Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 368:30de2d9b93c1
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 19 Jul 2020 03:24:39 +0900 |
| parents | f74681db63c7 |
| children | 1425104bb5d8 |
| rev | line source |
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| 190 | 1 open import Level |
| 236 | 2 open import Ordinals |
| 3 module filter {n : Level } (O : Ordinals {n}) where | |
| 4 | |
| 190 | 5 open import zf |
| 236 | 6 open import logic |
| 7 import OD | |
| 193 | 8 |
| 363 | 9 open import Relation.Nullary |
| 10 open import Relation.Binary | |
| 11 open import Data.Empty | |
| 190 | 12 open import Relation.Binary |
| 13 open import Relation.Binary.Core | |
| 363 | 14 open import Relation.Binary.PropositionalEquality |
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15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) |
| 363 | 16 import BAlgbra |
| 293 | 17 |
| 18 open BAlgbra O | |
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19 |
| 236 | 20 open inOrdinal O |
| 21 open OD O | |
| 22 open OD.OD | |
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23 open ODAxiom odAxiom |
| 190 | 24 |
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25 import ODC |
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26 |
| 236 | 27 open _∧_ |
| 28 open _∨_ | |
| 29 open Bool | |
| 30 | |
| 295 | 31 -- Kunen p.76 and p.53, we use ⊆ |
| 329 | 32 record Filter ( L : HOD ) : Set (suc n) where |
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33 field |
| 329 | 34 filter : HOD |
| 290 | 35 f⊆PL : filter ⊆ Power L |
| 329 | 36 filter1 : { p q : HOD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
| 37 filter2 : { p q : HOD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) | |
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38 |
| 292 | 39 open Filter |
| 40 | |
| 329 | 41 record prime-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
| 295 | 42 field |
| 43 proper : ¬ (filter P ∋ od∅) | |
| 329 | 44 prime : {p q : HOD } → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
| 292 | 45 |
| 329 | 46 record ultra-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where |
| 295 | 47 field |
| 48 proper : ¬ (filter P ∋ od∅) | |
| 329 | 49 ultra : {p : HOD } → p ⊆ L → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) |
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50 |
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51 open _⊆_ |
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52 |
| 329 | 53 trans-⊆ : { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C |
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54 trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) } |
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55 |
| 329 | 56 power→⊆ : ( A t : HOD) → Power A ∋ t → t ⊆ A |
| 331 | 57 power→⊆ A t PA∋t = record { incl = λ {x} t∋x → ODC.double-neg-eilm O (t1 t∋x) } where |
| 329 | 58 t1 : {x : HOD } → t ∋ x → ¬ ¬ (A ∋ x) |
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59 t1 = zf.IsZF.power→ isZF A t PA∋t |
| 292 | 60 |
| 329 | 61 ∈-filter : {L p : HOD} → (P : Filter L ) → filter P ∋ p → p ⊆ L |
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62 ∈-filter {L} {p} P lt = power→⊆ L p ( incl (f⊆PL P) lt ) |
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63 |
| 329 | 64 ∪-lemma1 : {L p q : HOD } → (p ∪ q) ⊆ L → p ⊆ L |
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65 ∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) } |
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66 |
| 329 | 67 ∪-lemma2 : {L p q : HOD } → (p ∪ q) ⊆ L → q ⊆ L |
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68 ∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) } |
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69 |
| 329 | 70 q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q |
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71 q∩q⊆q = record { incl = λ lt → proj1 lt } |
| 265 | 72 |
| 331 | 73 open HOD |
| 74 | |
| 295 | 75 ----- |
| 76 -- | |
| 77 -- ultra filter is prime | |
| 78 -- | |
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79 |
| 329 | 80 filter-lemma1 : {L : HOD} → (P : Filter L) → ∀ {p q : HOD } → ultra-filter P → prime-filter P |
| 295 | 81 filter-lemma1 {L} P u = record { |
| 82 proper = ultra-filter.proper u | |
| 83 ; prime = lemma3 | |
| 84 } where | |
| 329 | 85 lemma3 : {p q : HOD} → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) |
| 295 | 86 lemma3 {p} {q} lt with ultra-filter.ultra u (∪-lemma1 (∈-filter P lt) ) |
| 87 ... | case1 p∈P = case1 p∈P | |
| 88 ... | case2 ¬p∈P = case2 (filter1 P {q ∩ (L ＼ p)} (∪-lemma2 (∈-filter P lt)) lemma7 lemma8) where | |
| 331 | 89 lemma5 : ((p ∪ q ) ∩ (L ＼ p)) =h= (q ∩ (L ＼ p)) |
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90 lemma5 = record { eq→ = λ {x} lt → record { proj1 = lemma4 x lt ; proj2 = proj2 lt } |
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91 ; eq← = λ {x} lt → record { proj1 = case2 (proj1 lt) ; proj2 = proj2 lt } |
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92 } where |
| 331 | 93 lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (L ＼ p)) x → odef q x |
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94 lemma4 x lt with proj1 lt |
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95 lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px ) |
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96 lemma4 x lt | case2 qx = qx |
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97 lemma6 : filter P ∋ ((p ∪ q ) ∩ (L ＼ p)) |
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98 lemma6 = filter2 P lt ¬p∈P |
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99 lemma7 : filter P ∋ (q ∩ (L ＼ p)) |
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100 lemma7 = subst (λ k → filter P ∋ k ) (==→o≡ lemma5 ) lemma6 |
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101 lemma8 : (q ∩ (L ＼ p)) ⊆ q |
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102 lemma8 = q∩q⊆q |
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103 |
| 295 | 104 ----- |
| 105 -- | |
| 106 -- if Filter contains L, prime filter is ultra | |
| 107 -- | |
| 108 | |
| 329 | 109 filter-lemma2 : {L : HOD} → (P : Filter L) → filter P ∋ L → prime-filter P → ultra-filter P |
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110 filter-lemma2 {L} P f∋L prime = record { |
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111 proper = prime-filter.proper prime |
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112 ; ultra = λ {p} p⊆L → prime-filter.prime prime (lemma p p⊆L) |
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113 } where |
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114 open _==_ |
| 331 | 115 p+1-p=1 : {p : HOD} → p ⊆ L → L =h= (p ∪ (L ＼ p)) |
| 116 eq→ (p+1-p=1 {p} p⊆L) {x} lt with ODC.decp O (odef p x) | |
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117 eq→ (p+1-p=1 {p} p⊆L) {x} lt | yes p∋x = case1 p∋x |
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118 eq→ (p+1-p=1 {p} p⊆L) {x} lt | no ¬p = case2 (record { proj1 = lt ; proj2 = ¬p }) |
| 331 | 119 eq← (p+1-p=1 {p} p⊆L) {x} ( case1 p∋x ) = subst (λ k → odef L k ) diso (incl p⊆L ( subst (λ k → odef p k) (sym diso) p∋x )) |
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120 eq← (p+1-p=1 {p} p⊆L) {x} ( case2 ¬p ) = proj1 ¬p |
| 329 | 121 lemma : (p : HOD) → p ⊆ L → filter P ∋ (p ∪ (L ＼ p)) |
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122 lemma p p⊆L = subst (λ k → filter P ∋ k ) (==→o≡ (p+1-p=1 p⊆L)) f∋L |
| 293 | 123 |
| 329 | 124 record Dense (P : HOD ) : Set (suc n) where |
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125 field |
| 329 | 126 dense : HOD |
| 368 | 127 incl : dense ⊆ Power P |
| 329 | 128 dense-f : HOD → HOD |
| 368 | 129 dense-d : { p : HOD} → p ⊆ P → dense ∋ dense-f p |
| 130 dense-p : { p : HOD} → p ⊆ P → p ⊆ (dense-f p) | |
| 266 | 131 |
| 329 | 132 record Ideal ( L : HOD ) : Set (suc n) where |
| 293 | 133 field |
| 329 | 134 ideal : HOD |
| 293 | 135 i⊆PL : ideal ⊆ Power L |
| 329 | 136 ideal1 : { p q : HOD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q |
| 137 ideal2 : { p q : HOD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
| 293 | 138 |
| 139 open Ideal | |
| 140 | |
| 329 | 141 proper-ideal : {L : HOD} → (P : Ideal L ) → {p : HOD} → Set n |
| 293 | 142 proper-ideal {L} P {p} = ideal P ∋ od∅ |
| 143 | |
| 329 | 144 prime-ideal : {L : HOD} → Ideal L → ∀ {p q : HOD } → Set n |
| 293 | 145 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q ) |
| 146 | |
| 364 | 147 ------- |
| 363 | 148 -- the set of finite partial functions from ω to 2 |
| 149 -- | |
| 150 -- | |
| 151 | |
| 152 import OPair | |
| 153 open OPair O | |
| 154 | |
| 155 ODSuc : (y : HOD) → infinite ∋ y → HOD | |
| 156 ODSuc y lt = Union (y , (y , y)) | |
| 157 | |
| 366 | 158 data Hω2 : (i : Nat) ( x : Ordinal ) → Set n where |
| 159 hφ : Hω2 0 o∅ | |
| 160 h0 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 161 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 0 >) , ord→od x ))) | |
| 162 h1 : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 163 Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 1 >) , ord→od x ))) | |
| 164 he : {i : Nat} {x : Ordinal } → Hω2 i x → | |
| 165 Hω2 (Suc i) x | |
| 166 | |
| 167 record Hω2r (x : Ordinal) : Set n where | |
| 168 field | |
| 169 count : Nat | |
| 170 hω2 : Hω2 count x | |
| 171 | |
| 172 open Hω2r | |
| 363 | 173 |
| 174 HODω2 : HOD | |
| 366 | 175 HODω2 = record { od = record { def = λ x → Hω2r x } ; odmax = next o∅ ; <odmax = odmax0 } where |
| 365 | 176 ω<next : {y : Ordinal} → infinite-d y → y o< next o∅ |
| 177 ω<next = ω<next-o∅ ho< | |
| 366 | 178 lemma : {i j : Nat} {x : Ordinal } → od→ord (Union (< nat→ω i , nat→ω j > , ord→od x)) o< next x |
| 179 lemma = {!!} | |
| 180 odmax0 : {y : Ordinal} → Hω2r y → y o< next o∅ | |
| 181 odmax0 {y} r with hω2 r | |
| 182 ... | hφ = x<nx | |
| 183 ... | h0 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {0} {x}) | |
| 184 ... | h1 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {1} {x}) | |
| 185 ... | he {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) x<nx | |
| 363 | 186 |
| 368 | 187 data Two : Set n where |
| 188 i0 : Two | |
| 189 i1 : Two | |
| 190 | |
| 191 record ω2r : Set n where | |
| 192 field | |
| 193 func2 : Nat → Two | |
| 194 ω2 : {!!} | |
| 195 | |
| 196 ω→2 : HOD | |
| 197 ω→2 = Replace infinite (λ x → < x , {!!} > ) | |
| 198 | |
| 199 G : (Nat → Two) → Filter HODω2 | |
| 200 G f = record { | |
| 365 | 201 filter = {!!} |
| 202 ; f⊆PL = {!!} | |
| 203 ; filter1 = {!!} | |
| 204 ; filter2 = {!!} | |
| 205 } where | |
| 206 filter0 : HOD | |
| 207 filter0 = {!!} | |
| 208 f⊆PL1 : filter0 ⊆ Power HODω2 | |
| 209 f⊆PL1 = {!!} | |
| 210 filter11 : { p q : HOD } → q ⊆ HODω2 → filter0 ∋ p → p ⊆ q → filter0 ∋ q | |
| 211 filter11 = {!!} | |
| 212 filter12 : { p q : HOD } → filter0 ∋ p → filter0 ∋ q → filter0 ∋ (p ∩ q) | |
| 213 filter12 = {!!} | |
| 214 | |
| 363 | 215 -- the set of finite partial functions from ω to 2 |
| 216 | |
| 217 Hω2f : Set (suc n) | |
| 218 Hω2f = (Nat → Set n) → Two | |
| 219 | |
| 220 Hω2f→Hω2 : Hω2f → HOD | |
| 221 Hω2f→Hω2 p = record { od = record { def = λ x → (p {!!} ≡ i0 ) ∨ (p {!!} ≡ i1 )}; odmax = {!!} ; <odmax = {!!} } | |
| 222 | |
| 223 |