annotate filter.agda @ 290:359402cc6c3d

definition of filter
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 12 Jun 2020 19:19:16 +0900
parents d9d3654baee1
children ef93c56ad311
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 open import Level
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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2 open import Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 module filter {n : Level } (O : Ordinals {n}) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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4
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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5 open import zf
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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6 open import logic
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 import OD
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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8
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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9 open import Relation.Nullary
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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10 open import Relation.Binary
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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11 open import Data.Empty
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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12 open import Relation.Binary
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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13 open import Relation.Binary.Core
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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14 open import Relation.Binary.PropositionalEquality
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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17 open inOrdinal O
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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18 open OD O
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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19 open OD.OD
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d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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20 open ODAxiom odAxiom
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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21
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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22 open _∧_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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23 open _∨_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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24 open Bool
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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25
267
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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26 _∩_ : ( A B : OD ) → OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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27 A ∩ B = record { def = λ x → def A x ∧ def B x }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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28
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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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
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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32 _\_ : ( A B : OD ) → OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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33 A \ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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34
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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35
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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36 record Filter ( L : OD ) : Set (suc n) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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37 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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38 filter : OD
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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39 ¬f∋∅ : ¬ ( filter ∋ od∅ )
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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40 f∋L : filter ∋ L
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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41 f⊆PL : filter ⊆ Power L
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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42 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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43 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
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44
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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45 record Ideal ( L : OD ) : Set (suc n) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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46 field
359402cc6c3d definition of filter
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47 ideal : OD
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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48 i∋∅ : ideal ∋ od∅
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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49 ¬i∋L : ¬ ( ideal ∋ L )
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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50 i⊆PL : ideal ⊆ Power L
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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51 ideal1 : { p q : OD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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52 ideal2 : { p q : OD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q)
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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53
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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54 open Filter
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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55 open Ideal
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56
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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57 L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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58 L-filter {L} P {p} lt = {!!} -- filter1 P {p} {L} {!!} lt {!!}
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59
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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60 prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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61 prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )
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62
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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63 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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64 ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L \ p) )
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65
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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66
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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67 filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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68 filter-lemma1 {L} P {p} {q} u lt = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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69
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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70 filter-lemma2 : {L : OD} → (P : Filter L) → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) → ∀ (p : OD ) → ultra-filter {L} P {p}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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71 filter-lemma2 {L} P prime p with prime {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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72 ... | t = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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73
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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74 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } )
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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75 generated-filter {L} P p = record {
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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76 filter = {!!} ;
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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77 filter1 = {!!} ; filter2 = {!!}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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78 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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79
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80 record Dense (P : OD ) : Set (suc n) where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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81 field
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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82 dense : OD
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83 incl : dense ⊆ P
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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84 dense-f : OD → OD
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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85 dense-p : { p : OD} → P ∋ p → p ⊆ (dense-f p)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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86
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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87 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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88
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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89 infinite = ZF.infinite OD→ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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90
269
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
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91 module in-countable-ordinal {n : Level} where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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92
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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93 import ordinal
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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94
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6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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95 -- open ordinal.C-Ordinal-with-choice
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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96
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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97 Hω2 : Filter (Power (Power infinite))
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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98 Hω2 = {!!}
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99