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 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 

import OPair
open OPair O

record CountableModel (P : HOD) : Set (suc (suc n)) where
   field
       ctl-M : Ordinal
       ctl→ : Nat → Ordinal
       ctl← : (x : Ordinal )→  x o< ctl-M → Nat
       is-Model : (x : Nat) → ctl→ x o< ctl-M
       ctl-iso→ : { x : Ordinal } → (lt : x o< ctl-M)  → ctl→ (ctl← x lt ) ≡ x 
       ctl-iso← : { x : Nat }  →  ctl← (ctl→ x ) (is-Model x)  ≡ x
       ctl-P∈M : Power P ∈ * ctl-M

-- we expect ¬ G ∈ * ctl-M, so  ¬ P ∈ * ctl-M

open CountableModel 

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

---
--   p(n+1) = if (f n) != ∅ then (f n) otherwise p(n)
--
next-p :  (p : Ordinal) → (f : HOD → HOD) → Ordinal
next-p p f with is-o∅  ( & (f (* p)))  
next-p p f | yes y = p
next-p p f | no not = & (ODC.minimal O (f (* p) ) (λ eq → not (=od∅→≡o∅ eq)))  -- axiom of choice

---
--  search on p(n)
--
find-p :  (P : HOD ) (C : CountableModel P)  (i : Nat) → (x : Ordinal) → Ordinal
find-p P C Zero x = x
find-p P C (Suc i) x = find-p P C i ( next-p x (λ p → PGHOD i P C (& p) ))

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

open PDN

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

open PDN

----
--  Generic Filter on Power P for HOD's Countable Ordinal (G ⊆ Power P ≡ G i.e. Nat → 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

open _⊆_

find-an :{P p : HOD} → (C : CountableModel P ) → odef (Power P) (& p) → Nat
find-an = {!!}

found-an :{P p : HOD} → (C : CountableModel P ) → (pw : odef (Power P) (& p)) → * (ctl→ C (find-an C pw)) =h= p
found-an = {!!}

record ⊆-reduce ( p :  HOD ) : Set (suc n) where
   field
     next : HOD
     is-small :  next ⊆ p ∧ ( ¬ ( next =h= p ))

⊆-reduction : { ψ : (x : HOD) → Set (suc n) }
    → (p : HOD) → ψ p
    → (next : (x : HOD) → ψ x → ⊆-reduce x ) 
    → (ind  : (x : HOD) → (m : ψ x) → ψ ( ⊆-reduce.next (next x m )) )
    → ψ od∅
⊆-reduction {ψ} p ψp next ind = TransFinite {λ x → {!!} } {!!} o∅


P-GenericFilter : (P p0 : HOD ) → (C : CountableModel P) → GenericFilter P
P-GenericFilter P p0 C = record {
      genf = record { filter = PDHOD P p0 C ; f⊆PL =  f⊆PL ; filter1 = f1 ; filter2 = f2 }
    ; generic = λ D → {!!}
   } where
        PGHOD∈PL :  (i : Nat) → (x : Ordinal) →  PGHOD i P C x ⊆ Power P
        PGHOD∈PL i x = record { incl = λ {x} p → proj1 p }
        find-p-⊆P :  (i : Nat) → (x y : Ordinal)  → odef (Power P) x → odef (* (find-p P C i x)) y → odef P y 
        find-p-⊆P Zero x y Px Py = subst (λ k → odef P k ) &iso
            ( incl (ODC.power→⊆ O P (* x) (d→∋ (Power P)  Px)) (x<y→∋ Py))
        find-p-⊆P (Suc i) x y Px Py with is-o∅  ( & (PGHOD i P C (& (* x))))
        ... | yes y1 = find-p-⊆P i x y Px Py
        ... | no not = find-p-⊆P i (& fmin) y pg-01 Py where
            fmin : HOD
            fmin = ODC.minimal O (PGHOD i P C (& (* x))) (λ eq → not (=od∅→≡o∅ eq))
            fmin∈PGHOD : PGHOD i P C (& (* x)) ∋ fmin
            fmin∈PGHOD = ODC.x∋minimal O (PGHOD i P C (& (* x))) (λ eq → not (=od∅→≡o∅ eq))
            pg-01 : Power P ∋ fmin 
            pg-01 = incl (PGHOD∈PL i x ) (subst (λ k → PGHOD i P C k ∋ fmin ) &iso  fmin∈PGHOD )
        f⊆PL :  PDHOD P p0 C ⊆ Power P
        f⊆PL = record { incl = λ {x} lt → power← P x (λ {y} y<x →
             find-p-⊆P (gr lt) {!!} (& y) {!!} (pn<gr lt (& y) (subst (λ k → odef k (& y)) (sym *iso) y<x))) }
        f1 : {p q : HOD} → q ⊆ P → PDHOD P p0 C ∋ p → p ⊆ q → PDHOD P p0 C ∋ q
        f1 {p} {q}  q⊆P PD∋p p⊆q =  record { gr = {!!} ;  pn<gr = {!!} ; x∈PP = {!!} }  where
           --  reduction : {ψ : (x : HOD) → x =h= od∅ → P x} → 
           --  q ∈ a(∃ n) ⊆ P → ( p n ⊆ q →  PDHOD P p0 C ∋ q )
           --                 ∨ (¬ (p n ⊆ q) → induction on (p n - q) 
           PDNp :  {!!} -- PD⊆⊆N P C (& p)
           PDNp = PD∋p
           f02 : {x : Ordinal} → odef q x → odef P x
           f02 {x} lt = subst (λ k → def (od P) k) &iso (incl q⊆P (subst (λ k → def (od q) k) (sym &iso) lt) )
           f03 : {x : Ordinal} → odef p x → odef q x
           f03 {x} lt = subst (λ k → def (od q) k) &iso (incl p⊆q (subst (λ k → def (od p) k) (sym &iso) lt) )
           next1 : Ordinal
           next1 = {!!} -- find-p P C (gr PD) (next-p x (λ p₁ → PGHOD (gr PD) P C (& p₁))) 
           f05 : next1 o< ctl-M C
           f05 = {!!}
           f06 : odef (Power P) (& q)
           f06 = {!!}
           f07 :  (y : Ordinal) → odef (* (& q)) y → odef (* {!!} ) y
           f07 = {!!}
        f2 : {p q : HOD} → PDHOD P p0 C ∋ p → PDHOD P p0 C ∋ q → PDHOD P p0 C ∋ (p ∩ q)
        f2 {p} {q} PD∋p PD∋q = {!!}



open GenericFilter
open Filter

record Incompatible  (P : HOD ) : Set (suc (suc n)) where
   field
      q : {p : HOD } → Power P ∋ p → HOD 
      r : {p : HOD } → Power P ∋ p → HOD 
      incompatible : { p : HOD } →  (P∋p : Power P ∋ p)  →  Power P ∋ q P∋p  →  Power P ∋ r P∋p
          → ( p ⊆ q P∋p)   ∧ ( p ⊆ r P∋p)  
          → ∀ ( s : HOD ) →  Power P ∋ s → ¬ (( q P∋p  ⊆ s  ) ∧ ( r P∋p  ⊆ s ))

lemma725 : (P p : HOD ) (C : CountableModel P) 
    →  * (ctl-M C) ∋ Power P
    →  Incompatible P → ¬ ( * (ctl-M C) ∋ filter ( genf ( P-GenericFilter P p C )))
lemma725 = {!!}

open import PFOD O

-- HODω2 : HOD
-- 
-- ω→2 : HOD
-- ω→2 = Power infinite

lemma725-1 :   Incompatible HODω2
lemma725-1 = {!!}

lemma726 :  (C : CountableModel HODω2) 
    →  Union ( Replace HODω2 (λ p → filter ( genf ( P-GenericFilter HODω2 p C )))) =h= ω→2
lemma726 = {!!}

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

record valR (x : HOD) {P : HOD} (G : GenericFilter P) : Set (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 : HOD }
    →  (G : GenericFilter P)
    →  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 = {!!} }


--
--   W (ω , H ( ω , 2 )) = { p ∈ ( Nat → H (ω , 2) ) |  { i ∈ Nat → p i ≠ i1 } is finite }
--