{-# OPTIONS --allow-unsolved-metas #-}
import Level 
open import Ordinals
module generic-filter {n : Level.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 
import BAlgebra 

open BAlgebra O

open inOrdinal O
open OD O
open OD.OD
open ODAxiom odAxiom
import OrdUtil
import ODUtil
open Ordinals.Ordinals  O
open Ordinals.IsOrdinals isOrdinal
open Ordinals.IsNext isNext
open OrdUtil O
open ODUtil O


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 hiding (filter)
open import Data.Maybe 

open import ZProduct O

record CountableModel : Set (Level.suc (Level.suc n)) where
   field
       ctl-M : HOD
       ctl→ : ℕ → Ordinal
       ctl<M : (x : ℕ) → odef (ctl-M) (ctl→ x) 
       ctl← : (x : Ordinal )→  odef (ctl-M ) x → ℕ
       ctl-iso→ : { x : Ordinal } → (lt : odef (ctl-M) x )  → ctl→ (ctl← x lt ) ≡ x 
       -- we have no otherway round
       -- ctl-iso← : { x : ℕ }  →  ctl← (ctl→ x ) (ctl<M x)  ≡ x
--
-- almmost universe
-- find-p contains ∃ x : Ordinal → x o< & M → ∀ r ∈ M → ∈ Ord x
-- 

-- we expect  P ∈ * ctl-M ∧ G  ⊆ L ⊆ Power P  , ¬ G ∈ * ctl-M, 

open CountableModel 

----
--   a(n) ∈ M
--   ∃ q ∈ L ⊆ Power P → q ∈ a(n) ∧ p(n) ⊆ q
--
PGHOD :  (i : ℕ) (L : HOD) (C : CountableModel ) → (p : Ordinal) → HOD
PGHOD i L C p = record { od = record { def = λ x  →
       odef L x ∧ odef (* (ctl→ C i)) x  ∧  ( (y : Ordinal ) → odef (* p) y →  odef (* x) y ) }
   ; odmax = odmax L  ; <odmax = λ {y} lt → <odmax L (proj1 lt) }  

---
--   p(n+1) = if ({q | q ∈ a(n) ∧ p(n) ⊆ q)} != ∅ then q otherwise p(n)
--  
find-p :  (L : HOD ) (C : CountableModel )  (i : ℕ) → (x : Ordinal) → Ordinal
find-p L C zero x = x
find-p L C (suc i) x with is-o∅ ( & ( PGHOD i L C (find-p L C i x)) )
... | yes y  = find-p L C i x
... | no not  = & (ODC.minimal O ( PGHOD i L C (find-p L C i x)) (λ eq → not (=od∅→≡o∅ eq)))  -- axiom of choice

---
-- G = { r ∈ L ⊆ Power P | ∃ n → r ⊆ p(n) }
--
record PDN  (L p : HOD ) (C : CountableModel )  (x : Ordinal) : Set n where
   field
       gr : ℕ
       pn<gr : (y : Ordinal) → odef (* x) y → odef (* (find-p L C gr (& p))) y 
       x∈PP  : odef L x

open PDN

---
-- G as a HOD
--
PDHOD :  (L p : HOD ) (C : CountableModel  ) → HOD
PDHOD L p C  = record { od = record { def = λ x →  PDN L p C x }
    ; odmax = odmax L ; <odmax = λ {y} lt → <odmax L {y} (PDN.x∈PP lt)  } 

open PDN

----
--  Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. ℕ → P → Set )
--
--  p 0 ≡ ∅
--  p (suc n) = if ∃ q ∈ M ∧ p n ⊆ q → q  (by axiom of choice) ( q =  * ( ctl→ n ) )
---             else p n

P∅ : {P : HOD} → odef (Power P) o∅
P∅ {P} =  subst (λ k → odef (Power P) k ) ord-od∅ (lemma o∅  o∅≡od∅) where
    lemma : (x : Ordinal ) → * x ≡ od∅ → odef (Power P) (& od∅)
    lemma x eq = power← P od∅  (λ {x} lt → ⊥-elim (¬x<0 lt ))
x<y→∋ : {x y : Ordinal} → odef (* x) y → * x ∋ * y
x<y→∋ {x} {y} lt = subst (λ k → odef (* x) k ) (sym &iso) lt

gf05 : {a b : HOD} {x : Ordinal } → (odef (a ∪ b) x ) → ¬ odef a x → ¬ odef b x → ⊥
gf05 {a} {b} {x} (case1 ax) nax nbx = nax ax 
gf05 {a} {b} {x} (case2 bx) nax nbx = nbx bx

gf02 : {P a b : HOD } → (P ＼ a) ∩ (P ＼ b) ≡ ( P ＼ (a ∪ b) )
gf02 {P} {a} {b} = ==→o≡  record { eq→ = gf03 ; eq← = gf04 }where
       gf03 : {x : Ordinal} → odef ((P ＼ a) ∩ (P ＼ b)) x → odef (P ＼ (a ∪ b)) x
       gf03 {x} ⟪ ⟪ Px , ¬ax ⟫ , ⟪ _ , ¬bx ⟫ ⟫ = ⟪ Px , (λ pab → gf05 {a} {b} {x} pab ¬ax ¬bx )   ⟫
       gf04 : {x : Ordinal} → odef (P ＼ (a ∪ b)) x → odef ((P ＼ a) ∩ (P ＼ b)) x
       gf04 {x} ⟪ Px , abx ⟫  = ⟪ ⟪ Px , (λ ax → abx (case1 ax) ) ⟫ , ⟪ Px  , (λ bx → abx (case2 bx) ) ⟫ ⟫ 

open import Data.Nat.Properties
open import nat

p-monotonic1 :  (L p : HOD ) (C : CountableModel  ) → {n : ℕ} → (* (find-p L C n (& p))) ⊆ (* (find-p L C (suc n) (& p)))
p-monotonic1 L p C {n} {x} with is-o∅ (& (PGHOD n L C (find-p L C n (& p))))
... | yes y =  refl-⊆ {* (find-p L C n (& p))}
... | no not = λ  lt →   proj2 (proj2 fmin∈PGHOD) _ lt   where
    fmin : HOD
    fmin = ODC.minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq))
    fmin∈PGHOD : PGHOD n L C (find-p L C n (& p)) ∋ fmin
    fmin∈PGHOD = ODC.x∋minimal O (PGHOD n L C (find-p L C n (& p))) (λ eq → not (=od∅→≡o∅ eq))

p-monotonic :  (L p : HOD ) (C : CountableModel  ) → {n m : ℕ} → n ≤ m → (* (find-p L C n (& p))) ⊆ (* (find-p L C m (& p)))
p-monotonic L p C {zero} {zero} n≤m = refl-⊆ {* (find-p L C zero (& p))}
p-monotonic L p C {zero} {suc m} z≤n lt = p-monotonic1 L p C {m} (p-monotonic L p C {zero} {m} z≤n lt )
p-monotonic L p C {suc n} {suc m} (s≤s n≤m) with <-cmp n m
... | tri< a ¬b ¬c = λ lt → p-monotonic1 L p C {m} (p-monotonic L p C {suc n} {m} a lt) 
... | tri≈ ¬a refl ¬c = λ x → x
... | tri> ¬a ¬b c = ⊥-elim ( nat-≤> n≤m c )

record Dense  {L P : HOD } (LP : L ⊆ Power P)  : Set (Level.suc n) where
   field
       dense : HOD
       d⊆P :  dense ⊆ L
       dense-f : {p : HOD} → L ∋ p  → HOD
       dense-d :  { p : HOD} → (lt : L ∋ p) → dense ∋ dense-f lt
       dense-p :  { p : HOD} → (lt : L ∋ p) → (dense-f lt) ⊆ p  

record GenericFilter {L P : HOD} (LP : L ⊆ Power P) (M : HOD) : Set (Level.suc n) where
    field
       genf : Filter {L} {P} LP
       generic : (D : Dense {L} {P} LP ) → M ∋ Dense.dense D → ¬ ( (Dense.dense D ∩ Replace (Filter.filter genf) (λ x → P ＼ x )) ≡ od∅ )
    rgen : HOD
    rgen = Replace (Filter.filter genf) (λ x → P ＼ x )

-- ＼-⊆

P-GenericFilter : (P L p0 : HOD ) → (LP : L ⊆ Power P) → L ∋ p0 → ({p : HOD} → L ∋ p → L ∋ ( P ＼ p)) 
      → (C : CountableModel ) → GenericFilter {L} {P} LP ( ctl-M C )
P-GenericFilter P L p0 L⊆PP Lp0 NEG C = record {
      genf = record { filter = Replace (PDHOD L p0 C) (λ x → P ＼ x)  ; f⊆L =  gf01 ; filter1 = ? ; filter2 = ? }
    ; generic = λ D cd → subst (λ k → ¬ (Dense.dense D ∩ k) ≡ od∅ ) (sym gf00) (fdense D cd )
   } where
        GP =  Replace (PDHOD L p0 C) (λ x → P ＼ x) 
        f⊆PL :  PDHOD L p0 C ⊆ L 
        f⊆PL lt = x∈PP lt  
        gf01 : Replace (PDHOD L p0 C) (λ x → P ＼ x) ⊆ L
        gf01 {x} record { z = z ; az = az ; x=ψz = x=ψz } = subst (λ k → odef L k) (sym x=ψz) ( NEG (subst (λ k → odef L k) (sym &iso) (f⊆PL az)) )
        gf141 : {xp xq : Ordinal } → (Pp : PDN L p0 C xp) (Pq : PDN L p0 C xq) →  (* xp ∪ * xq) ⊆ P
        gf141 Pp Pq {x} (case1 xpx) = L⊆PP (PDN.x∈PP Pp)  _ xpx
        gf141 Pp Pq {x} (case2 xqx) = L⊆PP (PDN.x∈PP Pq)  _ xqx
        gf121 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q)  →  p ∩ q  ≡ P ＼ * (& (* (Replaced.z gp) ∪ * (Replaced.z gq)))
        gf121 {p} {q} gp gq = begin
               p ∩ q  ≡⟨ cong₂ (λ j k → j ∩ k ) (sym *iso) (sym *iso)  ⟩
               (* (& p)) ∩ (* (& q))  ≡⟨ cong₂ (λ j k → ( * j ) ∩ ( * k)) (Replaced.x=ψz gp) (Replaced.x=ψz gq) ⟩
               * (& (P ＼ (* xp ))) ∩ (* (& (P ＼ (* xq ))))  ≡⟨ cong₂ (λ j k → j ∩ k ) *iso *iso  ⟩
               (P ＼ (* xp )) ∩ (P ＼ (* xq ))  ≡⟨ gf02 {P} {* xp} {* xq}  ⟩
               P ＼ ((* xp) ∪ (* xq))  ≡⟨ cong (λ k → P ＼ k) (sym *iso) ⟩
               P ＼ * (& (* xp ∪ * xq))  ∎ where  
                  open ≡-Reasoning
                  xp = Replaced.z gp
                  xq = Replaced.z gq
        gf131 : {p q : HOD} (gp : GP ∋ p) (gq : GP ∋ q)  →  P ＼ (p ∩ q) ≡ * (Replaced.z gp) ∪ * (Replaced.z gq)
        gf131 {p} {q} gp gq = trans (cong (λ k → P ＼ k) (gf121 gp gq))
          (trans ( L＼Lx=x (subst (λ k → k ⊆ P) (sym *iso) (gf141 (Replaced.az gp) (Replaced.az gq))) ) *iso )

        f1 : {p q : HOD} → L ∋  q → PDHOD L p0 C ∋ p → p ⊆ q → PDHOD L p0 C ∋ q
        f1 {p} {q} L∋q PD∋p p⊆q =  ?
        f2 : {p q : HOD} → GP ∋ p → GP ∋ q → L ∋ (p ∩ q) → GP ∋ (p ∩ q)
        f2 {p} {q} record { z = xp ; az = Pp ; x=ψz = peq } 
                   record { z = xq ; az = Pq ; x=ψz = qeq } L∋pq with <-cmp (gr Pp) (gr Pq) 
        ... | tri< a ¬b ¬c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10  ; x=ψz = cong (&) (gf121 gp gq) } where
              gp = record { z = xp ; az = Pp ; x=ψz = peq } 
              gq = record { z = xq ; az = Pq ; x=ψz = qeq } 
              gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq))
              gf10 = record { gr = PDN.gr Pq ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where
                 gf16 : gr Pp ≤ gr Pq
                 gf16 = <to≤ a
                 gf15 :  (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pq) (& p0))) y
                 gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy 
                 ... | case1 xpy = p-monotonic L p0 C gf16 (PDN.pn<gr Pp y xpy )
                 ... | case2 xqy = PDN.pn<gr Pq _ xqy
        ... | tri≈ ¬a refl ¬c =  record { z = xp ; az = Pp  ; x=ψz = trans (cong (&) gf17) peq } where
              gf17 : p ∩ q ≡ p
              gf17 = ==→o≡ record { eq→ = proj1 ; eq← = λ {y} px → ⟪ px , ? ⟫   }
        ... | tri> ¬a ¬b c = record { z = & ( (* xp) ∪ (* xq) ) ; az = gf10  ; x=ψz = cong (&) (gf121 gp gq ) } where 
              gp = record { z = xp ; az = Pp ; x=ψz = peq } 
              gq = record { z = xq ; az = Pq ; x=ψz = qeq } 
              gf10 : odef (PDHOD L p0 C) (& (* xp ∪ * xq))
              gf10 = record { gr = PDN.gr Pp ; pn<gr = gf15 ; x∈PP = subst (λ k → odef L k) (cong (&) (gf131 gp gq)) ( NEG L∋pq ) } where
                 gf16 : gr Pq ≤ gr Pp
                 gf16 = <to≤ c
                 gf15 :  (y : Ordinal) → odef (* (& (* xp ∪ * xq))) y → odef (* (find-p L C (gr Pp) (& p0))) y
                 gf15 y gpqy with subst (λ k → odef k y ) *iso gpqy 
                 ... | case1 xpy = PDN.pn<gr Pp _ xpy
                 ... | case2 xqy = p-monotonic L p0 C gf16 (PDN.pn<gr Pq y xqy )
        gf00 : Replace (Replace (PDHOD L p0 C) (λ x → P ＼ x)) (_＼_ P) ≡ PDHOD L p0 C
        gf00 = ==→o≡ record { eq→ = gf20 ; eq← = gf22 } where
             gf20 : {x : Ordinal} → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x → PDN L p0 C x
             gf20 {x} record { z = z₁ ; az = record { z = z ; az = az ; x=ψz = x=ψz₁ } ; x=ψz = x=ψz } = 
                subst (λ k → PDN L p0 C k ) (begin
                  z ≡⟨ sym &iso ⟩ 
                  & (* z) ≡⟨ cong (&) (sym (L＼Lx=x gf21 ))  ⟩  
                  & (P ＼ ( P ＼ (* z) )) ≡⟨ cong (λ k →  & ( P ＼ k)) (sym *iso)   ⟩ 
                  & (P ＼ (* ( & (P ＼ (* z )))))  ≡⟨ cong (λ k → & (P ＼ (* k))) (sym x=ψz₁)  ⟩ 
                  & (P ＼ (* z₁))  ≡⟨  sym x=ψz  ⟩ 
                  x ∎ ) az where 
                  open ≡-Reasoning
                  gf21 : {x : Ordinal } → odef (* z) x → odef P x
                  gf21 {x} lt = L⊆PP ( PDN.x∈PP az) _ lt 
             gf22 : {x : Ordinal} → PDN L p0 C x → odef (Replace (Replace (PDHOD L p0 C) (λ x₁ → P ＼ x₁)) (_＼_ P)) x 
             gf22 {x} pdx = record { z = _ ; az = record { z = _ ; az = pdx ; x=ψz = refl } ; x=ψz = ( begin
               x ≡⟨ sym &iso ⟩ 
               & (* x)  ≡⟨ cong (&) (sym (L＼Lx=x gf21 ))  ⟩  
               & (P ＼ (P ＼ * x))  ≡⟨ cong (λ k →  & ( P ＼ k)) (sym *iso)   ⟩ 
               & (P ＼ * (& (P ＼ * x)))  ∎ ) } where 
                  open ≡-Reasoning
                  gf21 : {z : Ordinal } → odef (* x) z → odef P z
                  gf21 {z} lt = L⊆PP ( PDN.x∈PP pdx ) z lt
        fdense : (D : Dense {L} {P} L⊆PP ) → (ctl-M C ) ∋ Dense.dense D  → ¬ (Dense.dense D ∩ (PDHOD L p0 C)) ≡ od∅
        fdense D MD eq0  = ? where
           open Dense

open GenericFilter
open Filter

record NonAtomic  (L a : HOD ) (L∋a : L ∋ a ) : Set (Level.suc (Level.suc n)) where
   field
      b : HOD
      0<b : ¬ o∅ ≡ & b
      b<a : b ⊆ a

lemma232 : (P L p : HOD ) (C : CountableModel ) 
    →  (LP : L ⊆ Power P ) →  (Lp0 : L ∋ p  ) ( NEG : {p : HOD} → L ∋ p → L ∋ ( P ＼ p))
    →  ( {q : HOD} → (Lq : L ∋ q ) → NonAtomic L q Lq )
    →  ¬ ( (ctl-M C) ∋  rgen ( P-GenericFilter P L p LP Lp0 NEG  C )) 
lemma232 P L p C LP Lp0 NEG NA MG = {!!} where
    D : HOD  -- P - G
    D = ?

--
-- P-Generic Filter defines a countable model D ⊂ C from P
--

--
-- in D, we have V ≠ L
--

--
--   val x G = { val y G | ∃ p → G ∋ p → x ∋ < y , p > }
--

record valR (x : HOD) {P L : HOD} {LP : L ⊆ Power P} (C : CountableModel ) (G : GenericFilter {L} {P} LP (ctl-M C) ) : Set (Level.suc n) where
   field
     valx : HOD

record valS (ox oy oG : Ordinal) : Set n where
   field
     op : Ordinal
     p∈G : odef (* oG) op 
     is-val : odef (* ox) ( & < * oy , * op >  )

val : (x : HOD) {P L : HOD } {LP : L ⊆ Power P}
    →  (G : GenericFilter {L} {P} LP {!!} )
    →  HOD
val x G = TransFinite {λ x → HOD } ind (& x) where
  ind : (x : Ordinal) → ((y : Ordinal) → y o< x → HOD) → HOD
  ind x valy = record { od = record { def = λ y → valS x y (& (filter (genf G))) } ; odmax = {!!} ; <odmax = {!!} }