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comparison src/Tychonoff.agda @ 1201:03684784bc5f

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author kono
date Thu, 02 Mar 2023 11:09:02 +0800
parents 42000f20fdbe
children d6781ad8149e
comparison
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1200:42000f20fdbe 1201:03684784bc5f
33 open ODC O 33 open ODC O
34 34
35 open import filter O 35 open import filter O
36 open import OPair O 36 open import OPair O
37 open import Topology O 37 open import Topology O
38 open import maximum-filter O 38 -- open import maximum-filter O
39 39
40 open Filter 40 open Filter
41 open Topology 41 open Topology
42 42
43 -- FIP is UFL 43 -- FIP is UFL
60 60
61 UFLP→FIP : {P : HOD} (TP : Topology P) → 61 UFLP→FIP : {P : HOD} (TP : Topology P) →
62 ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP 62 ((F : Filter {Power P} {P} (λ x → x) ) (UF : ultra-filter F ) → UFLP TP F UF ) → FIP TP
63 UFLP→FIP {P} TP uflp with trio< (& P) o∅ 63 UFLP→FIP {P} TP uflp with trio< (& P) o∅
64 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a ) 64 ... | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a )
65 ... | tri≈ ¬a b ¬c = record { limit = ? ; is-limit = {!!} } where 65 ... | tri≈ ¬a P=0 ¬c = record { limit = λ CX fip → o∅ ; is-limit = fip03 } where
66 -- P is empty 66 -- P is empty
67 fip02 : {x : Ordinal } → ¬ odef P x 67 fip02 : {x : Ordinal } → ¬ odef P x
68 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym b) (∈∅< Px) ) 68 fip02 {x} Px = ⊥-elim ( o<¬≡ (sym P=0) (∈∅< Px) )
69 ... | tri> ¬a ¬b 0<P = record { limit = ? ; is-limit = uf01 } where 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 )
88 ... | tri> ¬a ¬b 0<P = record { limit = λ CSX fip → UFLP.limit (uflp (F CSX fip) (ultraf CSX fip)) ; is-limit = ? } where
70 fip : {X : Ordinal} → * X ⊆ CS TP → Set n 89 fip : {X : Ordinal} → * X ⊆ CS TP → Set n
71 fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x 90 fip {X} CSX = {x : Ordinal} → Subbase (* X) x → o∅ o< x
72 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD 91 N : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip CSX → HOD
73 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P) 92 N {X} CSX fp = record { od = record { def = λ u → FBase P X u } ; odmax = osuc (& P)
74 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) } 93 ; <odmax = λ {x} lt → subst₂ (λ j k → j o< osuc k) &iso refl (⊆→o≤ (FBase.u⊆P lt)) }
75 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P 94 N⊆PP : {X : Ordinal } → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → N CSX fp ⊆ Power P
76 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb 95 N⊆PP CSX fp nx b xb = FBase.u⊆P nx xb
77 nc : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → HOD 96 nc : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → HOD
78 nc = ? 97 nc {X} CSX fip with trio< o∅ X
79 N∋nc :{X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → odef (N CSX fp) (& (nc CSX fp) ) 98 ... | tri< 0<X ¬b ¬c = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) ) where
80 N∋nc = ? 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) )
103 ... | tri≈ ¬a b ¬c = od∅
104 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
105 N∋nc :{X : Ordinal} → (CSX : * X ⊆ CS TP) → (fip : fip CSX) → odef (N CSX fip) (& (nc CSX fip) )
106 N∋nc {X} CSX fip with trio< o∅ X
107 ... | tri< 0<X ¬b ¬c = record { b = ? ; x = ? ; b⊆X = ? ; sb = ? ; u⊆P = ? ; x⊆u = ? } where
108 e : HOD -- we have an element of X
109 e = ODC.minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) )
110 Xe : odef (* X) (& e)
111 Xe = ODC.x∋minimal O (* X) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) 0<X) )
112 nn01 = ODC.minimal O (* (& e)) (0<P→ne (subst (λ k → o∅ o< k) (sym &iso) (fip (gi Xe))) )
113 ... | tri≈ ¬a b ¬c = ? -- od∅
114 ... | tri> ¬a ¬b c = ⊥-elim ( ¬x<0 c )
81 0<PP : o∅ o< & (Power P) 115 0<PP : o∅ o< & (Power P)
82 0<PP = ? 116 0<PP = subst (λ k → k o< & (Power P)) &iso ( c<→o< (subst (λ k → odef (Power P) k) (sym &iso) nn00 )) where
117 nn00 : odef (Power P) o∅
118 nn00 x lt with subst (λ k → odef k x) o∅≡od∅ lt
119 ... | x<0 = ⊥-elim ( ¬x<0 x<0)
83 -- 120 --
84 -- FIP defines a filter 121 -- FIP defines a filter
85 -- 122 --
86 F : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip CSX) → Filter {Power P} {P} (λ x → x) 123 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 124 F {X} CSX fp = record { filter = N CSX fp ; f⊆L = N⊆PP CSX fp ; filter1 = f1 ; filter2 = f2 } where
113 proper = ? 150 proper = ?
114 -- 151 --
115 -- then we have maximum ultra filter 152 -- then we have maximum ultra filter
116 -- 153 --
117 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp) 154 maxf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → MaximumFilter (λ x → x) (F CSX fp)
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) 155 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)
119 mf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → Filter {Power P} {P} (λ x → x) 156 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) 157 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) 158 ultraf : {X : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX) → ultra-filter ( mf CSX fp)
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) 159 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)
123 -- 160 --
124 -- so i has a limit as a limit of UIP 161 -- so i has a limit as a limit of UIP
125 -- 162 --
126 limit : {X : Ordinal} → (CSX : * X ⊆ CS TP) → fip {X} CSX → Ordinal 163 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)) 164 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) 165 uf02 : {X v : Ordinal} → (CSX : * X ⊆ CS TP) → (fp : fip {X} CSX)
129 → Neighbor TP (limit CSX fp) v → * v ⊆ filter ( mf CSX fp ) 166 → 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 167 uf02 {X} {v} CSX fp nei {x} vx = UFLP.is-limit ( uflp ( mf CSX fp ) (ultraf CSX fp)) nei vx
131 -- 168 --
132 -- the limit is an element of entire elements of X 169 -- the limit is an limit of entire elements of X
133 -- 170 --
134 uf01 : {X : Ordinal} (CSX : * X ⊆ CS TP) (fp : fip {X} CSX) {x : Ordinal} → odef (* X) x → odef (* x) (limit CSX fp) 171 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)) 172 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 173 ... | yes y = subst (λ k → odef (* x) k) &iso y
137 ... | no nxl = ⊥-elim ( MaximumFilter.proper (maxf CSX fp) uf08 ) where 174 ... | no nxl = ⊥-elim ( MaximumFilter.proper (maxf CSX fp) uf08 ) where
149 uf08 = ? 186 uf08 = ?
150 187
151 188
152 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP 189 FIP→UFLP : {P : HOD} (TP : Topology P) → FIP TP
153 → (F : Filter {Power P} {P} (λ x → x)) (UF : ultra-filter F ) → UFLP {P} TP F UF 190 → (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 191 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
155 -- 192 --
156 -- take closure of given filter elements 193 -- take closure of given filter elements
157 -- 194 --
158 CF : HOD 195 CF : HOD
159 CF = Replace (filter F) (λ x → Cl TP x ) 196 CF = Replace (filter F) (λ x → Cl TP x )
166 ufl01 = ? 203 ufl01 = ?
167 -- 204 --
168 -- so we have a limit 205 -- so we have a limit
169 -- 206 --
170 limit : Ordinal 207 limit : Ordinal
171 limit = FIP.limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 208 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 209 ufl02 : {y : Ordinal } → odef (* (& CF)) y → odef (* y) limit
173 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ufl01 210 ufl02 = FIP.is-limit fip (subst (λ k → k ⊆ CS TP) (sym *iso) CF⊆CS) ? -- ufl01
174 -- 211 --
175 -- Neigbor of limit ⊆ Filter 212 -- Neigbor of limit ⊆ Filter
176 -- 213 --
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 214 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
178 ufl03 {f} {v} ff nei fv=0 = ? 215 ufl03 {f} {v} ff nei fv=0 = ?
191 → UFLP TP F UF 228 → UFLP TP F UF
192 uflP F UF = FIP→UFLP TP (Compact→FIP TP CP) F UF 229 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) 230 uflQ : (F : Filter {Power Q} {Q} (λ x → x)) (UF : ultra-filter F)
194 → UFLP TQ F UF 231 → UFLP TQ F UF
195 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF 232 uflQ F UF = FIP→UFLP TQ (Compact→FIP TQ CQ) F UF
196 -- Product of UFL has limit point 233 -- Product of UFL has a limit point
197 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F) 234 uflPQ : (F : Filter {Power (ZFP P Q)} {ZFP P Q} (λ x → x)) (UF : ultra-filter F)
198 → UFLP (ProductTopology TP TQ) F UF 235 → 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 236 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) 237 FP : Filter {Power P} {P} (λ x → x)
201 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where 238 FP = record { filter = Proj1 (filter F) (Power P) (Power Q) ; f⊆L = ty00 ; filter1 = ? ; filter2 = ? } where