Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate src/filter.agda @ 1096:55ab5de1ae02
recovery
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 23 Dec 2022 12:54:05 +0900 |
| parents | 9084a26445a7 |
| children | d122d0c1b094 |
| 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 |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
43 filter : HOD |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
44 f⊆L : filter ⊆ L |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
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 |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
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 |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
57 proper : ¬ (filter F ∋ od∅) |
| 461 | 58 ultra : {p : HOD } → L ∋ p → L ∋ ( P ＼ p) → ( filter F ∋ p ) ∨ ( filter F ∋ ( P ＼ p) ) |
| 431 | 59 |
| 458 | 60 ∈-filter : {L P p : HOD} → {LP : L ⊆ Power P} → (F : Filter LP ) → filter F ∋ p → L ∋ p |
| 1096 | 61 ∈-filter {L} {p} {LP} F lt = ( f⊆L F) lt |
| 458 | 62 |
| 63 ⊆-filter : {L P p q : HOD } → {LP : L ⊆ Power P } → (F : Filter LP) → L ∋ q → q ⊆ P | |
| 1096 | 64 ⊆-filter {L} {P} {p} {q} {LP} F lt = power→⊆ P q ( LP lt ) |
| 431 | 65 |
| 66 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L | |
| 1096 | 67 ∪-lemma1 {L} {p} {q} lt p∋x = lt (case1 p∋x) |
| 431 | 68 |
| 69 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L | |
| 1096 | 70 ∪-lemma2 {L} {p} {q} lt p∋x = lt (case2 p∋x) |
| 431 | 71 |
| 72 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q | |
| 1096 | 73 q∩q⊆q lt = proj1 lt |
| 431 | 74 |
| 75 open HOD | |
| 76 | |
| 77 ----- | |
| 78 -- | |
| 79 -- ultra filter is prime | |
| 80 -- | |
| 81 | |
| 461 | 82 filter-lemma1 : {P L : HOD} → (LP : L ⊆ Power P) |
| 83 → ({p : HOD} → L ∋ p → L ∋ (P ＼ p)) | |
| 84 → ({p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∩ q )) | |
| 85 → (F : Filter LP) → ultra-filter F → prime-filter F | |
| 86 filter-lemma1 {P} {L} LP NG IN F u = record { | |
| 431 | 87 proper = ultra-filter.proper u |
| 88 ; prime = lemma3 | |
| 89 } where | |
| 461 | 90 lemma3 : {p q : HOD} → L ∋ p → L ∋ q → L ∋ (p ∪ q) → filter F ∋ (p ∪ q) → ( filter F ∋ p ) ∨ ( filter F ∋ q ) |
| 91 lemma3 {p} {q} Lp Lq _ lt with ultra-filter.ultra u Lp (NG Lp) | |
| 458 | 92 ... | case1 p∈P = case1 p∈P |
| 93 ... | case2 ¬p∈P = case2 (filter1 F {q ∩ (P ＼ p)} Lq lemma7 lemma8) where | |
| 94 lemma5 : ((p ∪ q ) ∩ (P ＼ p)) =h= (q ∩ (P ＼ p)) | |
| 431 | 95 lemma5 = record { eq→ = λ {x} lt → ⟪ lemma4 x lt , proj2 lt ⟫ |
| 96 ; eq← = λ {x} lt → ⟪ case2 (proj1 lt) , proj2 lt ⟫ | |
| 97 } where | |
| 458 | 98 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (P ＼ p)) x → odef q x |
| 431 | 99 lemma4 x lt with proj1 lt |
| 100 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) | |
| 101 lemma4 x lt | case2 qx = qx | |
| 461 | 102 lemma9 : L ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 103 lemma9 = subst (λ k → L ∋ k ) (sym (==→o≡ lemma5)) (IN Lq (NG Lp)) | |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
104 lemma6 : filter F ∋ ((p ∪ q ) ∩ (P ＼ p)) |
| 461 | 105 lemma6 = filter2 F lt ¬p∈P lemma9 |
| 458 | 106 lemma7 : filter F ∋ (q ∩ (P ＼ p)) |
| 107 lemma7 = subst (λ k → filter F ∋ k ) (==→o≡ lemma5 ) lemma6 | |
| 108 lemma8 : (q ∩ (P ＼ p)) ⊆ q | |
| 431 | 109 lemma8 = q∩q⊆q |
| 110 | |
| 111 ----- | |
| 112 -- | |
| 113 -- if Filter contains L, prime filter is ultra | |
| 114 -- | |
| 115 | |
| 461 | 116 filter-lemma2 : {P L : HOD} → (LP : L ⊆ Power P) |
| 117 → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) | |
| 118 → (F : Filter LP) → filter F ∋ P → prime-filter F → ultra-filter F | |
| 459 | 119 filter-lemma2 {P} {L} LP Lm F f∋P prime = record { |
| 458 | 120 proper = prime-filter.proper prime |
| 1096 | 121 ; ultra = λ {p} L∋p _ → prime-filter.prime prime L∋p (Lm L∋p) (lemma10 L∋p ((f⊆L F) f∋P) ) (lemma p (p⊆L L∋p )) |
| 431 | 122 } where |
| 123 open _==_ | |
| 459 | 124 p⊆L : {p : HOD} → L ∋ p → p ⊆ P |
| 1096 | 125 p⊆L {p} lt = power→⊆ P p ( LP lt ) |
| 459 | 126 p+1-p=1 : {p : HOD} → p ⊆ P → P =h= (p ∪ (P ＼ p)) |
| 127 eq→ (p+1-p=1 {p} p⊆P) {x} lt with ODC.decp O (odef p x) | |
| 128 eq→ (p+1-p=1 {p} p⊆P) {x} lt | yes p∋x = case1 p∋x | |
| 129 eq→ (p+1-p=1 {p} p⊆P) {x} lt | no ¬p = case2 ⟪ lt , ¬p ⟫ | |
| 1096 | 130 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 | 131 eq← (p+1-p=1 {p} p⊆P) {x} ( case2 ¬p ) = proj1 ¬p |
| 458 | 132 lemma : (p : HOD) → p ⊆ P → filter F ∋ (p ∪ (P ＼ p)) |
| 459 | 133 lemma p p⊆P = subst (λ k → filter F ∋ k ) (==→o≡ (p+1-p=1 {p} p⊆P)) f∋P |
| 461 | 134 lemma10 : {p : HOD} → L ∋ p → L ∋ P → L ∋ (p ∪ (P ＼ p)) |
| 135 lemma10 {p} L∋p lt = subst (λ k → L ∋ k ) (==→o≡ (p+1-p=1 {p} (p⊆L L∋p))) lt | |
| 431 | 136 |
| 458 | 137 ----- |
| 138 -- | |
| 139 -- if there is a filter , there is a ultra filter under the axiom of choise | |
| 461 | 140 -- Zorn Lemma |
| 458 | 141 |
| 459 | 142 -- filter→ultra : {P L : HOD} → (LP : L ⊆ Power P) → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) → (F : Filter LP) → ultra-filter F |
| 143 -- filter→ultra {P} {L} LP Lm F = {!!} | |
| 458 | 144 |
| 145 record Dense {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where | |
| 431 | 146 field |
| 147 dense : HOD | |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
148 d⊆P : dense ⊆ L |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
149 dense-f : {p : HOD} → L ∋ p → HOD |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
150 dense-d : { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
151 dense-p : { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p |
| 431 | 152 |
| 458 | 153 record Ideal {L P : HOD } (LP : L ⊆ Power P) : Set (suc n) where |
| 431 | 154 field |
| 155 ideal : HOD | |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
156 i⊆L : ideal ⊆ L |
|
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
157 ideal1 : { p q : HOD } → L ∋ q → ideal ∋ p → q ⊆ p → ideal ∋ q |
| 431 | 158 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) |
| 159 | |
| 160 open Ideal | |
| 161 | |
| 458 | 162 proper-ideal : {L P : HOD} → (LP : L ⊆ Power P) → (P : Ideal LP ) → {p : HOD} → Set n |
| 163 proper-ideal {L} {P} LP I = ideal I ∋ od∅ | |
| 431 | 164 |
| 458 | 165 prime-ideal : {L P : HOD} → (LP : L ⊆ Power P) → Ideal LP → ∀ {p q : HOD } → Set n |
| 166 prime-ideal {L} {P} LP I {p} {q} = ideal I ∋ ( p ∩ q) → ( ideal I ∋ p ) ∨ ( ideal I ∋ q ) | |
| 431 | 167 |
| 168 | |
| 461 | 169 record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (suc n) where |
|
456
9207b0c3cfe9
fix filter on subset of Power P
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
455
diff
changeset
|
170 field |
| 458 | 171 genf : Filter LP |
| 461 | 172 generic : (D : Dense LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Filter.filter genf ) ≡ od∅ ) |
| 431 | 173 |
| 478 | 174 record MaximumFilter {L P : HOD} (LP : L ⊆ Power P) : Set (suc n) where |
| 175 field | |
| 176 mf : Filter LP | |
| 177 proper : ¬ (filter mf ∋ od∅) | |
| 481 | 178 is-maximum : ( f : Filter LP ) → ¬ (filter f ∋ od∅) → ¬ filter mf ≡ filter f → ¬ ( filter mf ⊆ filter f ) |
| 478 | 179 |
| 180 max→ultra : {L P : HOD} (LP : L ⊆ Power P) → (mx : MaximumFilter LP ) → ultra-filter ( MaximumFilter.mf mx ) | |
| 479 | 181 max→ultra {L} {P} LP mx = record { proper = MaximumFilter.proper mx ; ultra = ultra } where |
| 182 mf = MaximumFilter.mf mx | |
| 183 ultra : {p : HOD} → L ∋ p → L ∋ (P ＼ p) → (filter mf ∋ p) ∨ (filter mf ∋ (P ＼ p)) | |
| 184 ultra {p} lp lnp with ∋-p (filter mf) p | |
| 185 ... | yes y = case1 y | |
| 186 ... | no np with ∋-p (filter mf) (P ＼ p) | |
| 187 ... | yes y = case2 y | |
| 1096 | 188 ... | no n-p = ⊥-elim (MaximumFilter.is-maximum mx FisFilter FisProper {!!} ? ) where |
| 479 | 189 Y : (y : Ordinal) → (my : odef (filter mf) y ) → HOD |
| 190 Y y my = record { od = record { def = λ x → (x ≡ y) ∨ (x ≡ & p) } ; odmax = & L ; <odmax = {!!} } | |
| 191 F : HOD | |
| 192 F = record { od = record { def = λ x → (x ≡ & p) ∨ ((y : Ordinal) → (my : odef (filter mf) y ) → x ≡ & (Y y my) ) } ; odmax = & L ; <odmax = {!!} } | |
| 193 FisFilter : Filter LP | |
| 194 FisFilter = record { filter = F ; f⊆L = {!!} ; filter1 = {!!} ; filter2 = {!!} } | |
| 480 | 195 FisGreater : {x : HOD} → filter (MaximumFilter.mf mx) ∋ x → filter FisFilter ∋ x |
| 196 FisGreater = {!!} | |
| 197 FisProper : ¬ (filter FisFilter ∋ od∅) | |
| 198 FisProper = {!!} | |
| 478 | 199 |
| 480 | 200 open _==_ |
| 201 | |
| 202 open import Relation.Binary.Definitions | |
| 203 | |
| 204 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 | 205 → (U : Filter LP) → ultra-filter U → MaximumFilter LP |
| 206 ultra→max {L} {P} LP NG CAP U u = record { mf = U ; proper = ultra-filter.proper u ; is-maximum = is-maximum } where | |
| 207 is-maximum : (F : Filter LP) → (¬ (filter F ∋ od∅)) → ( U≠F : ¬ filter U ≡ filter F ) → (U⊆F : filter U ⊆ filter F ) → ⊥ | |
| 208 is-maximum F Prop U≠F U⊆F = Prop f0 where | |
| 480 | 209 GT : HOD |
| 210 GT = record { od = record { def = λ x → odef (filter F) x ∧ (¬ odef (filter U) x) } ; odmax = {!!} ; <odmax = {!!} } | |
| 211 GT≠∅ : ¬ (GT =h= od∅) | |
| 212 GT≠∅ eq = ⊥-elim (U≠F ( ==→o≡ (⊆→= U⊆F (U-F=∅→F⊆U gt01)))) where | |
| 213 gt01 : (x : Ordinal) → ¬ odef (filter F) x ∧ (¬ odef (filter U) x) | |
| 214 gt01 x not = ¬x<0 ( eq→ eq not ) | |
| 215 p : HOD | |
| 216 p = ODC.minimal O GT GT≠∅ | |
| 217 ¬U∋p : ¬ ( filter U ∋ p ) | |
| 218 ¬U∋p = proj2 (ODC.x∋minimal O GT GT≠∅) | |
| 219 U∋-p : filter U ∋ ( P ＼ p ) | |
| 220 U∋-p with ultra-filter.ultra u {p} {!!} {!!} | |
| 221 ... | case1 ux = ⊥-elim ( ¬U∋p ux ) | |
| 222 ... | case2 u-x = u-x | |
| 223 F∋p : filter F ∋ p | |
| 224 F∋p = proj1 (ODC.x∋minimal O GT GT≠∅) | |
| 225 F∋-p : filter F ∋ ( P ＼ p ) | |
| 1096 | 226 F∋-p = U⊆F U∋-p |
| 480 | 227 f0 : filter F ∋ od∅ |
| 228 f0 = subst (λ k → odef (filter F) k ) (trans (cong (&) ∩-comm) (cong (&) [a-b]∩b=0 ) ) ( filter2 F F∋p F∋-p ( CAP {!!} {!!}) ) | |
| 516 | 229 |
| 230 import zorn | |
| 1096 | 231 open zorn O _⊆_ |
| 574 | 232 |
| 233 open import Relation.Binary.Structures | |
| 481 | 234 |
| 516 | 235 MaximumSubset : {L P : HOD} |
| 1096 | 236 → o∅ o< & L → o∅ o< & P → P ⊆ L |
| 237 → (PO : IsStrictPartialOrder _≡_ _⊆_ ) | |
| 574 | 238 → ( (B : HOD) → B ⊆ P → IsTotalOrderSet PO B → SUP PO P B ) |
| 239 → Maximal PO P | |
| 240 MaximumSubset {L} {P} 0<L 0<P P⊆L PO C = Zorn-lemma PO 0<P {!!} | |
| 503 | 241 |
| 516 | 242 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)) |
| 243 → (F : Filter LP) → o∅ o< & L → o∅ o< & (filter F) → (¬ (filter F ∋ od∅)) → MaximumFilter LP | |
| 244 MaximumFilterExist {L} {P} LP NEG CAP F 0<L 0<F Fprop = record { mf = {!!} ; proper = {!!} ; is-maximum = {!!} } where | |
| 574 | 245 mf01 : Maximal {!!} {!!} |
| 246 mf01 = MaximumSubset 0<L {!!} {!!} {!!} {!!} | |
| 481 | 247 |
| 248 |