Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/filter.agda @ 503:1546541ed461
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 12 Apr 2022 18:52:14 +0900 |
| parents | 94586e4e0378 |
| children | 286016848403 |
| 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 | |
| 15 import BAlgbra | |
| 16 | |
| 17 open BAlgbra O | |
| 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 | |
| 40 -- Kunen p.76 and p.53, we use ⊆ | |
| 458 | 41 record Filter { L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 42 field |
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43 filter : HOD |
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44 f⊆L : filter ⊆ L |
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45 filter1 : { p q : HOD } → L ∋ q → filter ∋ p → p ⊆ q → filter ∋ q |
| 461 | 46 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → L ∋ (p ∩ q) → filter ∋ (p ∩ q) |
| 431 | 47 |
| 48 open Filter | |
| 49 | |
| 458 | 50 record prime-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter LP) : Set (suc (suc n)) where |
| 431 | 51 field |
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52 proper : ¬ (filter F ∋ od∅) |
| 461 | 53 prime : {p q : HOD } → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) |
| 431 | 54 |
| 458 | 55 record ultra-filter { L P : HOD } {LP : L ⊆ Power P} (F : Filter LP) : Set (suc (suc n)) where |
| 431 | 56 field |
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57 proper : ¬ (filter F ∋ od∅) |
| 461 | 58 ultra : {p : HOD } → L ∋ p → L ∋ ( P ＼ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) |
| 431 | 59 |
| 60 open _⊆_ | |
| 61 | |
| 458 | 62 ∈-filter : {L P p : HOD} → {LP : L ⊆ Power P} → (F : Filter LP ) → filter F ∋ p → L ∋ p |
| 63 ∈-filter {L} {p} {LP} F lt = incl ( f⊆L F) lt | |
| 64 | |
| 65 ⊆-filter : {L P p q : HOD } → {LP : L ⊆ Power P } → (F : Filter LP) → L ∋ q → q ⊆ P | |
| 66 ⊆-filter {L} {P} {p} {q} {LP} F lt = power→⊆ P q ( incl LP lt ) | |
| 431 | 67 |
| 68 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L | |
| 69 ∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) } | |
| 70 | |
| 71 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L | |
| 72 ∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) } | |
| 73 | |
| 74 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q | |
| 75 q∩q⊆q = record { incl = λ lt → proj1 lt } | |
| 76 | |
| 77 open HOD | |
| 78 | |
| 79 ----- | |
| 80 -- | |
| 81 -- ultra filter is prime | |
| 82 -- | |
| 83 | |
| 461 | 84 filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) |
| 85 → ({p : HOD} → L ∋ p → L ∋ (P ＼ p)) | |
| 86 → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) | |
| 87 → (F : Filter LP) → ultra-filter F → prime-filter F | |
| 88 filter-lemma1 {P} {L} LP NG IN F u = record { | |
| 431 | 89 proper = ultra-filter.proper u |
| 90 ; prime = lemma3 | |
| 91 } where | |
| 461 | 92 lemma3 : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) |
| 93 lemma3 {p} {q} Lp Lq _ lt with ultra-filter.ultra u Lp (NG Lp) | |
| 458 | 94 ... | case1 p∈P = case1 p∈P |
| 95 ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P ＼ p)} Lq lemma7 lemma8) where | |
| 96 lemma5 : ((p ∪ q ) ∩ (P ＼ p)) =h= (q ∩ (P ＼ p)) | |
| 431 | 97 lemma5 = record { eq→ = λ {x} lt → ⟪ lemma4 x lt , proj2 lt ⟫ |
| 98 ; eq← = λ {x} lt → ⟪ case2 (proj1 lt) , proj2 lt ⟫ | |
| 99 } where | |
| 458 | 100 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (P ＼ p)) x → odef q x |
| 431 | 101 lemma4 x lt with proj1 lt |
| 102 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) | |
| 103 lemma4 x lt | case2 qx = qx | |
| 461 | 104 lemma9 : L ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 105 lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (IN Lq (NG Lp)) | |
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parents:
455
diff
changeset
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106 lemma6 : filter F ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 461 | 107 lemma6 = filter2 F lt ¬p∈P lemma9 |
| 458 | 108 lemma7 : filter F ∋ (q ∩ (P ＼ p)) |
| 109 lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 | |
| 110 lemma8 : (q ∩ (P ＼ p)) ⊆ q | |
| 431 | 111 lemma8 = q∩q⊆q |
| 112 | |
| 113 ----- | |
| 114 -- | |
| 115 -- if Filter contains L, prime filter is ultra | |
| 116 -- | |
| 117 | |
| 461 | 118 filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) |
| 119 → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) | |
| 120 → (F : Filter LP) → filter F ∋ P → prime-filter F → ultra-filter F | |
| 459 | 121 filter-lemma2 {P} {L} LP Lm F f∋P prime = record { |
| 458 | 122 proper = prime-filter.proper prime |
| 461 | 123 ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (Lm L∋p) (lemma10 L∋p (incl (f⊆L F) f∋P) ) (lemma p (p⊆L L∋p )) |
| 431 | 124 } where |
| 125 open _==_ | |
| 459 | 126 p⊆L : {p : HOD} → L ∋ p → p ⊆ P |
| 127 p⊆L {p} lt = power→⊆ P p ( incl LP lt ) | |
| 128 p+1-p=1 : {p : HOD} → p ⊆ P → P =h= (p ∪ (P ＼ p)) | |
| 129 eq→ (p+1-p=1 {p} p⊆P) {x} lt with ODC.decp O (odef p x) | |
| 130 eq→ (p+1-p=1 {p} p⊆P) {x} lt | yes p∋x = case1 p∋x | |
| 131 eq→ (p+1-p=1 {p} p⊆P) {x} lt | no ¬p = case2 ⟪ lt , ¬p ⟫ | |
| 132 eq← (p+1-p=1 {p} p⊆P) {x} ( case1 p∋x ) = subst (λ k → odef P k ) &iso (incl p⊆P ( subst (λ k → odef p k) (sym &iso) p∋x )) | |
| 133 eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p | |
| 458 | 134 lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P ＼ p)) |
| 459 | 135 lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P |
| 461 | 136 lemma10 : {p : HOD} → L ∋ p → L ∋ P → L ∋ (p ∪ (P ＼ p)) |
| 137 lemma10 {p} L∋p lt = subst (λ k → L ∋ k ) (==→o≡ (p+1-p=1 {p} (p⊆L L∋p))) lt | |
| 431 | 138 |
| 458 | 139 ----- |
| 140 -- | |
| 141 -- if there is a filter , there is a ultra filter under the axiom of choise | |
| 461 | 142 -- Zorn Lemma |
| 458 | 143 |
| 459 | 144 -- filter→ultra : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → (F : Filter LP) → ultra-filter F |
| 145 -- filter→ultra {P} {L} LP Lm F = {!!} | |
| 458 | 146 |
| 147 record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where | |
| 431 | 148 field |
| 149 dense : HOD | |
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parents:
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diff
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150 d⊆P : dense ⊆ L |
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151 dense-f : {p : HOD} → L ∋ p → HOD |
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152 dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt |
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parents:
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153 dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p |
| 431 | 154 |
| 458 | 155 record Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 156 field |
| 157 ideal : HOD | |
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158 i⊆L : ideal ⊆ L |
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159 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
| 431 | 160 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) |
| 161 | |
| 162 open Ideal | |
| 163 | |
| 458 | 164 proper-ideal : {L P : HOD} → (LP : L ⊆ Power P) → (P : Ideal LP ) → {p : HOD} → Set n |
| 165 proper-ideal {L} {P} LP I = ideal I ∋ od∅ | |
| 431 | 166 |
| 458 | 167 prime-ideal : {L P : HOD} → (LP : L ⊆ Power P) → Ideal LP → ∀ {p q : HOD } → Set n |
| 168 prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) | |
| 431 | 169 |
| 170 | |
| 461 | 171 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where |
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172 field |
| 458 | 173 genf : Filter LP |
| 461 | 174 generic : (D : Dense LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) |
| 431 | 175 |
| 478 | 176 record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) : Set (suc n) where |
| 177 field | |
| 178 mf : Filter LP | |
| 179 proper : ¬ (filter mf ∋ od∅) | |
| 481 | 180 is-maximum : ( f : Filter LP ) → ¬ (filter f ∋ od∅) → ¬ filter mf ≡ filter f → ¬ ( filter mf ⊆ filter f ) |
| 478 | 181 |
| 182 max→ultra : {L P : HOD} (LP : L ⊆ Power P) → (mx : MaximumFilter LP ) → ultra-filter ( MaximumFilter.mf mx ) | |
| 479 | 183 max→ultra {L} {P} LP mx = record { proper = MaximumFilter.proper mx ; ultra = ultra } where |
| 184 mf = MaximumFilter.mf mx | |
| 185 ultra : {p : HOD} → L ∋ p → L ∋ (P ＼ p) → (filter mf ∋ p) ∨ (filter mf ∋ (P ＼ p)) | |
| 186 ultra {p} lp lnp with ∋-p (filter mf) p | |
| 187 ... | yes y = case1 y | |
| 188 ... | no np with ∋-p (filter mf) (P ＼ p) | |
| 189 ... | yes y = case2 y | |
| 481 | 190 ... | no n-p = ⊥-elim (MaximumFilter.is-maximum mx FisFilter FisProper {!!} record { incl = FisGreater } ) where |
| 479 | 191 Y : (y : Ordinal) → (my : odef (filter mf) y ) → HOD |
| 192 Y y my = record { od = record { def = λ x → (x ≡ y) ∨ (x ≡ & p) } ; odmax = & L ; <odmax = {!!} } | |
| 193 F : HOD | |
| 194 F = record { od = record { def = λ x → (x ≡ & p) ∨ ((y : Ordinal) → (my : odef (filter mf) y ) → x ≡ & (Y y my) ) } ; odmax = & L ; <odmax = {!!} } | |
| 195 FisFilter : Filter LP | |
| 196 FisFilter = record { filter = F ; f⊆L = {!!} ; filter1 = {!!} ; filter2 = {!!} } | |
| 480 | 197 FisGreater : {x : HOD} → filter (MaximumFilter.mf mx) ∋ x → filter FisFilter ∋ x |
| 198 FisGreater = {!!} | |
| 199 FisProper : ¬ (filter FisFilter ∋ od∅) | |
| 200 FisProper = {!!} | |
| 478 | 201 |
| 480 | 202 open _==_ |
| 203 | |
| 204 open import Relation.Binary.Definitions | |
| 205 | |
| 206 ultra→max : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) | |
| 481 | 207 → (U : Filter LP) → ultra-filter U → MaximumFilter LP |
| 208 ultra→max {L} {P} LP NG CAP U u = record { mf = U ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where | |
| 209 is-maximum : (F : Filter LP) → (¬ (filter F ∋ od∅)) → ( U≠F : ¬ filter U ≡ filter F ) → (U⊆F : filter U ⊆ filter F ) → ⊥ | |
| 210 is-maximum F Prop U≠F U⊆F = Prop f0 where | |
| 480 | 211 GT : HOD |
| 212 GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = {!!} ; <odmax = {!!} } | |
| 213 GT≠∅ : ¬ (GT =h= od∅) | |
| 214 GT≠∅ eq = ⊥-elim (U≠F ( ==→o≡ (⊆→= U⊆F (U-F=∅→F⊆U gt01)))) where | |
| 215 gt01 : (x : Ordinal) → ¬ odef (filter F) x ∧ (¬ odef (filter U) x) | |
| 216 gt01 x not = ¬x<0 ( eq→ eq not ) | |
| 217 p : HOD | |
| 218 p = ODC.minimal O GT GT≠∅ | |
| 219 ¬U∋p : ¬ ( filter U ∋ p ) | |
| 220 ¬U∋p = proj2 (ODC.x∋minimal O GT GT≠∅) | |
| 221 U∋-p : filter U ∋ ( P ＼ p ) | |
| 222 U∋-p with ultra-filter.ultra u {p} {!!} {!!} | |
| 223 ... | case1 ux = ⊥-elim ( ¬U∋p ux ) | |
| 224 ... | case2 u-x = u-x | |
| 225 F∋p : filter F ∋ p | |
| 226 F∋p = proj1 (ODC.x∋minimal O GT GT≠∅) | |
| 227 F∋-p : filter F ∋ ( P ＼ p ) | |
| 228 F∋-p = incl U⊆F U∋-p | |
| 229 f0 : filter F ∋ od∅ | |
| 230 f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP {!!} {!!}) ) | |
| 478 | 231 |
| 503 | 232 -- open import zorn |
| 481 | 233 |
| 503 | 234 -- MaximumSubset : {L P : HOD} |
| 235 -- → o∅ o< & L → o∅ o< & P → P ⊆ L | |
| 236 -- → IsPartialOrderSet P _⊆'_ | |
| 237 -- → ( (B : HOD) → B ⊆ P → IsTotalOrderSet B _⊆'_ → SUP P B _⊆'_ ) | |
| 238 -- → Maximal P (_⊆'_) | |
| 239 -- MaximumSubset {L} {P} 0<L 0<P P⊆L PO SP = Zorn-lemma {P} {_⊆'_} 0<P PO SP | |
| 240 | |
| 241 -- MaximumFilterExist : {L P : HOD} (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q)) | |
| 242 -- → (F : Filter LP) → o∅ o< & L → o∅ o< & (filter F) → (¬ (filter F ∋ od∅)) → MaximumFilter LP | |
| 243 -- MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = {!!} where | |
| 244 -- mf01 : Maximal O P (_⊆'_ O) | |
| 245 -- mf01 = MaximumSubset O 0<L {!!} {!!} {!!} {!!} | |
| 481 | 246 |
| 247 |