annotate filter.agda @ 293:9972bd4a0d6f

...
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 14 Jun 2020 08:57:14 +0900
parents 773e03dfd6ed
children 4340ffcfa83d
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:
diff changeset
1 open import Level
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
2 open import Ordinals
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
3 module filter {n : Level } (O : Ordinals {n}) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
4
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 open import zf
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
6 open import logic
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
7 import OD
193
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
8
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 open import Relation.Nullary
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 open import Relation.Binary
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 open import Data.Empty
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 open import Relation.Binary
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 open import Relation.Binary.Core
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
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
diff changeset
15 open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ )
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
16 import BAlgbra
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
17
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
18 open BAlgbra O
191
9eb6a8691f02 choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
19
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
20 open inOrdinal O
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
21 open OD O
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
22 open OD.OD
277
d9d3654baee1 seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 276
diff changeset
23 open ODAxiom odAxiom
190
6e778b0a7202 add filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
24
236
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
25 open _∧_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
26 open _∨_
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
27 open Bool
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
28
292
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
29 -- Kunen p.76 and p.53
265
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 236
diff changeset
30 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
diff changeset
31 field
290
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
32 filter : OD
359402cc6c3d definition of filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 277
diff changeset
33 f⊆PL : filter ⊆ Power L
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
diff changeset
34 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
35 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
36
292
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
37 open Filter
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
38
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
39 -- should use inhabit?
292
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
40 proper-filter : {L : OD} → (P : Filter L ) → {p : OD} → Set n
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
41 proper-filter {L} P {p} = ¬ (filter P ∋ od∅)
292
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
42
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
43 prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
44 prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 291
diff changeset
45
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
46 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set (suc n )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
47 ultra-filter {L} P {p} = p ⊆ L → ( filter P ∋ p ) ∨ ( filter P ∋ ( L \ p) )
265
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 236
diff changeset
48
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
49 filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q}
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
50 filter-lemma1 {L} P {p} {q} u lt with lt
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
51 ... | t = {!!}
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
52
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
53 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: 292
diff changeset
54 filter-lemma2 {L} P prime p = {!!}
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
55
267
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 266
diff changeset
56 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } )
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
57 generated-filter {L} P p = {!!}
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
58
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
59 record Dense (P : OD ) : Set (suc n) where
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
60 field
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
61 dense : OD
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
diff changeset
62 incl : dense ⊆ P
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
63 dense-f : OD → OD
271
2169d948159b fix incl
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 270
diff changeset
64 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
65
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
66 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω))
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
67
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
68 infinite = ZF.infinite OD→ZF
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
69
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
70 module in-countable-ordinal {n : Level} where
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
71
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
72 import ordinal
266
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 265
diff changeset
73
276
6f10c47e4e7a separate choice
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 272
diff changeset
74 -- open ordinal.C-Ordinal-with-choice
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
75
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
76 Hω2 : Filter (Power (Power infinite))
270
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 269
diff changeset
77 Hω2 = {!!}
269
30e419a2be24 disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 268
diff changeset
78
293
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
79
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
80 record Ideal ( L : OD ) : Set (suc n) where
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
81 field
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
82 ideal : OD
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
83 i⊆PL : ideal ⊆ Power L
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
84 ideal1 : { p q : OD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
85 ideal2 : { p q : OD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
86
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
87 open Ideal
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
88
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
89 proper-ideal : {L : OD} → (P : Ideal L ) → {p : OD} → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
90 proper-ideal {L} P {p} = ideal P ∋ od∅
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
91
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
92 prime-ideal : {L : OD} → Ideal L → ∀ {p q : OD } → Set n
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
93 prime-ideal {L} P {p} {q} = ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal P ∋ q )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 292
diff changeset
94