open import Level
open import Ordinals
module generic-filter {n : Level } (O : Ordinals {n})   where

import filter 
open import zf
open import logic
open import partfunc {n} O
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 filter O

open _∧_
open _∨_
open Bool


open HOD

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

open import Data.List 
open import Data.Maybe 

import OPair
open OPair O

open PFunc

_f∩_ : (f g : PFunc (Lift n Nat) (Lift n Two) ) →  PFunc (Lift n Nat) (Lift n Two)
f f∩ g = record { dom = λ x → (dom f x ) ∧ (dom g x ) ∧ ((fr : dom f x ) → (gr : dom g x ) → pmap f x fr ≡ pmap g x gr)
              ; pmap = λ x p →  pmap f x (proj1  p) ; meq = meq f }

_↑_ :  (Nat → Two) → Nat →  PFunc (Lift n Nat) (Lift n Two)
_↑_  f i = record { dom = λ x → Lift n (lower x ≤ i) ; pmap = λ x _ → lift (f (lower x)) ; meq = λ {x} {p} {q} → refl }

record _f⊆_ (f g : PFunc (Lift n Nat) (Lift n Two)  ) : Set (suc n) where
  field
     extend : {x : Nat} → (fr : dom f (lift x) ) →  dom g (lift x  )
     feq : {x : Nat} → {fr : dom f (lift x) } →  pmap f (lift x) fr ≡ pmap g (lift x) (extend fr)

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

record FiniteF (p :  PFunc (Lift n Nat) (Lift n Two) ) : Set (suc n) where
   field
     f-max :  Nat 
     f-func :  Nat → Two
     f-p⊆f :  p f⊆ (f-func ↑ f-max)
     f-f⊆p :  (f-func ↑ f-max) f⊆ p 

open FiniteF


-- Dense-Gf : {n : Level} → F-Dense (PFunc {n}) (λ x → Lift (suc n) (One {n})) _f⊆_ _f∩_
-- Dense-Gf = record {
--        dense =  λ x → FiniteF x
--     ;  d⊆P = lift OneObj
--     ;  dense-f = λ x → record { dom = {!!} ; pmap = {!!} }
--     ;  dense-d = λ {p} d → {!!}
--     ;  dense-p = λ {p} d → {!!}
--   }

open Gf
open _f⊆_ 
open import Data.Nat.Properties

GF :  (Nat → Two ) → F-Filter (PFunc (Lift n Nat) (Lift n Two)) (λ x → Lift (suc n) (One {n})  ) _f⊆_ _f∩_
GF  f = record {  
       filter = λ p → Gf f p
     ; f⊆P = lift 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 = min (gn fp) (gn fq) ; f<n = f2 fp fq }
 } where
     f1 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf f p ) → ( p⊆q : p f⊆ q ) → (f ↑ gn fp) f⊆ q
     f1 {p} {q} fp p⊆q = record { extend = λ {x} x<g → extend p⊆q  (extend (f<n fp )  x<g) ; feq = λ {x} {fr} → lemma {x} {fr}  } where
         lemma : {x : Nat} {fr : Lift n (x ≤ gn fp)} → pmap (f ↑ gn fp) (lift x) fr ≡ pmap q (lift x) (extend p⊆q (extend (f<n fp) fr))
         lemma {x} {fr} = begin
             pmap (f ↑ gn fp) (lift x) fr
            ≡⟨ feq (f<n fp )  ⟩
             pmap p (lift x) (extend (f<n fp)  fr)
            ≡⟨ feq p⊆q  ⟩
             pmap q (lift x) (extend p⊆q  (extend (f<n fp)  fr))
            ∎  where open ≡-Reasoning 
     f2 : {p q : PFunc (Lift n Nat) (Lift n Two) } → (fp : Gf  f p ) → (fq : Gf  f q ) → (f ↑ (min (gn fp) (gn fq)))  f⊆ (p f∩ q)
     f2 {p} {q} fp fq  = record { extend = λ  {x} x<g → lemma2 x<g ; feq = λ {x} {fr} → lemma3 fr } where
            fmin : PFunc (Lift n Nat) (Lift n Two)
            fmin =  f ↑  (min (gn fp) (gn fq)) 
            lemma1 : {x : Nat}  → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → (fr : dom p (lift x)) (gr : dom q (lift x)) → pmap p (lift x) fr ≡ pmap q (lift x) gr
            lemma1 {x} x<g fr gr = begin
                   pmap p (lift x) fr 
                ≡⟨ meq p ⟩
                   pmap p (lift x) (extend (f<n fp) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))))
                ≡⟨ sym (feq (f<n fp)) ⟩
                   pmap (f ↑  (min (gn fp) (gn fq))) (lift x) x<g
                ≡⟨ feq (f<n fq) ⟩
                   pmap q (lift x) (extend (f<n fq) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))))
                ≡⟨ meq q ⟩
                   pmap q (lift x) gr
                ∎  where open ≡-Reasoning 
            lemma2  : {x : Nat}  → ( x<g : Lift n (x ≤ min (gn fp) (gn fq)) ) → dom (p f∩ q) (lift x)
            lemma2 x<g = record { proj1 = extend (f<n fp ) (lift ( ≤-trans (lower x<g) (m⊓n≤m _ _))) ;
                     proj2 = record {proj1 = extend (f<n fq ) (lift ( ≤-trans (lower x<g) (m⊓n≤n _ _))) ; proj2 = lemma1 x<g }}
            f∩→⊆ : (p q : PFunc (Lift n Nat) (Lift n Two) ) → (p f∩ q ) f⊆ q
            f∩→⊆ p q = record {
                   extend = λ {x} pq → proj1 (proj2 pq)
                 ; feq = λ {x} {fr} → (proj2 (proj2 fr)) (proj1 fr) (proj1 (proj2 fr)) 
                }
            lemma3 :  {x : Nat}  → ( fr : Lift n (x ≤ min (gn fp) (gn fq)))  → pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr ≡ pmap (p f∩ q) (lift x) (lemma2 fr)
            lemma3 {x} fr = 
                  pmap (f ↑ min (gn fp) (gn fq)) (lift x) fr
                ≡⟨ feq (f<n fq) ⟩
                  pmap q (lift x) (extend (f<n fq) ( lift (≤-trans (lower fr) (m⊓n≤n _ _)) ))
                ≡⟨ sym (feq (f∩→⊆ p q ) {x} {lemma2 fr} )   ⟩
                  pmap (p f∩ q) (lift x) (lemma2 fr)
                ∎  where open ≡-Reasoning 


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

3→Hω2 : List (Maybe Two) → HOD
3→Hω2 t = list→hod t 0 where
   list→hod : List (Maybe Two) → Nat → HOD
   list→hod [] _ = od∅
   list→hod (just i0 ∷ t) i = Union (< nat→ω i , nat→ω 0 > , ( list→hod t (Suc i) )) 
   list→hod (just i1 ∷ t) i = Union (< nat→ω i , nat→ω 1 > , ( list→hod t (Suc i) )) 
   list→hod (nothing ∷ t) i = list→hod t (Suc i ) 

Hω2→3 : (x :  HOD) → HODω2 ∋ x → List (Maybe Two) 
Hω2→3 x = lemma where
   lemma : { y : Ordinal } →  Hω2r y → List (Maybe Two)
   lemma record { count = 0 ; hω2 = hφ } = []
   lemma record { count = (Suc i) ; hω2 = (h0 hω3) } = just i0 ∷ lemma record { count = i ; hω2 =  hω3 }
   lemma record { count = (Suc i) ; hω2 = (h1 hω3) } = just i1 ∷ lemma record { count = i ; hω2 =  hω3 }
   lemma record { count = (Suc i) ; hω2 = (he hω3) } = nothing ∷ lemma record { count = i ; hω2 =  hω3 }

ω→2 : HOD
ω→2 = Replace (Power infinite) (λ p  → Replace infinite (λ 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

ω→2f : (x : HOD) → ω→2 ∋ x → Nat → Two
ω→2f x = {!!}

↑n : (f n : HOD) → ((ω→2 ∋ f ) ∧ (infinite ∋ n)) → HOD
↑n f n lt = 3→Hω2 ( ω→2f f (proj1 lt) 3↑ (ω→nat n (proj2 lt) ))

record Gfo (x : Ordinal) : Set n where
  field
     gfunc : Ordinal
     gmax : Ordinal
     gcond : (odef ω→2  gfunc) ∧ (odef infinite gmax)
     gfdef : {!!} -- ( ↑n (ord→od gfunc) (ord→od gmax) (subst₂ ? ? ? gcond) )  ⊆ ord→od x 
     pcond : odef HODω2 x

open Gfo

HODGf : HOD
HODGf = record { od = record { def = λ x → Gfo x } ; odmax = next o∅ ; <odmax = {!!} }

G : (Nat → Two) → Filter HODω2
G f = record {
       filter = HODGf
     ; 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 = {!!} }


record CountableOrdinal : Set (suc (suc n)) where
   field
       ctl→ : Nat → Ordinal
       ctl← : Ordinal → Nat
       ctl-iso→ : { x : Ordinal } → ctl→ (ctl← x ) ≡ x 
       ctl-iso← : { x : Nat }  → ctl← (ctl→ x ) ≡ x
       
open CountableOrdinal 

PGOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Set n
PGOD  i C p q =  ¬ ( odef (ord→od (ctl→ C i)) q ∧ ( (x : Ordinal ) → odef (ord→od p) x →  odef (ord→od q) x ))

PGHOD : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → HOD
PGHOD i C p = record { od = record { def = λ x → PGOD i C {!!} {!!} } ; odmax = {!!} ; <odmax = {!!} } 

ord-compare : (i : Nat) → (C : CountableOrdinal) → (p : Ordinal) → ( q : Ordinal ) → Ordinal
ord-compare i C p q with ODC.p∨¬p O ( (q : Ordinal ) → PGOD i C p q )
ord-compare i C p q | case1 y = p
ord-compare i C p q | case2 n = od→ord (ODC.minimal O (PGHOD i C p ) (∅< (subst₂ (λ j k → odef j {!!} ) refl {!!} n)) ) 

data PD (P : HOD) (C : CountableOrdinal) : (x : Ordinal) (i : Nat) →  Set (suc n) where
    pd0 : PD P C o∅ 0 
    -- pdq : {q pnx : Ordinal } {n : Nat}  → (pn : PD P C pnx n ) → odef (ctl→ C n) q → ord→od p0x ⊆ ord→od q → PD P C q (suc n) 

P-GenericFilter : {P : HOD} → (C : CountableOrdinal) → GenericFilter P
P-GenericFilter {P} C = record {
      genf = record { filter = {!!} ; f⊆PL = {!!} ; filter1 = {!!} ; filter2 = {!!} }
    ; generic = λ D → {!!}
   }