -- | |
| 1170 | 197 limit : Ordinal |
| 1201 | 198 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ? -- ufl01 |
| 1170 | 199 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit |
| 1201 | 200 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ? -- ufl01 |
| 1174 | 201 -- |
| 202 -- Neigbor of limit ⊆ Filter | |
| 203 -- | |
| 1171 | 204 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 | 205 ufl03 {f} {v} ff nei fv=0 = ? |
| 206 pp : {v x : Ordinal} → Neighbor TP limit v → odef (* v) x → Power P ∋ (* x) | |
| 207 pp {v} {x} nei vx z pz = ? | |
| 208 ufl00 : {v : Ordinal} → Neighbor TP limit v → * v ⊆ filter F | |
| 209 ufl00 {v} nei {x} fx with ultra-filter.ultra UF (pp nei fx) (NEG P (pp nei fx)) | |
| 210 ... | case1 fv = subst (λ k → odef (filter F) k) &iso fv | |
| 1171 | 211 ... | case2 nfv = ? -- will contradicts ufl03 |
| 1163 | 212 |
| 1124 | 213 -- product topology of compact topology is compact |
| 431 | 214 |
| 1142 | 215 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 216 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1169 | 217 uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) |
| 218 → UFLP TP F UF | |
| 219 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF | |
| 220 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) | |
| 221 → UFLP TQ F UF | |
| 222 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF | |
| 1201 | 223 -- Product of UFL has a limit point |
| 1169 | 224 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) |
| 225 → UFLP (ProductTopology TP TQ) F UF | |
| 226 uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where | |
| 227 FP : Filter {Power P} {P} (λ x → x) | |
| 1164 | 228 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
| 1169 | 229 ty00 : Proj1 (filter F) (Power P) (Power Q) ⊆ Power P |
| 230 ty00 {x} ⟪ PPx , ppf ⟫ = PPx | |
| 1161 | 231 UFP : ultra-filter FP |
| 1159 | 232 UFP = record { proper = ? ; ultra = ? } |
| 1169 | 233 uflp : UFLP TP FP UFP |
| 234 uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP | |
| 1154 | 235 |
| 1169 | 236 FQ : Filter {Power Q} {Q} (λ x → x) |
| 1166 | 237 FQ = record { filter = Proj2 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where |
| 1169 | 238 ty00 : Proj2 (filter F) (Power P) (Power Q) ⊆ Power Q |
| 239 ty00 {x} ⟪ QPx , ppf ⟫ = QPx | |
| 1166 | 240 UFQ : ultra-filter FQ |
| 241 UFQ = record { proper = ? ; ultra = ? } | |
| 1169 | 242 uflq : UFLP TQ FQ UFQ |
| 243 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ | |
| 1154 | 244 |
| 1166 | 245 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 246 Pf = ? | |
| 1171 | 247 pq⊆F : {p q : HOD} → Neighbor TP (& p) (UFLP.limit uflp) → Neighbor TP (& q) (UFLP.limit uflq) → ? ⊆ filter F |
| 1170 | 248 pq⊆F = ? |
| 249 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → * v ⊆ filter F | |
| 1173 | 250 isL {v} npq {x} fx = ? where |
| 1172 | 251 bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u npq) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 252 bpq = Neighbor.ou npq (Neighbor.ux npq) | |
| 253 pqb : Subbase (pbase TP TQ) (Base.b bpq ) | |
| 254 pqb = Base.sb bpq | |
| 1173 | 255 pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u npq ) |
| 256 pqb⊆opq = Base.b⊆u bpq | |
| 257 base⊆F : {b : Ordinal } → Subbase (pbase TP TQ) b → * b ⊆ * (Neighbor.u npq) → * b ⊆ filter F | |
| 258 base⊆F (gi (case1 px)) b⊆u {z} bz = fz where | |
| 259 -- F contains no od∅, because it projection contains no od∅ | |
| 260 -- x is an element of BaseP, which is a subset of Neighbor npq | |
| 261 -- x is also an elment of Proj1 F because Proj1 F has UFLP (uflp) | |
| 262 -- BaseP ∩ F is not empty | |
| 263 -- (Base P ∩ F) ⊆ F , (Base P ) ⊆ F , | |
| 264 il1 : odef (Power P) z ∧ ZProj1 (filter F) z | |
| 265 il1 = UFLP.is-limit uflp ? bz | |
| 266 nei1 : HOD | |
| 267 nei1 = Proj1 (* (Neighbor.u npq)) (Power P) (Power Q) | |
| 268 plimit : Ordinal | |
| 1174 | 269 plimit = UFLP.limit uflp |
| 1173 | 270 nproper : {b : Ordinal } → * b ⊆ nei1 → o∅ o< b |
| 271 nproper = ? | |
| 272 b∋z : odef nei1 z | |
| 273 b∋z = ? | |
| 274 bp : BaseP {P} TP Q z | |
| 1187 | 275 bp = record { p = ? ; op = ? ; prod = ? } |
| 1173 | 276 neip : {p : Ordinal } → ( bp : BaseP TP Q p ) → * p ⊆ filter F |
| 277 neip = ? | |
| 278 il2 : * z ⊆ ZFP (Power P) (Power Q) | |
| 279 il2 = ? | |
| 280 il3 : filter F ∋ ? | |
| 281 il3 = ? | |
| 282 fz : odef (filter F) z | |
| 283 fz = subst (λ k → odef (filter F) k) &iso (filter1 F ? ? ?) | |
| 284 base⊆F (gi (case2 qx)) b⊆u {z} bz = ? | |
| 285 base⊆F (g∩ b1 b2) b⊆u {z} bz = ? | |
| 1154 | 286 |
| 1170 | 287 |
| 288 | |
| 289 | |
| 290 | |
| 291 |