### view filter.agda @ 287:5de8905a5a2b

...
author Shinji KONO Sun, 07 Jun 2020 20:29:12 +0900 d9d3654baee1 ef93c56ad311
line wrap: on
line source
```
open import Level
open import Ordinals
module filter {n : Level } (O : Ordinals {n})   where

open import zf
open import logic
import OD

open import Relation.Nullary
open import Relation.Binary
open import Data.Empty
open import Relation.Binary
open import Relation.Binary.Core
open import  Relation.Binary.PropositionalEquality
open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ )

open inOrdinal O
open OD O
open OD.OD
open ODAxiom odAxiom

open _∧_
open _∨_
open Bool

_∩_ : ( A B : OD  ) → OD
A ∩ B = record { def = λ x → def A x ∧ def B x }

_∪_ : ( A B : OD  ) → OD
A ∪ B = record { def = λ x → def A x ∨ def B x }

_＼_ : ( A B : OD  ) → OD
A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }

record Filter  ( L : OD  ) : Set (suc n) where
field
filter : OD
proper : ¬ ( filter ∋ od∅ )
inL :  filter ⊆ L
filter1 : { p q : OD } →  q ⊆ L  → filter ∋ p →  p ⊆ q  → filter ∋ q
filter2 : { p q : OD } → filter ∋ p →  filter ∋ q  → filter ∋ (p ∩ q)

open Filter

L⊆L : (L : OD) → L ⊆ L
L⊆L L = record { incl = λ {x} lt → lt }

L-filter : {L : OD} → (P : Filter L ) → {p : OD} → filter P ∋ p → filter P ∋ L
L-filter {L} P {p} lt = filter1 P {p} {L} (L⊆L L) lt {!!}

prime-filter : {L : OD} → Filter L → ∀ {p q : OD } → Set n
prime-filter {L} P {p} {q} =  filter P ∋ ( p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )

ultra-filter :  {L : OD} → Filter L → ∀ {p : OD } → Set n
ultra-filter {L} P {p} = L ∋ p →  ( filter P ∋ p ) ∨ (  filter P ∋ ( L ＼ p) )

filter-lemma1 :  {L : OD} → (P : Filter L)  → ∀ {p q : OD } → ( ∀ (p : OD ) → ultra-filter {L} P {p} ) → prime-filter {L} P {p} {q}
filter-lemma1 {L} P {p} {q} u lt = {!!}

filter-lemma2 :  {L : OD} → (P : Filter L)  → ( ∀ {p q : OD } → prime-filter {L} P {p} {q}) →  ∀ (p : OD ) → ultra-filter {L} P {p}
filter-lemma2 {L} P prime p with prime {!!}
... | t = {!!}

generated-filter : {L : OD} → Filter L → (p : OD ) → Filter ( record { def = λ x → def L x ∨ (x ≡ od→ord p) } )
generated-filter {L} P p = record {
proper = {!!} ;
filter = {!!}  ; inL = {!!} ;
filter1 = {!!} ; filter2 = {!!}
}

record Dense  (P : OD ) : Set (suc n) where
field
dense : OD
incl :  dense ⊆ P
dense-f : OD → OD
dense-p :  { p : OD} → P ∋ p  → p ⊆ (dense-f p)

-- H(ω,2) = Power ( Power ω ) = Def ( Def ω))

infinite = ZF.infinite OD→ZF

module in-countable-ordinal {n : Level} where

import ordinal

-- open  ordinal.C-Ordinal-with-choice
-- both Power and infinite is too ZF, it is better to use simpler one
Hω2 : Filter (Power (Power infinite))
Hω2 = {!!}

```