Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/Tychonoff.agda @ 1215:881e85e12e38
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 05 Mar 2023 14:46:53 +0900 |
| parents | 93e1869bb57c |
| children | 6861b48c1e08 |
| 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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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 |
| 1213 | 38 -- open import maximum-filter O |
| 431 | 39 |
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40 open Filter |
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41 open Topology |
| 1169 | 42 |
| 431 | 43 -- FIP is UFL |
| 44 | |
| 1159 | 45 -- filter Base |
| 1205 | 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 | |
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59 is-limit : {v : Ordinal} → Neighbor TP limit v → filter F ∋ (* v) |
| 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 |
| 1205 | 96 -- filter base is not empty necessary for generating ultra filter |
| 1204 | 97 nc : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD |
| 1205 | 98 nc {X} 0<X CSX fip = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) -- an element of X |
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99 N∋nc :{X : Ordinal} → (0<X : o∅ o< X) → (CSX : * X ⊆ CS TP) |
| 1204 | 100 → (fip : fip CSX) → odef (N CSX fip) (& (nc 0<X CSX fip) ) |
| 1205 | 101 N∋nc {X} 0<X CSX fip = record { b = X ; x = & e ; b⊆X = λ x → x ; sb = gi Xe ; u⊆P = nn02 ; x⊆u = λ x → x } where |
| 1201 | 102 e : HOD -- we have an element of X |
| 103 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
| 104 Xe : odef (* X) (& e) | |
| 105 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) ) | |
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106 nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) |
| 1205 | 107 nn02 : * (& (nc 0<X CSX fip)) ⊆ P |
| 108 nn02 = subst (λ k → k ⊆ P ) (sym *iso) (cs⊆L TP (CSX Xe ) ) | |
| 109 | |
| 1165 | 110 0<PP : o∅ o< & (Power P) |
| 1201 | 111 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where |
| 112 nn00 : odef (Power P) o∅ | |
| 113 nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt | |
| 114 ... | x<0 = ⊥-elim ( ¬x<0 x<0) | |
| 1174 | 115 -- |
| 116 -- FIP defines a filter | |
| 117 -- | |
| 1158 | 118 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) |
| 119 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where | |
| 120 f1 : {p q : HOD} → Power P ∋ q → N CSX fp ∋ p → p ⊆ q → N CSX fp ∋ q | |
| 1161 | 121 f1 {p} {q} Xq record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = Xp ; x⊆u = x⊆p } p⊆q = |
| 122 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 | 123 subst (λ k → odef k z) (sym *iso) (p⊆q (subst (λ k → odef k z) *iso (x⊆p xp))) } where |
| 124 f10 : q ⊆ P | |
| 125 f10 {x} qx = subst (λ k → odef P k) &iso (power→ P _ Xq (subst (λ k → odef q k) (sym &iso) qx )) | |
| 126 f2 : {p q : HOD} → N CSX fp ∋ p → N CSX fp ∋ q → Power P ∋ (p ∩ q) → N CSX fp ∋ (p ∩ q) | |
| 127 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 | |
| 128 p∩q⊆p : * (& (p ∩ q)) ⊆ P | |
| 129 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 | 130 Np+Nq = * (FBase.b Np) ∪ * (FBase.b Nq) |
| 131 xp∧xq = * (FBase.x Np) ∩ * (FBase.x Nq) | |
| 1155 | 132 sbpq : Subbase (* (& Np+Nq)) (& xp∧xq) |
| 1159 | 133 sbpq = subst₂ (λ j k → Subbase j k ) (sym *iso) refl ( g∩ (sb⊆ case1 (FBase.sb Np)) (sb⊆ case2 (FBase.sb Nq))) |
| 1155 | 134 f20 : * (& Np+Nq) ⊆ * X |
| 135 f20 {x} npq with subst (λ k → odef k x) *iso npq | |
| 1159 | 136 ... | case1 np = FBase.b⊆X Np np |
| 137 ... | case2 nq = FBase.b⊆X Nq nq | |
| 1155 | 138 f22 : * (& xp∧xq) ⊆ * (& (p ∩ q)) |
| 1161 | 139 f22 = subst₂ ( λ j k → j ⊆ k ) (sym *iso) (sym *iso) (λ {w} xpq |
| 1159 | 140 → ⟪ 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 | 141 -- |
| 1207 | 142 -- it contains no empty sets becase it is a finite intersection ( Subbase (* X) x ) |
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143 -- |
| 1207 | 144 proper : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip {X} CSX) → ¬ (N CSX fip ∋ od∅) |
| 145 proper {X} CSX fip record { b = b ; x = x ; b⊆X = b⊆X ; sb = sb ; u⊆P = u⊆P ; x⊆u = x⊆u } = o≤> x≤0 (fip (fp00 _ _ b⊆X sb)) where | |
| 146 x≤0 : x o< osuc o∅ | |
| 147 x≤0 = subst₂ (λ j k → j o< osuc k) &iso (trans (cong (&) *iso) ord-od∅ ) (⊆→o≤ (x⊆u)) | |
| 148 fp00 : (b x : Ordinal) → * b ⊆ * X → Subbase (* b) x → Subbase (* X) x | |
| 149 fp00 b y b<X (gi by ) = gi ( b<X by ) | |
| 150 fp00 b _ b<X (g∩ {y} {z} sy sz ) = g∩ (fp00 _ _ b<X sy) (fp00 _ _ b<X sz) | |
| 1174 | 151 -- |
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152 -- then we have maximum ultra filter |
| 1174 | 153 -- |
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154 maxf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) |
| 1213 | 155 maxf {X} 0<X CSX fp = {!!} -- F→Maximum {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc 0<X CSX fp) (proper CSX fp) |
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156 mf : {X : Ordinal} → o∅ o< X → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) |
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157 mf {X} 0<X CSX fp = MaximumFilter.mf (maxf 0<X CSX fp) |
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158 ultraf : {X : Ordinal} → (0<X : o∅ o< X ) → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf 0<X CSX fp) |
| 1213 | 159 ultraf {X} 0<X CSX fp = {!!} -- F→ultra {Power P} {P} (λ x → x) (CAP P) (F CSX fp) 0<PP (N∋nc 0<X CSX fp) (proper CSX fp) |
| 1174 | 160 -- |
| 161 -- so i has a limit as a limit of UIP | |
| 162 -- | |
| 1170 | 163 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal |
| 1203 | 164 limit {X} CSX fp with trio< o∅ X |
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165 ... | tri< 0<X ¬b ¬c = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) |
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166 ... | tri≈ ¬a 0=X ¬c = o∅ |
| 1203 | 167 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1174 | 168 -- |
| 1201 | 169 -- the limit is an limit of entire elements of X |
| 1174 | 170 -- |
| 1170 | 171 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) |
| 1204 | 172 uf01 {X} CSX fp {x} xx with trio< o∅ X |
| 1206 | 173 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c ) |
| 1205 | 174 ... | tri≈ ¬a 0=X ¬c = ⊥-elim ( ¬a (subst (λ k → o∅ o< k) &iso ( ∈∅< xx ))) |
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175 ... | tri< 0<X ¬b ¬c with ∨L＼X {P} {* x} {UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))} |
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176 (UFLP.P∋limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp))) |
| 1206 | 177 ... | case1 lt = lt -- odef (* x) y |
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178 ... | case2 nlxy = ⊥-elim (MaximumFilter.proper (maxf 0<X CSX fp) uf11 ) where |
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179 y = UFLP.limit ( uflp ( mf 0<X CSX fp ) (ultraf 0<X CSX fp)) |
| 1207 | 180 x⊆P : * x ⊆ P |
| 181 x⊆P = cs⊆L TP (CSX (subst (λ k → odef (* X) k) (sym &iso) xx)) | |
| 1206 | 182 uf10 : odef (P ＼ * x ) y |
| 183 uf10 = nlxy | |
| 184 uf03 : Neighbor TP y (& (P ＼ * x )) | |
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185 uf03 = record { u = _ ; ou = P＼CS=OS TP (CSX (subst (λ k → odef (* X) k ) (sym &iso) xx)) |
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186 ; ux = subst (λ k → odef k y) (sym *iso) uf10 |
| 1206 | 187 ; v⊆P = λ {z} xz → proj1 (subst(λ k → odef k z) *iso xz ) |
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188 ; u⊆v = λ x → x } |
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189 uf07 : * (& (* x , * x)) ⊆ * X |
| 1207 | 190 uf07 {y} lt with subst (λ k → odef k y) *iso lt |
| 191 ... | case1 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
| 192 ... | case2 refl = subst (λ k → odef (* X) k ) (sym &iso) xx | |
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193 uf05 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) x |
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194 uf05 = MaximumFilter.F⊆mf (maxf 0<X CSX fp) record { b = & (* x , * x) ; b⊆X = uf07 |
| 1207 | 195 ; sb = gi (subst (λ k → odef k x) (sym *iso) (case1 (sym &iso)) ) ; u⊆P = x⊆P ; x⊆u = λ x → x } |
| 1213 | 196 uf061 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (* (& (P ＼ * x )))) |
| 197 uf061 = UFLP.is-limit ( uflp (mf 0<X CSX fp) (ultraf 0<X CSX fp)) {& (P ＼ * x)} uf03 | |
| 198 -- uf06 (same as uf061) have yellow if zorn lemma is not imported | |
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199 uf06 : odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) (& (P ＼ * x )) |
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200 uf06 = subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k) &iso (UFLP.is-limit ( uflp (mf 0<X CSX fp) (ultraf 0<X CSX fp)) {& (P ＼ * x)} uf03 ) |
| 1206 | 201 uf13 : & ((* x) ∩ (P ＼ * x )) ≡ o∅ |
| 1207 | 202 uf13 = subst₂ (λ j k → j ≡ k ) refl ord-od∅ (cong (&) ( ==→o≡ record { eq→ = uf14 ; eq← = λ {x} lt → ⊥-elim (¬x<0 lt) } ) ) where |
| 203 uf14 : {y : Ordinal} → odef (* x ∩ (P ＼ * x)) y → odef od∅ y | |
| 204 uf14 {y} ⟪ xy , ⟪ Px , ¬xy ⟫ ⟫ = ⊥-elim ( ¬xy xy ) | |
| 1206 | 205 uf12 : odef (Power P) (& ((* x) ∩ (P ＼ * x ))) |
| 1207 | 206 uf12 z pz with subst (λ k → odef k z) *iso pz |
| 207 ... | ⟪ xz , ⟪ Pz , ¬xz ⟫ ⟫ = Pz | |
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208 uf11 : filter (MaximumFilter.mf (maxf 0<X CSX fp)) ∋ od∅ |
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209 uf11 = subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k ) (trans uf13 (sym ord-od∅)) |
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210 ( filter2 (MaximumFilter.mf (maxf 0<X CSX fp)) (subst (λ k → odef (filter (MaximumFilter.mf (maxf 0<X CSX fp))) k) (sym &iso) uf05) uf06 uf12 ) |
| 1142 | 211 |
| 1208 | 212 x⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → x ⊆ Cl TP x |
| 213 x⊆Clx {P} TP {x} x<p {y} xy = ⟪ x<p xy , (λ c csc x<c → x<c xy ) ⟫ | |
| 214 P⊆Clx : {P : HOD} (TP : Topology P) → {x : HOD} → x ⊆ P → Cl TP x ⊆ P | |
| 215 P⊆Clx {P} TP {x} x<p {y} xy = proj1 xy | |
| 216 | |
| 1158 | 217 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP |
| 1169 | 218 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF |
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219 FIP→UFLP {P} TP fip F UF = record { limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1209 | 220 ; P∋limit = ufl10 ; is-limit = ufl00 } where |
| 221 F∋P : odef (filter F) (& P) | |
| 222 F∋P with ultra-filter.ultra UF {od∅} (λ z az → ⊥-elim (¬x<0 (subst (λ k → odef k z) *iso az)) ) (λ z az → proj1 (subst (λ k → odef k z) *iso az ) ) | |
| 223 ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) | |
| 224 ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where | |
| 225 fl20 : (P ＼ Ord o∅) =h= P | |
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226 fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } |
| 1209 | 227 0<P : o∅ o< & P |
| 228 0<P with trio< o∅ (& P) | |
| 229 ... | tri< a ¬b ¬c = a | |
| 230 ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋P) ) | |
| 231 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1174 | 232 -- |
| 233 -- take closure of given filter elements | |
| 234 -- | |
| 1160 | 235 CF : HOD |
| 1188 | 236 CF = Replace (filter F) (λ x → Cl TP x ) |
| 1160 | 237 CF⊆CS : CF ⊆ CS TP |
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238 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 | 239 -- |
| 240 -- it is set of closed set and has FIP ( F is proper ) | |
| 241 -- | |
| 1208 | 242 ufl08 : {z : Ordinal} → odef (Power P) (& (Cl TP (* z))) |
| 243 ufl08 {z} w zw with subst (λ k → odef k w ) *iso zw | |
| 244 ... | t = proj1 t | |
| 245 fx→px : {x : Ordinal} → odef (filter F) x → Power P ∋ * x | |
| 246 fx→px {x} fx z xz = f⊆L F fx _ (subst (λ k → odef k z) *iso xz ) | |
| 247 F∋sb : {x : Ordinal} → Subbase CF x → odef (filter F) x | |
| 248 F∋sb {x} (gi record { z = z ; az = az ; x=ψz = x=ψz }) = ufl07 where | |
| 249 ufl09 : * z ⊆ P | |
| 250 ufl09 {y} zy = f⊆L F az _ zy | |
| 251 ufl07 : odef (filter F) x | |
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252 ufl07 = subst (λ k → odef (filter F) k) &iso ( filter1 F (subst (λ k → odef (Power P) k) (trans (sym x=ψz) (sym &iso)) ufl08 ) |
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253 (subst (λ k → odef (filter F) k) (sym &iso) az) |
| 1208 | 254 (subst (λ k → * z ⊆ k ) (trans (sym *iso) (sym (cong (*) x=ψz)) ) (x⊆Clx TP {* z} ufl09 ) )) |
| 255 F∋sb (g∩ {x} {y} sx sy) = filter2 F (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sx)) | |
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256 (subst (λ k → odef (filter F) k) (sym &iso) (F∋sb sy)) |
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257 (λ z xz → fx→px (F∋sb sx) _ (subst (λ k → odef k _) (sym *iso) (proj1 (subst (λ k → odef k z) *iso xz) ))) |
| 1187 | 258 ufl01 : {x : Ordinal} → Subbase (* (& CF)) x → o∅ o< x |
| 1208 | 259 ufl01 {x} sb = ufl04 where |
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260 ufl04 : o∅ o< x |
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261 ufl04 with trio< o∅ x |
| 1208 | 262 ... | tri< a ¬b ¬c = a |
| 263 ... | tri≈ ¬a b ¬c = ⊥-elim ( ultra-filter.proper UF (subst (λ k → odef (filter F) k) ( | |
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264 begin |
| 1208 | 265 x ≡⟨ sym b ⟩ |
| 266 o∅ ≡⟨ sym ord-od∅ ⟩ | |
| 267 & od∅ ∎ ) (F∋sb (subst (λ k → Subbase k x) *iso sb )) )) where open ≡-Reasoning | |
| 268 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 1209 | 269 ufl10 : odef P (FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01) |
| 270 ufl10 = FIP.L∋limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 {& P} ufl11 where | |
| 271 ufl11 : odef (* (& CF)) (& P) | |
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272 ufl11 = subst (λ k → odef k (& P)) (sym *iso) record { z = _ ; az = F∋P ; x=ψz = sym (cong (&) (trans (cong (Cl TP) *iso) (ClL TP))) } |
| 1174 | 273 -- |
| 274 -- so we have a limit | |
| 275 -- | |
| 1170 | 276 limit : Ordinal |
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277 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1170 | 278 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit |
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279 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 |
| 1174 | 280 -- |
| 281 -- Neigbor of limit ⊆ Filter | |
| 282 -- | |
| 1210 | 283 -- if odef (* X) x, Cl TP x contains limit (ufl02) |
| 284 -- in (nei : Neighbor TP limit v) , there is an open Set u which contains the limit | |
| 285 -- F contains u or P ＼ u because F is ultra | |
| 286 -- if F ∋ u, then F ∋ v from u ⊆ v which is a propetery of Neighbor | |
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287 -- if F ∋ P ＼ u, it is a closed set (Cl (P ＼ u) ≡ P ＼ u) and it does not contains the limit |
| 1210 | 288 -- this contradicts ufl02 |
| 289 pp : {v : Ordinal} → (nei : Neighbor TP limit v ) → Power P ∋ (* (Neighbor.u nei)) | |
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290 pp {v} record { u = u ; ou = ou ; ux = ux ; v⊆P = v⊆P ; u⊆v = u⊆v } z pz |
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291 = os⊆L TP (subst (λ k → odef (OS TP) k) (sym &iso) ou ) (subst (λ k → odef k z) *iso pz ) |
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292 ufl00 : {v : Ordinal} → Neighbor TP limit v → filter F ∋ * v |
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293 ufl00 {v} nei with ultra-filter.ultra UF (pp nei ) (NEG P (pp nei )) |
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294 ... | case1 fu = subst (λ k → odef (filter F) k) &iso |
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295 ( filter1 F (subst (λ k → odef (Power P) k ) (sym &iso) px) fu (subst (λ k → u ⊆ k ) (sym *iso) (Neighbor.u⊆v nei))) where |
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296 u = * (Neighbor.u nei) |
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297 px : Power P ∋ * v |
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298 px z vz = Neighbor.v⊆P nei (subst (λ k → odef k z) *iso vz ) |
| 1210 | 299 ... | case2 nfu = ⊥-elim ( ¬P\u∋limit P\u∋limit ) where |
| 300 u = * (Neighbor.u nei) | |
| 301 P\u : HOD | |
| 302 P\u = P ＼ u | |
| 303 P\u∋limit : odef P\u limit | |
| 304 P\u∋limit = subst (λ k → odef k limit) *iso ( ufl02 (subst (λ k → odef k (& P\u)) (sym *iso) ufl03 )) where | |
| 305 ufl04 : & P\u ≡ & (Cl TP (* (& P\u))) | |
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306 ufl04 = cong (&) (sym (trans (cong (Cl TP) *iso) |
| 1210 | 307 (CS∋x→Clx=x TP (P＼OS=CS TP (subst (λ k → odef (OS TP) k) (sym &iso) (Neighbor.ou nei) ))))) |
| 308 ufl03 : odef CF (& P\u ) | |
| 309 ufl03 = record { z = _ ; az = nfu ; x=ψz = ufl04 } | |
| 310 ¬P\u∋limit : ¬ odef P\u limit | |
| 311 ¬P\u∋limit ⟪ Pl , nul ⟫ = nul ( Neighbor.ux nei ) | |
| 1163 | 312 |
| 1124 | 313 -- product topology of compact topology is compact |
| 431 | 314 |
| 1142 | 315 Tychonoff : {P Q : HOD } → (TP : Topology P) → (TQ : Topology Q) → Compact TP → Compact TQ → Compact (ProductTopology TP TQ) |
| 1158 | 316 Tychonoff {P} {Q} TP TQ CP CQ = FIP→Compact (ProductTopology TP TQ) (UFLP→FIP (ProductTopology TP TQ) uflPQ ) where |
| 1169 | 317 uflP : (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F) |
| 318 → UFLP TP F UF | |
| 319 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF | |
| 320 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F) | |
| 321 → UFLP TQ F UF | |
| 322 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF | |
| 1201 | 323 -- Product of UFL has a limit point |
| 1169 | 324 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) |
| 325 → UFLP (ProductTopology TP TQ) F UF | |
| 326 uflPQ F UF = record { limit = & < * ( UFLP.limit uflp ) , * ( UFLP.limit uflq ) > ; P∋limit = Pf ; is-limit = isL } where | |
| 1213 | 327 F∋PQ : odef (filter F) (& (ZFP P Q)) |
| 328 F∋PQ with ultra-filter.ultra UF {od∅} (λ z az → ⊥-elim (¬x<0 (subst (λ k → odef k z) *iso az)) ) (λ z az → proj1 (subst (λ k → odef k z) *iso az ) ) | |
| 329 ... | case1 fp = ⊥-elim ( ultra-filter.proper UF fp ) | |
| 330 ... | case2 flp = subst (λ k → odef (filter F) k) (cong (&) (==→o≡ fl20)) flp where | |
| 331 fl20 : (ZFP P Q ＼ Ord o∅) =h= ZFP P Q | |
| 332 fl20 = record { eq→ = λ {x} lt → proj1 lt ; eq← = λ {x} lt → ⟪ lt , (λ lt → ⊥-elim (¬x<0 lt) ) ⟫ } | |
| 333 0<P : o∅ o< & (ZFP P Q) | |
| 334 0<P with trio< o∅ (& (ZFP P Q)) | |
| 335 ... | tri< a ¬b ¬c = a | |
| 336 ... | tri≈ ¬a b ¬c = ⊥-elim (ultra-filter.proper UF (subst (λ k → odef (filter F) k) (trans (sym b) (sym ord-od∅)) F∋PQ) ) | |
| 337 ... | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) | |
| 338 isP→PxQ : {x : HOD} → (x⊆P : x ⊆ P ) → ZFP x Q ⊆ ZFP P Q | |
| 339 isP→PxQ {x} x⊆P (ab-pair p q) = ab-pair (x⊆P p) q | |
| 340 F⊆pxq : {x : HOD } → filter F ∋ x → x ⊆ ZFP P Q | |
| 341 F⊆pxq {x} fx {y} xy = f⊆L F fx y (subst (λ k → odef k y) (sym *iso) xy) | |
| 342 FPSet : HOD | |
| 343 FPSet = Replace' (filter F) (λ x fx → Replace' x ( λ y xy → * (zπ1 (F⊆pxq fx xy) ))) | |
| 1215 | 344 FPSet∋zpq : {q : HOD} → q ⊆ P → filter F ∋ ZFP q Q → FPSet ∋ q |
| 345 FPSet∋zpq {q} q⊆P fq = record { z = _ ; az = ? ; x=ψz = ? } where | |
| 346 ty04 : q ≡ Replace' (* (& (ZFP q Q))) (λ y xy → * (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq ) xy))) | |
| 347 ty04 = ==→o≡ record { eq→ = ? ; eq← = ? } where | |
| 348 ty041 : {x : Ordinal} → odef q x → odef (Replace' (* (& (ZFP q Q))) | |
| 349 (λ y xy → * (zπ1 (F⊆pxq (subst (odef (filter F)) (sym &iso) fq) xy)))) x | |
| 350 ty041 {x} qx = record { z = _ ; az = subst (λ k → odef k (& < * x , * ? > )) (sym *iso) (ab-pair qx ? ) ; x=ψz = ? } | |
| 1213 | 351 FPSet⊆PP : FPSet ⊆ Power P |
| 352 FPSet⊆PP {x} record { z = z ; az = fz ; x=ψz = x=ψz } w xw with subst (λ k → odef k w) (trans (cong (*) x=ψz) *iso) xw | |
| 353 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } | |
| 354 = subst (λ k → odef P k) (sym (trans x=ψz1 &iso)) | |
| 355 (zp1 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fz) (subst (λ k → odef (* z) k) (sym &iso) az1)) ) | |
| 1169 | 356 FP : Filter {Power P} {P} (λ x → x) |
| 1213 | 357 FP = record { filter = FPSet ; f⊆L = FPSet⊆PP ; filter1 = ty01 ; filter2 = {!!} } where |
| 358 ty01 : {p q : HOD} → Power P ∋ q → FPSet ∋ p → p ⊆ q → FPSet ∋ q | |
| 1215 | 359 ty01 {p} {q} Pq record { z = x ; az = fx ; x=ψz = x=ψz } p⊆q = FPSet∋zpq q⊆P (ty10 ty05 ty06 ) |
| 360 where | |
| 1213 | 361 -- p ≡ (Replace' (* x) (λ y xy → (zπ1 (F⊆pxq (subst (odef (filter.Filter.filter F)) (sym &iso) fx) xy)) |
| 362 -- x = ( px , qx ) , px ⊆ q | |
| 363 ty03 : Power (ZFP P Q) ∋ ZFP q Q | |
| 364 ty03 z zpq = isP→PxQ {* (& q)} (Pq _) ( subst (λ k → odef k z ) (trans *iso (cong (λ k → ZFP k Q) (sym *iso))) zpq ) | |
| 365 p⊆P : p ⊆ P | |
| 366 p⊆P {w} pw with subst (λ k → odef k w) (subst₂ (λ j k → j ≡ k ) *iso *iso (cong (*) x=ψz)) pw | |
| 367 ... | record { z = z1 ; az = az1 ; x=ψz = x=ψz1 } = subst (λ k → odef P k) (sym (trans x=ψz1 &iso)) | |
| 368 (zp1 (F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) (subst (λ k → odef (* x) k) (sym &iso) az1 )) ) | |
| 1215 | 369 q⊆P : q ⊆ P |
| 370 q⊆P {w} qw = Pq _ (subst (λ k → odef k w ) (sym *iso) qw ) | |
| 1213 | 371 x⊆pxq : * x ⊆ ZFP p Q |
| 1214 | 372 x⊆pxq {w} xw with F⊆pxq (subst (λ k → odef (filter F) k) (sym &iso) fx) xw |
| 373 ... | ab-pair {a} {b} pw qw = ab-pair ty08 qw where | |
| 374 ty07 : odef (* x) (& < * a , * b >) | |
| 375 ty07 = xw | |
| 376 ty08 : odef p a | |
| 377 ty08 = subst (λ k → odef k a ) (subst₂ (λ j k → j ≡ k) *iso *iso (sym (cong (*) x=ψz))) | |
| 378 record { z = _ ; az = xw ; x=ψz = ? } | |
| 1213 | 379 ty05 : filter F ∋ ZFP p Q |
| 380 ty05 = filter1 F (λ z wz → isP→PxQ p⊆P (subst (λ k → odef k z) *iso wz)) (subst (λ k → odef (filter F) k) (sym &iso) fx) x⊆pxq | |
| 381 ty06 : ZFP p Q ⊆ ZFP q Q | |
| 382 ty06 (ab-pair wp wq ) = ab-pair (p⊆q wp) wq | |
| 1215 | 383 ty10 : filter F ∋ ZFP p Q → ZFP p Q ⊆ ZFP q Q → filter F ∋ ZFP q Q |
| 384 ty10 fzp zp⊆zq = filter1 F ty03 fzp zp⊆zq | |
| 1213 | 385 |
| 1161 | 386 UFP : ultra-filter FP |
| 1213 | 387 UFP = record { proper = {!!} ; ultra = {!!} } |
| 1169 | 388 uflp : UFLP TP FP UFP |
| 389 uflp = FIP→UFLP TP (Compact→FIP TP CP) FP UFP | |
| 1154 | 390 |
| 1169 | 391 FQ : Filter {Power Q} {Q} (λ x → x) |
| 1213 | 392 FQ = record { filter = Proj2 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = {!!} ; filter2 = {!!} } where |
| 1169 | 393 ty00 : Proj2 (filter F) (Power P) (Power Q) ⊆ Power Q |
| 394 ty00 {x} ⟪ QPx , ppf ⟫ = QPx | |
| 1166 | 395 UFQ : ultra-filter FQ |
| 1213 | 396 UFQ = record { proper = {!!} ; ultra = {!!} } |
| 1169 | 397 uflq : UFLP TQ FQ UFQ |
| 398 uflq = FIP→UFLP TQ (Compact→FIP TQ CQ) FQ UFQ | |
| 1154 | 399 |
| 1166 | 400 Pf : odef (ZFP P Q) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 1213 | 401 Pf = {!!} |
| 402 pq⊆F : {p q : HOD} → Neighbor TP (& p) (UFLP.limit uflp) → Neighbor TP (& q) (UFLP.limit uflq) → {!!} ⊆ filter F | |
| 403 pq⊆F = {!!} | |
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404 isL : {v : Ordinal} → Neighbor (ProductTopology TP TQ) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) v → filter F ∋ * v |
| 1213 | 405 isL {v} npq = {!!} where |
| 1172 | 406 bpq : Base (ZFP P Q) (pbase TP TQ) (Neighbor.u npq) (& < * (UFLP.limit uflp) , * (UFLP.limit uflq) >) |
| 407 bpq = Neighbor.ou npq (Neighbor.ux npq) | |
| 408 pqb : Subbase (pbase TP TQ) (Base.b bpq ) | |
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409 pqb = Base.sb bpq |
| 1173 | 410 pqb⊆opq : * (Base.b bpq) ⊆ * ( Neighbor.u npq ) |
| 411 pqb⊆opq = Base.b⊆u bpq | |
| 412 base⊆F : {b : Ordinal } → Subbase (pbase TP TQ) b → * b ⊆ * (Neighbor.u npq) → * b ⊆ filter F | |
| 413 base⊆F (gi (case1 px)) b⊆u {z} bz = fz where | |
| 414 -- F contains no od∅, because it projection contains no od∅ | |
| 415 -- x is an element of BaseP, which is a subset of Neighbor npq | |
| 416 -- x is also an elment of Proj1 F because Proj1 F has UFLP (uflp) | |
| 417 -- BaseP ∩ F is not empty | |
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418 -- (Base P ∩ F) ⊆ F , (Base P ) ⊆ F , |
| 1173 | 419 il1 : odef (Power P) z ∧ ZProj1 (filter F) z |
| 1213 | 420 il1 = {!!} -- UFLP.is-limit uflp ? bz |
| 1173 | 421 nei1 : HOD |
| 422 nei1 = Proj1 (* (Neighbor.u npq)) (Power P) (Power Q) | |
| 423 plimit : Ordinal | |
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424 plimit = UFLP.limit uflp |
| 1173 | 425 nproper : {b : Ordinal } → * b ⊆ nei1 → o∅ o< b |
| 1213 | 426 nproper = {!!} |
| 1173 | 427 b∋z : odef nei1 z |
| 1213 | 428 b∋z = {!!} |
| 1173 | 429 bp : BaseP {P} TP Q z |
| 1213 | 430 bp = record { p = {!!} ; op = {!!} ; prod = {!!} } |
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431 neip : {p : Ordinal } → ( bp : BaseP TP Q p ) → * p ⊆ filter F |
| 1213 | 432 neip = {!!} |
| 1173 | 433 il2 : * z ⊆ ZFP (Power P) (Power Q) |
| 1213 | 434 il2 = {!!} |
| 435 il3 : filter F ∋ {!!} | |
| 436 il3 = {!!} | |
| 1173 | 437 fz : odef (filter F) z |
| 1213 | 438 fz = subst (λ k → odef (filter F) k) &iso (filter1 F {!!} {!!} {!!}) |
| 439 base⊆F (gi (case2 qx)) b⊆u {z} bz = {!!} | |
| 440 base⊆F (g∩ b1 b2) b⊆u {z} bz = {!!} | |
| 1154 | 441 |
| 1170 | 442 |
| 443 | |
| 444 | |
| 445 | |
| 446 |