annotate filter.agda @ 289:9f926b2210bc release

Added tag current for changeset 4fcac1eebc74
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 07 Jun 2020 20:35:14 +0900
parents 5de8905a5a2b
children ef93c56ad311
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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1 open import Level
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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2 open import Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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3 module filter {n : Level } (O : Ordinals {n}) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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4
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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5 open import zf
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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6 open import logic
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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7 import OD
193
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
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8
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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9 open import Relation.Nullary
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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10 open import Relation.Binary
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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11 open import Data.Empty
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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12 open import Relation.Binary
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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13 open import Relation.Binary.Core
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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14 open import Relation.Binary.PropositionalEquality
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
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15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
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16
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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17 open inOrdinal O
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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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
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
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20 open ODAxiom odAxiom
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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21
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
22 open _∧_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
23 open _∨_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
24 open Bool
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
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25
267
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 266
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26 _∩_ : ( A B : OD ) → OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 266
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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>
parents: 266
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28
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 266
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29 _∪_ : ( A B : OD ) → OD
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
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30 A ∪ B = record { def = λ x → def A x ∨ def B x }
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
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31
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
32 _\_ : ( A B : OD ) → OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
33 A \ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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34
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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35
265
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 236
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36 record Filter ( L : OD ) : Set (suc n) where
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
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37 field
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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38 filter : OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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39 proper : ¬ ( filter ∋ od∅ )
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
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40 inL : filter ⊆ L
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
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41 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
42 filter2 : { p q : OD } → filter ∋ p → filter ∋ q → filter ∋ (p ∩ q)
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
43
265
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 236
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44 open Filter
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
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45
287
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
46 L⊆L : (L : OD) → L ⊆ L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
47 L⊆L L = record { incl = λ {x} lt → lt }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
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48
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
49 L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L
287
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
50 L-filter {L} P {p} lt = filter1 P {p} {L} (L⊆L L) lt {!!}
190
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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51
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
52 prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
53 prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )
190
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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54
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
55 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
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56 ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L \ p) )
190
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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57
265
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 236
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58
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
59 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>
parents: 269
diff changeset
60 filter-lemma1 {L} P {p} {q} u lt = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
61
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
62 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>
parents: 269
diff changeset
63 filter-lemma2 {L} P prime p with prime {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
64 ... | t = {!!}
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
65
267
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 266
diff changeset
66 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } )
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
67 generated-filter {L} P p = record {
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
diff changeset
68 proper = {!!} ;
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
69 filter = {!!} ; inL = {!!} ;
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
70 filter1 = {!!} ; filter2 = {!!}
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
71 }
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
72
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
73 record Dense (P : OD ) : Set (suc n) where
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
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74 field
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
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75 dense : OD
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
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76 incl : dense ⊆ P
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
77 dense-f : OD → OD
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
diff changeset
78 dense-p : { p : OD} → P ∋ p → p ⊆ (dense-f p)
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
79
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
80 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
81
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
82 infinite = ZF.infinite OD→ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
83
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
84 module in-countable-ordinal {n : Level} where
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
85
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
86 import ordinal
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
87
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
88 -- open ordinal.C-Ordinal-with-choice
287
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
89 -- both Power and infinite is too ZF, it is better to use simpler one
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
90 Hω2 : Filter (Power (Power infinite))
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
91 Hω2 = {!!}
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
92