Mercurial > hg > Members > kono > Proof > ZF-in-agda
annotate filter.agda @ 266:0d7d6e4da36f
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 30 Sep 2019 17:07:40 +0900 |
| parents | 9bf100ae50ac |
| children | e469de3ae7cc |
| 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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9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
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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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9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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changeset
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16 |
| 236 | 17 open inOrdinal O |
| 18 open OD O | |
| 19 open OD.OD | |
| 190 | 20 |
| 236 | 21 open _∧_ |
| 22 open _∨_ | |
| 23 open Bool | |
| 24 | |
| 265 | 25 record Filter ( L : OD ) : Set (suc n) where |
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choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
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26 field |
| 266 | 27 F1 : { p q : Ordinal } → def L p → p o< osuc q → def L q |
| 28 F2 : { p q : Ordinal } → def L p → def L q → def L (minα p q) | |
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191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
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29 |
| 265 | 30 open Filter |
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191
9eb6a8691f02
choice function cannot jump between ordinal level
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
190
diff
changeset
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31 |
| 265 | 32 proper-filter : {L : OD} → Filter L → Set n |
| 266 | 33 proper-filter {L} P = ¬ ( def L o∅ ) |
| 190 | 34 |
| 266 | 35 prime-filter : {L : OD} → Filter L → {p q : Ordinal } → Set n |
| 36 prime-filter {L} P {p} {q} = def L ( maxα p q) → ( def L p ) ∨ ( def L q ) | |
| 190 | 37 |
| 266 | 38 ultra-filter : {L : OD} → Filter L → {p : Ordinal } → Set n |
| 39 ultra-filter {L} P {p} = ( def L p ) ∨ ( ¬ ( def L p )) | |
| 190 | 40 |
| 265 | 41 postulate |
| 42 dist-ord : {p q r : Ordinal } → minα p ( maxα q r ) ≡ maxα ( minα p q ) ( minα p r ) | |
| 43 | |
| 266 | 44 filter-lemma1 : {L : OD} → (P : Filter L) → {p q : Ordinal } → ( (p : Ordinal ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q} |
| 45 filter-lemma1 {L} P {p} {q} u lt with u p | u q | |
| 46 filter-lemma1 {L} P {p} {q} u lt | case1 x | case1 y = case1 x | |
| 47 filter-lemma1 {L} P {p} {q} u lt | case1 x | case2 y = case1 x | |
| 48 filter-lemma1 {L} P {p} {q} u lt | case2 x | case1 y = case2 y | |
| 49 filter-lemma1 {L} P {p} {q} u lt | case2 x | case2 y with trio< p q | |
| 50 filter-lemma1 {L} P {p} {q} u lt | case2 x | case2 y | tri< a ¬b ¬c = ⊥-elim ( y lt ) | |
| 51 filter-lemma1 {L} P {p} {q} u lt | case2 x | case2 y | tri≈ ¬a refl ¬c = ⊥-elim ( y lt ) | |
| 52 filter-lemma1 {L} P {p} {q} u lt | case2 x | case2 y | tri> ¬a ¬b c = ⊥-elim ( x lt ) | |
| 53 | |
| 54 generated-filter : {L : OD} → Filter L → (p : Ordinal ) → Filter ( record { def = λ x → def L x ∨ (x ≡ p) } ) | |
| 55 generated-filter {L} P p = record { | |
| 56 F1 = {!!} ; F2 = {!!} | |
| 57 } | |
| 58 | |
| 59 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω)) | |
| 60 | |
| 61 infinite = ZF.infinite OD→ZF | |
| 62 | |
| 63 Hω2 : Filter (Power (Power infinite)) | |
| 64 Hω2 = {!!} | |
| 65 |