### changeset 290:359402cc6c3d

definition of filter
author Shinji KONO Fri, 12 Jun 2020 19:19:16 +0900 4ae48eed654a ef93c56ad311 OD.agda filter.agda 2 files changed, 28 insertions(+), 8 deletions(-) [+]
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```--- a/OD.agda	Sun May 10 09:25:18 2020 +0900
+++ b/OD.agda	Fri Jun 12 19:19:16 2020 +0900
@@ -53,9 +53,19 @@
eq← ( ⇔→==  {x} {y}  eq ) {z} m = proj2 eq m

-- next assumptions are our axiom
---  it defines a subset of OD, which is called HOD, usually defined as
+--  In classical Set Theory, HOD is used, as a subset of OD,
--     HOD = { x | TC x ⊆ OD }
---  where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x
+--  where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
+--  This is not possible because we don't have V yet.
+--  We simply assume V=OD here.
+--
+--  We also assumes ODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
+--  ODs have an ovbious maximum, but Ordinals are not. This means, od→ord is not an on-to mapping.
+--
+--  ==→o≡ is necessary to prove axiom of extensionality.
+--
+--  In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
+--  we need explict assumption on sup.

record ODAxiom : Set (suc n) where
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )```
```--- a/filter.agda	Sun May 10 09:25:18 2020 +0900
+++ b/filter.agda	Fri Jun 12 19:19:16 2020 +0900
@@ -35,16 +35,27 @@

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

+record Ideal  ( L : OD  ) : Set (suc n) where
+   field
+       ideal : OD
+       i∋∅ : ideal ∋ od∅
+       ¬i∋L : ¬ ( ideal ∋ L )
+       i⊆PL :  ideal ⊆ Power L
+       ideal1 : { p q : OD } →  q ⊆ L  → ideal ∋ p →  q ⊆ p  → ideal ∋ q
+       ideal2 : { p q : OD } → ideal ∋ p →  ideal ∋ q  → ideal ∋ (p ∪ q)
+
open Filter
+open Ideal

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} {!!} lt {!!}
+L-filter {L} P {p} lt = {!!} -- filter1 P {p} {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 )
@@ -62,8 +73,7 @@

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 = {!!} ;
+     filter = {!!}  ;
filter1 = {!!} ; filter2 = {!!}
}
```