Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 292:773e03dfd6ed
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 13 Jun 2020 15:59:10 +0900 |
| parents | ef93c56ad311 |
| children | 9972bd4a0d6f |
| 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 |
| 190 | 9 open import Relation.Nullary |
| 10 open import Relation.Binary | |
| 11 open import Data.Empty | |
| 12 open import Relation.Binary | |
| 13 open import Relation.Binary.Core | |
| 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⊔_ ) |
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16 |
| 236 | 17 open inOrdinal O |
| 18 open OD O | |
| 19 open OD.OD | |
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20 open ODAxiom odAxiom |
| 190 | 21 |
| 236 | 22 open _∧_ |
| 23 open _∨_ | |
| 24 open Bool | |
| 25 | |
| 267 | 26 _∩_ : ( A B : OD ) → OD |
| 27 A ∩ B = record { def = λ x → def A x ∧ def B x } | |
| 28 | |
| 29 _∪_ : ( A B : OD ) → OD | |
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30 A ∪ B = record { def = λ x → def A x ∨ def B x } |
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31 |
| 270 | 32 _＼_ : ( A B : OD ) → OD |
| 33 A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) } | |
| 34 | |
| 35 | |
| 292 | 36 -- Kunen p.76 and p.53 |
| 265 | 37 record Filter ( L : OD ) : Set (suc n) where |
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38 field |
| 290 | 39 filter : OD |
| 40 f⊆PL : filter ⊆ Power L | |
| 271 | 41 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
| 270 | 42 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q) |
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43 |
| 292 | 44 open Filter |
| 45 | |
| 46 proper-filter : {L : OD} → (P : Filter L ) → {p : OD} → Set n | |
| 47 proper-filter {L} P {p} = filter P ∋ L | |
| 48 | |
| 49 prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n | |
| 50 prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) | |
| 51 | |
| 290 | 52 record Ideal ( L : OD ) : Set (suc n) where |
| 53 field | |
| 54 ideal : OD | |
| 55 i⊆PL : ideal ⊆ Power L | |
| 56 ideal1 : { p q : OD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q | |
| 57 ideal2 : { p q : OD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
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58 |
| 290 | 59 open Ideal |
| 287 | 60 |
| 292 | 61 proper-ideal : {L : OD} → (P : Ideal L ) → {p : OD} → Set n |
| 62 proper-ideal {L} P {p} = ideal P ∋ od∅ | |
| 190 | 63 |
| 292 | 64 prime-ideal : {L : OD} → Ideal L → ∀ {p q : OD } → Set n |
| 65 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q ) | |
| 66 | |
| 190 | 67 |
| 270 | 68 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n |
| 69 ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) | |
| 190 | 70 |
| 265 | 71 |
| 270 | 72 filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} |
| 73 filter-lemma1 {L} P {p} {q} u lt = {!!} | |
| 74 | |
| 75 filter-lemma2 : {L : OD} → (P : Filter L) → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) → ∀ (p : OD ) → ultra-filter {L} P {p} | |
| 76 filter-lemma2 {L} P prime p with prime {!!} | |
| 77 ... | t = {!!} | |
| 266 | 78 |
| 267 | 79 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } ) |
| 266 | 80 generated-filter {L} P p = record { |
| 290 | 81 filter = {!!} ; |
| 270 | 82 filter1 = {!!} ; filter2 = {!!} |
| 266 | 83 } |
| 84 | |
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85 record Dense (P : OD ) : Set (suc n) where |
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86 field |
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87 dense : OD |
| 271 | 88 incl : dense ⊆ P |
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89 dense-f : OD → OD |
| 271 | 90 dense-p : { p : OD} → P ∋ p → p ⊆ (dense-f p) |
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91 |
| 266 | 92 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) |
| 93 | |
| 94 infinite = ZF.infinite OD→ZF | |
| 95 | |
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96 module in-countable-ordinal {n : Level} where |
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97 |
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98 import ordinal |
| 266 | 99 |
| 276 | 100 -- open ordinal.C-Ordinal-with-choice |
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101 |
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102 Hω2 : Filter (Power (Power infinite)) |
| 270 | 103 Hω2 = {!!} |
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104 |