view filter.agda @ 372:8c3b59f583f2

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author Shinji KONO Sun, 19 Jul 2020 19:14:12 +0900 e75402b1f7d4 b4a247f9d940
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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⊔_ )
import BAlgbra

open BAlgbra O

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

import ODC

open _∧_
open _∨_
open Bool

-- Kunen p.76 and p.53, we use ⊆
record Filter  ( L : HOD  ) : Set (suc n) where
field
filter : HOD
f⊆PL :  filter ⊆ Power L
filter1 : { p q : HOD } →  q ⊆ L  → filter ∋ p →  p ⊆ q  → filter ∋ q
filter2 : { p q : HOD } → filter ∋ p →  filter ∋ q  → filter ∋ (p ∩ q)

open Filter

record prime-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where
field
proper  : ¬ (filter P ∋ od∅)
prime   : {p q : HOD } →  filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )

record ultra-filter { L : HOD } (P : Filter L) : Set (suc (suc n)) where
field
proper  : ¬ (filter P ∋ od∅)
ultra   : {p : HOD } → p ⊆ L →  ( filter P ∋ p ) ∨ (  filter P ∋ ( L ＼ p) )

open _⊆_

trans-⊆ :  { A B C : HOD} → A ⊆ B → B ⊆ C → A ⊆ C
trans-⊆ A⊆B B⊆C = record { incl = λ x → incl B⊆C (incl A⊆B x) }

power→⊆ :  ( A t : HOD) → Power A ∋ t → t ⊆ A
power→⊆ A t  PA∋t = record { incl = λ {x} t∋x → ODC.double-neg-eilm O (t1 t∋x) } where
t1 : {x : HOD }  → t ∋ x → ¬ ¬ (A ∋ x)
t1 = zf.IsZF.power→ isZF A t PA∋t

∈-filter : {L p : HOD} → (P : Filter L ) → filter P ∋ p → p ⊆ L
∈-filter {L} {p} P lt = power→⊆ L p ( incl (f⊆PL P) lt )

∪-lemma1 : {L p q : HOD } → (p ∪ q)  ⊆ L → p ⊆ L
∪-lemma1 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case1 p∋x) }

∪-lemma2 : {L p q : HOD } → (p ∪ q)  ⊆ L → q ⊆ L
∪-lemma2 {L} {p} {q} lt = record { incl = λ {x} p∋x → incl lt (case2 p∋x) }

q∩q⊆q : {p q : HOD } → (q ∩ p) ⊆ q
q∩q⊆q = record { incl = λ lt → proj1 lt }

open HOD

-----
--
--  ultra filter is prime
--

filter-lemma1 :  {L : HOD} → (P : Filter L)  → ∀ {p q : HOD } → ultra-filter P  → prime-filter P
filter-lemma1 {L} P u = record {
proper = ultra-filter.proper u
; prime = lemma3
} where
lemma3 : {p q : HOD} → filter P ∋ (p ∪ q) → ( filter P ∋ p ) ∨ ( filter P ∋ q )
lemma3 {p} {q} lt with ultra-filter.ultra u (∪-lemma1 (∈-filter P lt) )
... | case1 p∈P  = case1 p∈P
... | case2 ¬p∈P = case2 (filter1 P {q ∩ (L ＼ p)} (∪-lemma2 (∈-filter P lt)) lemma7 lemma8) where
lemma5 : ((p ∪ q ) ∩ (L ＼ p)) =h=  (q ∩ (L ＼ p))
lemma5 = record { eq→ = λ {x} lt → record { proj1 = lemma4 x lt ; proj2 = proj2 lt  }
;  eq← = λ {x} lt → record { proj1 = case2 (proj1 lt) ; proj2 = proj2 lt }
} where
lemma4 : (x : Ordinal ) → odef ((p ∪ q) ∩ (L ＼ p)) x → odef q x
lemma4 x lt with proj1 lt
lemma4 x lt | case1 px = ⊥-elim ( proj2 (proj2 lt) px )
lemma4 x lt | case2 qx = qx
lemma6 : filter P ∋ ((p ∪ q ) ∩ (L ＼ p))
lemma6 = filter2 P lt ¬p∈P
lemma7 : filter P ∋ (q ∩ (L ＼ p))
lemma7 =  subst (λ k → filter P ∋ k ) (==→o≡ lemma5 ) lemma6
lemma8 : (q ∩ (L ＼ p)) ⊆ q
lemma8 = q∩q⊆q

-----
--
--  if Filter contains L, prime filter is ultra
--

filter-lemma2 :  {L : HOD} → (P : Filter L)  → filter P ∋ L → prime-filter P → ultra-filter P
filter-lemma2 {L} P f∋L prime = record {
proper = prime-filter.proper prime
; ultra = λ {p}  p⊆L → prime-filter.prime prime (lemma p  p⊆L)
} where
open _==_
p+1-p=1 : {p : HOD} → p ⊆ L → L =h= (p ∪ (L ＼ p))
eq→ (p+1-p=1 {p} p⊆L) {x} lt with ODC.decp O (odef p x)
eq→ (p+1-p=1 {p} p⊆L) {x} lt | yes p∋x = case1 p∋x
eq→ (p+1-p=1 {p} p⊆L) {x} lt | no ¬p = case2 (record { proj1 = lt ; proj2 = ¬p })
eq← (p+1-p=1 {p} p⊆L) {x} ( case1 p∋x ) = subst (λ k → odef L k ) diso (incl p⊆L ( subst (λ k → odef p k) (sym diso) p∋x  ))
eq← (p+1-p=1 {p} p⊆L) {x} ( case2 ¬p  ) = proj1 ¬p
lemma : (p : HOD) → p ⊆ L   →  filter P ∋ (p ∪ (L ＼ p))
lemma p p⊆L = subst (λ k → filter P ∋ k ) (==→o≡ (p+1-p=1 p⊆L)) f∋L

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

record Ideal  ( L : HOD  ) : Set (suc n) where
field
ideal : HOD
i⊆PL :  ideal ⊆ Power L
ideal1 : { p q : HOD } →  q ⊆ L  → ideal ∋ p →  q ⊆ p  → ideal ∋ q
ideal2 : { p q : HOD } → ideal ∋ p →  ideal ∋ q  → ideal ∋ (p ∪ q)

open Ideal

proper-ideal : {L : HOD} → (P : Ideal L ) → {p : HOD} → Set n
proper-ideal {L} P {p} = ideal P ∋ od∅

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

-------
--    the set of finite partial functions from ω to 2
--
--

import OPair
open OPair O

data Two : Set n where
i0 : Two
i1 : Two

record PFunc : Set (suc n) where
field
restrict : Nat → Set n
map : (x : Nat ) → restrict x → Two

open PFunc

record _f⊆_ (f g : PFunc) : Set (suc n) where
field
extend : (x : Nat) → (fr : restrict f x ) →  restrict g x
feq : (x : Nat) → (fr : restrict f x ) →  map f x fr ≡ map g x (extend x fr)

open _f⊆_

_f∩_ : (f g : PFunc) → PFunc
f f∩ g = record { restrict = λ x → (restrict f x ) ∧ (restrict g x ) ∧ ((fr : restrict f x ) → (gr : restrict g x ) → map f x fr ≡ map g x gr)
; map = λ x p →  map f x (proj1 p) }

_↑_ : (Nat → Two) → Nat → PFunc
f ↑ i = record { restrict = λ x → Lift n (x ≤ i) ; map = λ x _ → f x }

record Gf (f : Nat → Two) (p : PFunc ) : Set (suc n) where
field
gn  : Nat
f<n :  p f⊆ (f ↑ gn)

open Gf

record F-Filter (L : Set (suc n)) (PL : (L → Set (suc n)) → Set n) ( _⊆_ : L  → L → Set (suc n))  (_∩_ : L → L → L ) : Set (suc (suc n)) where
field
filter : L → Set (suc n)
f⊆P :  PL filter
filter1 : { p q : L } → filter p →  p ⊆ q  → filter q
filter2 : { p q : L } → filter p →  filter q  → filter (p ∩ q)

GF : (Nat → Two ) → F-Filter PFunc (λ x → One) _f⊆_ _f∩_
GF f = record {
filter = λ p → Gf f p
; f⊆P = OneObj
; filter1 = λ {p} {q} fp p⊆q → record { gn = gn fp ; f<n = f1 fp p⊆q }
; filter2 = λ {p} {q} fp fq  → record { gn = {!!} ; f<n = {!!} }
} where
f1 : {p q : PFunc } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → q f⊆ (f ↑ gn fp)
f1 {p} {q} fp p⊆q = record { extend = λ x fr → extend (f<n fp) x {!!}  ; feq = λ x fr → {!!} } where
f2 : (x : Nat) →  x ≤ gn fp
f2 x = ? -- extend (f<n fp) x  ?

ODSuc : (y : HOD) → infinite ∋ y → HOD
ODSuc y lt = Union (y , (y , y))

data Hω2 :  (i : Nat) ( x : Ordinal  ) → Set n where
hφ :  Hω2 0 o∅
h0 : {i : Nat} {x : Ordinal  } → Hω2 i x  →
Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 0 >) ,  ord→od x )))
h1 : {i : Nat} {x : Ordinal  } → Hω2 i x  →
Hω2 (Suc i) (od→ord (Union ((< nat→ω i , nat→ω 1 >) ,  ord→od x )))
he : {i : Nat} {x : Ordinal  } → Hω2 i x  →
Hω2 (Suc i) x

record  Hω2r (x : Ordinal) : Set n where
field
count : Nat
hω2 : Hω2 count x

open Hω2r

HODω2 :  HOD
HODω2 = record { od = record { def = λ x → Hω2r x } ; odmax = next o∅ ; <odmax = odmax0 } where
ω<next : {y : Ordinal} → infinite-d y → y o< next o∅
ω<next = ω<next-o∅ ho<
lemma : {i j : Nat} {x : Ordinal } → od→ord (Union (< nat→ω i , nat→ω j > , ord→od x)) o< next x
lemma = {!!}
odmax0 :  {y : Ordinal} → Hω2r y → y o< next o∅
odmax0 {y} r with hω2 r
... | hφ = x<nx
... | h0 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {0} {x})
... | h1 {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) (lemma {i} {1} {x})
... | he {i} {x} t = next< (odmax0 record { count = i ; hω2 = t }) x<nx

ω→2 : HOD
ω→2 = Replace (Power infinite) (λ p p⊆i → Replace infinite (λ x i∋x → < x , repl p x > )) where
repl : HOD → HOD → HOD
repl p x with ODC.∋-p O p x
... | yes _  = nat→ω 1
... | no _  = nat→ω 0

record _↑n (f : HOD) (ω→2∋f : ω→2 ∋ f ) : Set n where
-- field
-- n : HOD
-- ? : Select f (λ x f∋x → ω→nat (π1 f∋x) < ω→nat n

-- Gf : {f : HOD} → ω→2 ∋ f  → HOD
-- Gf {f} lt = Select HODω2 (λ x H∋x → {!!}   )

G : (Nat → Two) → Filter HODω2
G f = record {
filter = {!!}
; f⊆PL = {!!}
; filter1 = {!!}
; filter2 = {!!}
} where
filter0 : HOD
filter0 = {!!}
f⊆PL1 :  filter0 ⊆ Power HODω2
f⊆PL1 = {!!}
filter11 : { p q : HOD } →  q ⊆ HODω2  → filter0 ∋ p →  p ⊆ q  → filter0 ∋ q
filter11 = {!!}
filter12 : { p q : HOD } → filter0 ∋ p →  filter0 ∋ q  → filter0 ∋ (p ∩ q)
filter12 = {!!}

-- the set of finite partial functions from ω to 2

Hω2f : Set (suc n)
Hω2f = (Nat → Set n) → Two

Hω2f→Hω2 : Hω2f  → HOD
Hω2f→Hω2 p = record { od = record { def = λ x → (p {!!} ≡ i0 ) ∨ (p {!!} ≡ i1 )}; odmax = {!!} ; <odmax = {!!} }

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