--- a/filter.agda	Fri Jul 26 21:08:06 2019 +0900
+++ b/filter.agda	Sun Jul 28 14:07:08 2019 +0900
@@ -10,6 +10,8 @@
 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⊔_ ) 
+
 
 record Filter {n : Level} ( P max : OD {suc n} )  : Set (suc (suc n)) where
    field
@@ -24,6 +26,20 @@
 dense :  {n : Level} → Set (suc (suc n))
 dense {n} = { D P p : OD {suc n} } → ({x : OD {suc n}} → P ∋ p → ¬ ( ( q : OD {suc n}) → D ∋ q → od→ord p o< od→ord q ))
 
+record NatFilter {n : Level} ( P : Nat → Set n)  : Set (suc n) where
+   field
+       GN : Nat → Set n
+       GN∋1 : GN 0
+       GNmax : { p : Nat } → P p →  0 ≤ p 
+       GNless : { p q : Nat } → GN p → P q →  q ≤ p  → GN q
+       Gr : (  p q : Nat ) →  GN p → GN q  →  Nat
+       GNcompat : { p q : Nat }  → (gp : GN p) → (gq : GN q ) → (  Gr p q gp gq ≤  p ) ∨ (  Gr p q gp gq ≤ q )
+
+record NatDense {n : Level} ( P : Nat → Set n)  : Set (suc n) where
+   field
+       Gmid : { p : Nat } → P p  → Nat
+       GDense : { D : Nat → Set n } → {x p : Nat } → (pp : P p ) → D (Gmid {p} pp) →  Gmid pp  ≤ p 
+
 open OD.OD
 
 -- H(ω,2) = Power ( Power ω ) = Def ( Def ω))
@@ -36,12 +52,9 @@
 Hω2 : {n : Level} →  OD {suc n}
 Hω2 {n} = record { def = λ x → {dom : Ordinal {suc n}} → x ≡ od→ord ( Pred ( ord→od dom )) }
 
-_⊆_ : {n : Level} → ( A B : OD {suc n}  ) → ∀{ x : OD {suc n} } →  Set (suc n)
-_⊆_ A B {x} = A ∋ x →  B ∋ x
-
 Hω2Filter :   {n : Level} → Filter {n} Hω2 od∅
 Hω2Filter {n} = record {
-       _⊇_ = {!!}
+       _⊇_ = _⊇_
      ; G = {!!}
      ; G∋1 = {!!}
      ; Gmax = {!!}