### changeset 292:773e03dfd6ed

...
author Shinji KONO Sat, 13 Jun 2020 15:59:10 +0900 ef93c56ad311 9972bd4a0d6f filter.agda 1 files changed, 14 insertions(+), 9 deletions(-) [+]
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```--- a/filter.agda	Fri Jun 12 19:21:14 2020 +0900
+++ b/filter.agda	Sat Jun 13 15:59:10 2020 +0900
@@ -33,32 +33,37 @@
A ＼ B = record { def = λ x → def A x ∧ ( ¬ ( def B x ) ) }

+-- Kunen p.76 and p.53
record Filter  ( L : OD  ) : Set (suc n) where
field
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)

+open Filter
+
+proper-filter : {L : OD} → (P : Filter L ) → {p : OD} → Set n
+proper-filter {L} P {p} = filter P ∋ L
+
+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 )
+
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 {!!}
+proper-ideal : {L : OD} → (P : Ideal L ) → {p : OD} → Set n
+proper-ideal {L} P {p} = ideal P ∋ od∅

-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 )
+prime-ideal : {L : OD} → Ideal L → ∀ {p q : OD } → Set n
+prime-ideal {L} P {p} {q} =  ideal P ∋ ( p ∩ q) → ( ideal P ∋ p ) ∨ ( ideal 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) )```