Mercurial > hg > Members > kono > Proof > ZF-in-agda
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 |
| 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 | |
|
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⊔_ ) |
|
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
16 |
| 236 | 17 open inOrdinal O |
| 18 open OD O | |
| 19 open OD.OD | |
|
277
d9d3654baee1
seperate choice from LEM
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
276
diff
changeset
|
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 | |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
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
diff
changeset
|
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 |
|
191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
37 field |
| 290 | 38 filter : OD |
| 39 ¬f∋∅ : ¬ ( filter ∋ od∅ ) | |
| 40 f∋L : filter ∋ L | |
| 41 f⊆PL : filter ⊆ Power L | |
| 271 | 42 filter1 : { p q : OD } → q ⊆ L → filter ∋ p → p ⊆ q → filter ∋ q |
| 270 | 43 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
|
44 |
| 290 | 45 record Ideal ( L : OD ) : Set (suc n) where |
| 46 field | |
| 47 ideal : OD | |
| 48 i∋∅ : ideal ∋ od∅ | |
| 49 ¬i∋L : ¬ ( ideal ∋ L ) | |
| 50 i⊆PL : ideal ⊆ Power L | |
| 51 ideal1 : { p q : OD } → q ⊆ L → ideal ∋ p → q ⊆ p → ideal ∋ q | |
| 52 ideal2 : { p q : OD } → ideal ∋ p → ideal ∋ q → ideal ∋ (p ∪ q) | |
| 53 | |
| 265 | 54 open Filter |
| 290 | 55 open Ideal |
|
191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
|
56 |
| 270 | 57 L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L |
| 290 | 58 L-filter {L} P {p} lt = {!!} -- filter1 P {p} {L} {!!} lt {!!} |
| 190 | 59 |
| 270 | 60 prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n |
| 61 prime-filter {L} P {p} {q} = filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q ) | |
| 190 | 62 |
| 270 | 63 ultra-filter : {L : OD} → Filter L → ∀ {p : OD } → Set n |
| 64 ultra-filter {L} P {p} = L ∋ p → ( filter P ∋ p ) ∨ ( filter P ∋ ( L ＼ p) ) | |
| 190 | 65 |
| 265 | 66 |
| 270 | 67 filter-lemma1 : {L : OD} → (P : Filter L) → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} |
| 68 filter-lemma1 {L} P {p} {q} u lt = {!!} | |
| 69 | |
| 70 filter-lemma2 : {L : OD} → (P : Filter L) → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) → ∀ (p : OD ) → ultra-filter {L} P {p} | |
| 71 filter-lemma2 {L} P prime p with prime {!!} | |
| 72 ... | t = {!!} | |
| 266 | 73 |
| 267 | 74 generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } ) |
| 266 | 75 generated-filter {L} P p = record { |
| 290 | 76 filter = {!!} ; |
| 270 | 77 filter1 = {!!} ; filter2 = {!!} |
| 266 | 78 } |
| 79 | |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
80 record Dense (P : OD ) : Set (suc n) where |
|
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
81 field |
|
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
82 dense : OD |
| 271 | 83 incl : dense ⊆ P |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
84 dense-f : OD → OD |
| 271 | 85 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
|
86 |
| 266 | 87 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) |
| 88 | |
| 89 infinite = ZF.infinite OD→ZF | |
| 90 | |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
91 module in-countable-ordinal {n : Level} where |
|
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
92 |
|
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
93 import ordinal |
| 266 | 94 |
| 276 | 95 -- open ordinal.C-Ordinal-with-choice |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
96 |
|
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
97 Hω2 : Filter (Power (Power infinite)) |
| 270 | 98 Hω2 = {!!} |
|
269
30e419a2be24
disjunction and conjunction
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
268
diff
changeset
|
99 |