Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/filter.agda @ 1205:83ac320583f8
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| author | kono |
|---|---|
| date | Fri, 03 Mar 2023 10:42:58 +0800 |
| parents | 6216562a2bce |
| children | 5223f0b40d91 |
| rev | line source |
|---|---|
| 457 | 1 {-# OPTIONS --allow-unsolved-metas #-} |
| 2 | |
| 431 | 3 open import Level |
| 4 open import Ordinals | |
| 5 module filter {n : Level } (O : Ordinals {n}) where | |
| 6 | |
| 7 open import zf | |
| 8 open import logic | |
| 9 import OD | |
| 10 | |
| 11 open import Relation.Nullary | |
| 12 open import Data.Empty | |
| 13 open import Relation.Binary.Core | |
| 14 open import Relation.Binary.PropositionalEquality | |
| 1124 | 15 import BAlgebra |
| 431 | 16 |
| 1124 | 17 open BAlgebra O |
| 431 | 18 |
| 19 open inOrdinal O | |
| 20 open OD O | |
| 21 open OD.OD | |
| 22 open ODAxiom odAxiom | |
| 23 | |
| 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 | |
| 33 import ODC | |
| 34 open ODC O | |
| 35 | |
| 36 open _∧_ | |
| 37 open _∨_ | |
| 38 open Bool | |
| 39 | |
| 1133 | 40 -- L is a boolean algebra, but we don't assume this explicitly |
| 41 -- | |
| 42 -- NEG : {p : HOD} → L ∋ p → L ∋ (P ＼ p) | |
| 43 -- CAP : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q ) | |
| 44 -- | |
| 431 | 45 -- Kunen p.76 and p.53, we use ⊆ |
| 1124 | 46 record Filter { L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 47 field |
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48 filter : HOD |
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49 f⊆L : filter ⊆ L |
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50 filter1 : { p q : HOD } → L ∋ q → filter ∋ p → p ⊆ q → filter ∋ q |
| 461 | 51 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → L ∋ (p ∩ q) → filter ∋ (p ∩ q) |
| 431 | 52 |
| 53 open Filter | |
| 54 | |
| 1124 | 55 record prime-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where |
| 431 | 56 field |
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57 proper : ¬ (filter F ∋ od∅) |
| 1157 | 58 prime : {p q : HOD } → L ∋ p → L ∋ q → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) |
| 431 | 59 |
| 1124 | 60 record ultra-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter {L} {P} LP) : Set (suc (suc n)) where |
| 431 | 61 field |
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62 proper : ¬ (filter F ∋ od∅) |
| 461 | 63 ultra : {p : HOD } → L ∋ p → L ∋ ( P ＼ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) |
| 431 | 64 |
| 1124 | 65 ∈-filter : {L P p : HOD} → {LP : L ⊆ Power P} → (F : Filter {L} {P} LP ) → filter F ∋ p → L ∋ p |
| 1096 | 66 ∈-filter {L} {p} {LP} F lt = ( f⊆L F) lt |
| 458 | 67 |
| 1124 | 68 ⊆-filter : {L P p q : HOD } → {LP : L ⊆ Power P } → (F : Filter {L} {P} LP) → L ∋ q → q ⊆ P |
| 1096 | 69 ⊆-filter {L} {P} {p} {q} {LP} F lt = power→⊆ P q ( LP lt ) |
| 431 | 70 |
| 71 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L | |
| 1096 | 72 ∪-lemma1 {L} {p} {q} lt p∋x = lt (case1 p∋x) |
| 431 | 73 |
| 74 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L | |
| 1096 | 75 ∪-lemma2 {L} {p} {q} lt p∋x = lt (case2 p∋x) |
| 431 | 76 |
| 77 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q | |
| 1096 | 78 q∩q⊆q lt = proj1 lt |
| 431 | 79 |
| 1140 | 80 ∩→≡1 : {p q : HOD } → p ⊆ q → (q ∩ p) ≡ p |
| 81 ∩→≡1 {p} {q} p⊆q = ==→o≡ record { eq→ = c00 ; eq← = c01 } where | |
| 82 c00 : {x : Ordinal} → odef (q ∩ p) x → odef p x | |
| 83 c00 {x} qpx = proj2 qpx | |
| 84 c01 : {x : Ordinal} → odef p x → odef (q ∩ p) x | |
| 85 c01 {x} px = ⟪ p⊆q px , px ⟫ | |
| 86 | |
| 87 ∩→≡2 : {p q : HOD } → q ⊆ p → (q ∩ p) ≡ q | |
| 88 ∩→≡2 {p} {q} q⊆p = ==→o≡ record { eq→ = c00 ; eq← = c01 } where | |
| 89 c00 : {x : Ordinal} → odef (q ∩ p) x → odef q x | |
| 90 c00 {x} qpx = proj1 qpx | |
| 91 c01 : {x : Ordinal} → odef q x → odef (q ∩ p) x | |
| 92 c01 {x} qx = ⟪ qx , q⊆p qx ⟫ | |
| 93 | |
| 431 | 94 open HOD |
| 95 | |
| 96 ----- | |
| 97 -- | |
| 98 -- ultra filter is prime | |
| 99 -- | |
| 100 | |
| 461 | 101 filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) |
| 102 → ({p : HOD} → L ∋ p → L ∋ (P ＼ p)) | |
| 103 → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) | |
| 1124 | 104 → (F : Filter {L} {P} LP) → ultra-filter F → prime-filter F |
| 1157 | 105 filter-lemma1 {P} {L} LNEG NEG CAP F u = record { |
| 431 | 106 proper = ultra-filter.proper u |
| 107 ; prime = lemma3 | |
| 108 } where | |
| 1157 | 109 lemma3 : {p q : HOD} → L ∋ p → L ∋ q → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) |
| 110 lemma3 {p} {q} Lp Lq lt with ultra-filter.ultra u Lp (NEG Lp) | |
| 458 | 111 ... | case1 p∈P = case1 p∈P |
| 112 ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P ＼ p)} Lq lemma7 lemma8) where | |
| 113 lemma5 : ((p ∪ q ) ∩ (P ＼ p)) =h= (q ∩ (P ＼ p)) | |
| 431 | 114 lemma5 = record { eq→ = λ {x} lt → ⟪ lemma4 x lt , proj2 lt ⟫ |
| 115 ; eq← = λ {x} lt → ⟪ case2 (proj1 lt) , proj2 lt ⟫ | |
| 116 } where | |
| 458 | 117 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (P ＼ p)) x → odef q x |
| 431 | 118 lemma4 x lt with proj1 lt |
| 119 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) | |
| 120 lemma4 x lt | case2 qx = qx | |
| 461 | 121 lemma9 : L ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 1157 | 122 lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (CAP Lq (NEG Lp)) |
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123 lemma6 : filter F ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 461 | 124 lemma6 = filter2 F lt ¬p∈P lemma9 |
| 458 | 125 lemma7 : filter F ∋ (q ∩ (P ＼ p)) |
| 126 lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 | |
| 127 lemma8 : (q ∩ (P ＼ p)) ⊆ q | |
| 1124 | 128 lemma8 lt = proj1 lt |
| 431 | 129 |
| 130 ----- | |
| 131 -- | |
| 1124 | 132 -- if Filter {L} {P} contains L, prime filter is ultra |
| 431 | 133 -- |
| 134 | |
| 461 | 135 filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) |
| 136 → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) | |
| 1124 | 137 → (F : Filter {L} {P} LP) → filter F ∋ P → prime-filter F → ultra-filter F |
| 1157 | 138 filter-lemma2 {P} {L} LP NEG F f∋P prime = record { |
| 458 | 139 proper = prime-filter.proper prime |
| 1157 | 140 ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (NEG L∋p) (lemma p (p⊆L L∋p )) |
| 431 | 141 } where |
| 142 open _==_ | |
| 459 | 143 p⊆L : {p : HOD} → L ∋ p → p ⊆ P |
| 1096 | 144 p⊆L {p} lt = power→⊆ P p ( LP lt ) |
| 459 | 145 p+1-p=1 : {p : HOD} → p ⊆ P → P =h= (p ∪ (P ＼ p)) |
| 146 eq→ (p+1-p=1 {p} p⊆P) {x} lt with ODC.decp O (odef p x) | |
| 147 eq→ (p+1-p=1 {p} p⊆P) {x} lt | yes p∋x = case1 p∋x | |
| 148 eq→ (p+1-p=1 {p} p⊆P) {x} lt | no ¬p = case2 ⟪ lt , ¬p ⟫ | |
| 1096 | 149 eq← (p+1-p=1 {p} p⊆P) {x} ( case1 p∋x ) = subst (λ k → odef P k ) &iso (p⊆P ( subst (λ k → odef p k) (sym &iso) p∋x )) |
| 459 | 150 eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p |
| 458 | 151 lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P ＼ p)) |
| 459 | 152 lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P |
| 431 | 153 |
| 458 | 154 record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 155 field |
| 156 dense : HOD | |
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157 d⊆P : dense ⊆ L |
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158 dense-f : {p : HOD} → L ∋ p → HOD |
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159 dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt |
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160 dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p |
| 431 | 161 |
| 458 | 162 record Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 163 field |
| 164 ideal : HOD | |
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165 i⊆L : ideal ⊆ L |
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166 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
| 431 | 167 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) |
| 168 | |
| 169 open Ideal | |
| 170 | |
| 1124 | 171 proper-ideal : {L P : HOD} → (LP : L ⊆ Power P) → (P : Ideal {L} {P} LP ) → {p : HOD} → Set n |
| 458 | 172 proper-ideal {L} {P} LP I = ideal I ∋ od∅ |
| 431 | 173 |
| 1124 | 174 prime-ideal : {L P : HOD} → (LP : L ⊆ Power P) → Ideal {L} {P} LP → ∀ {p q : HOD } → Set n |
| 458 | 175 prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) |
| 431 | 176 |
| 1131 | 177 open import Relation.Binary.Definitions |
| 431 | 178 |
| 461 | 179 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where |
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180 field |
| 1124 | 181 genf : Filter {L} {P} LP |
| 182 generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) | |
| 431 | 183 |
| 1135 | 184 record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) (F : Filter {L} {P} LP) : Set (suc n) where |
| 478 | 185 field |
| 1124 | 186 mf : Filter {L} {P} LP |
| 1136 | 187 F⊆mf : filter F ⊆ filter mf |
| 478 | 188 proper : ¬ (filter mf ∋ od∅) |
| 1140 | 189 is-maximum : ( f : Filter {L} {P} LP ) → ¬ (filter f ∋ od∅) → filter F ⊆ filter f → ¬ ( filter mf ⊂ filter f ) |
| 478 | 190 |
| 1135 | 191 record Fp {L P : HOD} (LP : L ⊆ Power P) (F : Filter {L} {P} LP) (mx : MaximumFilter {L} {P} LP F ) (p x : Ordinal ) : Set n where |
| 1131 | 192 field |
| 193 y : Ordinal | |
| 194 mfy : odef (filter (MaximumFilter.mf mx)) y | |
| 195 y-p⊂x : ( * y ＼ * p ) ⊆ * x | |
| 196 | |
| 1125 | 197 max→ultra : {L P : HOD} (LP : L ⊆ Power P) |
| 1127 | 198 → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) |
| 1136 | 199 → (F0 : Filter {L} {P} LP) → {y : Ordinal } → odef (filter F0) y |
| 1135 | 200 → (mx : MaximumFilter {L} {P} LP F0 ) → ultra-filter ( MaximumFilter.mf mx ) |
| 1136 | 201 max→ultra {L} {P} LP CAP F0 {y} mfy mx = record { proper = MaximumFilter.proper mx ; ultra = ultra } where |
| 479 | 202 mf = MaximumFilter.mf mx |
| 203 ultra : {p : HOD} → L ∋ p → L ∋ (P ＼ p) → (filter mf ∋ p) ∨ (filter mf ∋ (P ＼ p)) | |
| 1131 | 204 ultra {p} Lp Lnp with ODC.∋-p O (filter mf) p |
| 479 | 205 ... | yes y = case1 y |
| 1132 | 206 ... | no np = case2 (subst (λ k → k ∋ (P ＼ p)) F=mf F∋P-p) where |
| 1129 | 207 F : HOD |
| 1135 | 208 F = record { od = record { def = λ x → odef L x ∧ Fp {L} {P} LP F0 mx (& p) x } |
| 1129 | 209 ; odmax = & L ; <odmax = λ lt → odef< (proj1 lt) } |
| 1124 | 210 mu01 : {r : HOD} {q : HOD} → L ∋ q → F ∋ r → r ⊆ q → F ∋ q |
| 1130 | 211 mu01 {r} {q} Lq ⟪ Lr , record { y = y ; mfy = mfy ; y-p⊂x = y-p⊂x } ⟫ r⊆q = ⟪ Lq , record { y = y ; mfy = mfy ; y-p⊂x = mu03 } ⟫ where |
| 212 mu05 : (* y ＼ p) ⊆ r | |
| 1131 | 213 mu05 = subst₂ (λ j k → (* y ＼ j ) ⊆ k ) *iso *iso y-p⊂x |
| 1130 | 214 mu04 : (* y ＼ * (& p)) ⊆ * (& q) |
| 215 mu04 {x} ⟪ yx , npx ⟫ = subst (λ k → odef k x ) (sym *iso) (r⊆q (mu05 ⟪ yx , (λ px1 → npx (subst (λ k → odef k x) (sym *iso) px1 )) ⟫ ) ) | |
| 1131 | 216 mu03 : (* y ＼ * (& p)) ⊆ * (& q) |
| 217 mu03 = mu04 | |
| 1125 | 218 mu02 : {r : HOD} {q : HOD} → F ∋ r → F ∋ q → L ∋ (r ∩ q) → F ∋ (r ∩ q) |
| 1129 | 219 mu02 {r} {q} ⟪ Lr , record { y = ry ; mfy = mfry ; y-p⊂x = ry-p⊂x } ⟫ |
| 1130 | 220 ⟪ Lq , record { y = qy ; mfy = mfqy ; y-p⊂x = qy-p⊂x } ⟫ Lrq = ⟪ Lrq , record { y = & (* qy ∩ * ry) ; mfy = mu20 ; y-p⊂x = mu22 } ⟫ where |
| 221 mu21 : L ∋ (* qy ∩ * ry) | |
| 222 mu21 = CAP (subst (λ k → odef L k ) (sym &iso) (f⊆L mf mfqy)) (subst (λ k → odef L k ) (sym &iso) (f⊆L mf mfry)) | |
| 223 mu20 : odef (filter mf) (& (* qy ∩ * ry)) | |
| 224 mu20 = filter2 mf (subst (λ k → odef (filter mf) k) (sym &iso) mfqy) (subst (λ k → odef (filter mf) k) (sym &iso) mfry) mu21 | |
| 225 mu24 : ((* qy ∩ * ry) ＼ * (& p)) ⊆ (r ∩ q) | |
| 1131 | 226 mu24 {x} ⟪ qry , npx ⟫ = ⟪ subst (λ k → odef k x) *iso ( ry-p⊂x ⟪ proj2 qry , npx ⟫ ) |
| 227 , subst (λ k → odef k x) *iso ( qy-p⊂x ⟪ proj1 qry , npx ⟫ ) ⟫ | |
| 228 mu23 : ((* qy ∩ * ry) ＼ * (& p) ) ⊆ (r ∩ q) | |
| 229 mu23 = mu24 | |
| 230 mu22 : (* (& (* qy ∩ * ry)) ＼ * (& p)) ⊆ * (& (r ∩ q)) | |
| 231 mu22 = subst₂ (λ j k → (j ＼ * (& p)) ⊆ k ) (sym *iso) (sym *iso) mu23 | |
| 1124 | 232 FisFilter : Filter {L} {P} LP |
| 1131 | 233 FisFilter = record { filter = F ; f⊆L = λ {x} lt → proj1 lt ; filter1 = mu01 ; filter2 = mu02 } |
| 1124 | 234 FisGreater : {x : Ordinal } → odef (filter (MaximumFilter.mf mx)) x → odef (filter FisFilter ) x |
| 1129 | 235 FisGreater {x} mfx = ⟪ f⊆L mf mfx , record { y = x ; mfy = mfx ; y-p⊂x = mu03 } ⟫ where |
| 1131 | 236 mu03 : (* x ＼ * (& p)) ⊆ * x |
| 237 mu03 {z} ⟪ xz , _ ⟫ = xz | |
| 238 F∋P-p : F ∋ (P ＼ p ) | |
| 1136 | 239 F∋P-p = ⟪ Lnp , record { y = y ; mfy = mxy ; y-p⊂x = mu30 } ⟫ where |
| 240 mxy : odef (filter (MaximumFilter.mf mx)) y | |
| 241 mxy = MaximumFilter.F⊆mf mx mfy | |
| 242 mu30 : (* y ＼ * (& p)) ⊆ * (& (P ＼ p)) | |
| 1132 | 243 mu30 {z} ⟪ yz , ¬pz ⟫ = subst (λ k → odef k z) (sym *iso) ( ⟪ Pz , (λ pz → ¬pz (subst (λ k → odef k z) (sym *iso) pz )) ⟫ ) where |
| 244 Pz : odef P z | |
| 1136 | 245 Pz = LP (f⊆L mf mxy ) _ yz |
| 1133 | 246 FisProper : ¬ (filter FisFilter ∋ od∅) -- if F contains p, p is in mf which contract np |
| 247 FisProper ⟪ L0 , record { y = z ; mfy = mfz ; y-p⊂x = z-p⊂x } ⟫ = | |
| 248 ⊥-elim ( np (filter1 mf Lp (subst (λ k → odef (filter mf) k) (sym &iso) mfz) mu31) ) where | |
| 249 mu31 : * z ⊆ p | |
| 250 mu31 {x} zx with ODC.decp O (odef p x) | |
| 251 ... | yes px = px | |
| 252 ... | no npx = ⊥-elim ( ¬x<0 (subst (λ k → odef k x) *iso (z-p⊂x ⟪ zx , (λ px → npx (subst (λ k → odef k x) *iso px) ) ⟫ ) ) ) | |
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253 F0⊆F : filter F0 ⊆ F |
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254 F0⊆F {x} fx = ⟪ f⊆L F0 fx , record { y = _ ; mfy = MaximumFilter.F⊆mf mx fx ; y-p⊂x = mu42 } ⟫ where |
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255 mu42 : (* x ＼ * (& p)) ⊆ * x |
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256 mu42 {z} ⟪ xz , ¬p ⟫ = xz |
| 1132 | 257 F=mf : F ≡ filter mf |
| 258 F=mf with osuc-≡< ( ⊆→o≤ FisGreater ) | |
| 259 ... | case1 eq = &≡&→≡ (sym eq) | |
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260 ... | case2 lt = ⊥-elim ( MaximumFilter.is-maximum mx FisFilter FisProper F0⊆F ⟪ lt , FisGreater ⟫ ) |
| 478 | 261 |
| 480 | 262 open _==_ |
| 263 | |
| 1124 | 264 ultra→max : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} |
| 265 → L ∋ p → L ∋ ( P ＼ p)) | |
| 266 → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) | |
| 1135 | 267 → (U : Filter {L} {P} LP) → ultra-filter U → MaximumFilter LP U |
| 1157 | 268 ultra→max {L} {P} LP NEG CAP U u = record { mf = U ; F⊆mf = λ x → x ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where |
| 1140 | 269 is-maximum : (F : Filter {L} {P} LP) → (¬ (filter F ∋ od∅)) → filter U ⊆ filter F → (U⊂F : filter U ⊂ filter F ) → ⊥ |
| 270 is-maximum F Prop F⊆U ⟪ U<F , U⊆F ⟫ = Prop f0 where | |
| 480 | 271 GT : HOD |
| 1124 | 272 GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = & L ; <odmax = um02 } where |
| 273 um02 : {y : Ordinal } → odef (filter F) y ∧ (¬ odef (filter U) y) → y o< & L | |
| 274 um02 {y} Fy = odef< ( f⊆L F (proj1 Fy ) ) | |
| 480 | 275 GT≠∅ : ¬ (GT =h= od∅) |
| 1124 | 276 GT≠∅ eq = ⊥-elim (U≠F ( ==→o≡ ((⊆→= {filter U} {filter F}) U⊆F (U-F=∅→F⊆U {filter F} {filter U} gt01)))) where |
| 277 U≠F : ¬ ( filter U ≡ filter F ) | |
| 278 U≠F eq = o<¬≡ (cong (&) eq) U<F | |
| 279 gt01 : (x : Ordinal) → ¬ ( odef (filter F) x ∧ (¬ odef (filter U) x)) | |
| 480 | 280 gt01 x not = ¬x<0 ( eq→ eq not ) |
| 281 p : HOD | |
| 282 p = ODC.minimal O GT GT≠∅ | |
| 283 ¬U∋p : ¬ ( filter U ∋ p ) | |
| 284 ¬U∋p = proj2 (ODC.x∋minimal O GT GT≠∅) | |
| 1124 | 285 L∋p : L ∋ p |
| 286 L∋p = f⊆L F ( proj1 (ODC.x∋minimal O GT GT≠∅)) | |
| 287 um00 : ¬ odef (filter U) (& p) | |
| 288 um00 = proj2 (ODC.x∋minimal O GT GT≠∅) | |
| 289 L∋-p : L ∋ ( P ＼ p ) | |
| 1157 | 290 L∋-p = NEG L∋p |
| 480 | 291 U∋-p : filter U ∋ ( P ＼ p ) |
| 1124 | 292 U∋-p with ultra-filter.ultra u {p} L∋p L∋-p |
| 480 | 293 ... | case1 ux = ⊥-elim ( ¬U∋p ux ) |
| 294 ... | case2 u-x = u-x | |
| 295 F∋p : filter F ∋ p | |
| 296 F∋p = proj1 (ODC.x∋minimal O GT GT≠∅) | |
| 297 F∋-p : filter F ∋ ( P ＼ p ) | |
| 1096 | 298 F∋-p = U⊆F U∋-p |
| 480 | 299 f0 : filter F ∋ od∅ |
| 1124 | 300 f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP L∋p L∋-p) ) |
| 516 | 301 |
| 1133 | 302 -- if there is a filter , there is a ultra filter under the axiom of choise |
| 303 -- Zorn Lemma | |
| 304 | |
| 1138 | 305 record IsFilter { L P : HOD } (LP : L ⊆ Power P) (filter : Ordinal ) : Set n where |
| 1133 | 306 field |
| 307 f⊆L : (* filter) ⊆ L | |
| 308 filter1 : { p q : Ordinal } → odef L q → odef (* filter) p → (* p) ⊆ (* q) → odef (* filter) q | |
| 309 filter2 : { p q : Ordinal } → odef (* filter) p → odef (* filter) q → odef L (& ((* p) ∩ (* q))) → odef (* filter) (& ((* p) ∩ (* q))) | |
| 310 proper : ¬ (odef (* filter ) o∅) | |
| 1126 | 311 |
| 1140 | 312 Filter-is-Filter : { L P : HOD } (LP : L ⊆ Power P) → (F : Filter {L} {P} LP) → (proper : ¬ (filter F) ∋ od∅ ) → IsFilter {L} {P} LP (& (filter F)) |
| 313 Filter-is-Filter {L} {P} LP F proper = record { | |
| 314 f⊆L = subst (λ k → k ⊆ L ) (sym *iso) (f⊆L F) | |
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315 ; filter1 = λ {p} {q} Lq Fp p⊆q → subst₂ (λ j k → odef j k ) (sym *iso) &iso |
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316 ( filter1 F (subst (λ k → odef L k) (sym &iso) Lq) (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) p⊆q ) |
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317 ; filter2 = λ {p} {q} Fp Fq Lpq → subst₂ (λ j k → odef j k ) (sym *iso) refl ( filter2 F (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fp) |
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318 (subst₂ (λ j k → odef j k ) *iso (sym &iso) Fq) Lpq ) |
| 1140 | 319 ; proper = subst₂ (λ j k → ¬ odef j k ) (sym *iso) ord-od∅ proper |
| 320 } |