Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 277:d9d3654baee1
seperate choice from LEM
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 09 May 2020 09:38:21 +0900 |
| parents | 6f10c47e4e7a |
| children | 5de8905a5a2b 359402cc6c3d |
| rev | line source |
|---|---|
| 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 | |
| 265 | 36 record Filter ( L : OD ) : Set (suc n) where |
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37 field |
| 270 | 38 filter : OD |
| 39 proper : ¬ ( filter ∋ od∅ ) | |
| 271 | 40 inL : filter ⊆ L |
| 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 |
| 265 | 44 open Filter |
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45 |
| 270 | 46 L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L |
| 47 L-filter {L} P {p} lt = filter1 P {p} {L} {!!} lt {!!} | |
| 190 | 48 |
| 270 | 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 ) | |
| 190 | 51 |
| 270 | 52 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n |
| 53 ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) | |
| 190 | 54 |
| 265 | 55 |
| 270 | 56 filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} |
| 57 filter-lemma1 {L} P {p} {q} u lt = {!!} | |
| 58 | |
| 59 filter-lemma2 : {L : OD} → (P : Filter L) → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) → ∀ (p : OD ) → ultra-filter {L} P {p} | |
| 60 filter-lemma2 {L} P prime p with prime {!!} | |
| 61 ... | t = {!!} | |
| 266 | 62 |
| 267 | 63 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } ) |
| 266 | 64 generated-filter {L} P p = record { |
| 271 | 65 proper = {!!} ; |
| 270 | 66 filter = {!!} ; inL = {!!} ; |
| 67 filter1 = {!!} ; filter2 = {!!} | |
| 266 | 68 } |
| 69 | |
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70 record Dense (P : OD ) : Set (suc n) where |
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71 field |
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72 dense : OD |
| 271 | 73 incl : dense ⊆ P |
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74 dense-f : OD → OD |
| 271 | 75 dense-p : { p : OD} → P ∋ p → p ⊆ (dense-f p) |
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76 |
| 266 | 77 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) |
| 78 | |
| 79 infinite = ZF.infinite OD→ZF | |
| 80 | |
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81 module in-countable-ordinal {n : Level} where |
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82 |
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83 import ordinal |
| 266 | 84 |
| 276 | 85 -- open ordinal.C-Ordinal-with-choice |
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86 |
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87 Hω2 : Filter (Power (Power infinite)) |
| 270 | 88 Hω2 = {!!} |
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89 |