open import Level
module ordinal-definable where

open import zf
open import ordinal

open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 

open import  Relation.Binary.PropositionalEquality

open import Data.Nat.Properties 
open import Data.Empty
open import Relation.Nullary

open import Relation.Binary
open import Relation.Binary.Core

-- Ordinal Definable Set

record OD {n : Level}  : Set (suc n) where
  field
    def : (x : Ordinal {n} ) → Set n

open OD
open import Data.Unit

postulate      
  od→ord : {n : Level} → OD {n} → Ordinal {n}
  ord→od : {n : Level} → Ordinal {n} → OD {n} 

_∋_ : { n : Level } → ( a x : OD {n} ) → Set n
_∋_ {n} a x  = def a ( od→ord x )

_c<_ : { n : Level } → ( a x : OD {n} ) → Set n
x c< a = a ∋ x 

_c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )

od∅ : {n : Level} → OD {n} 
od∅ {n} = record { def = λ _ → Lift n ⊥ }

postulate      
  c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y
  o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y
  oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
  diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
  sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
  sup-c< : {n : Level } → ( ψ : OD {n} →  OD {n}) → ∀ {x : OD {n}} → ψ x  c< sup-od ψ
  ∅-base-def : {n : Level} → def ( ord→od (o∅ {n}) ) ≡ def (od∅ {n})

∅1 : {n : Level} →  ( x : OD {n} )  → ¬ ( x c< od∅ {n} )
∅1 {n} x (lift ())

∅3 : {n : Level} →  { x : Ordinal {n}}  → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
∅3 {n} {x} = TransFinite {n} c1 c2 c3 x where
   c0 : Nat →  Ordinal {n}  → Set n
   c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x))  → x ≡ o∅ {n}
   c1 : ∀ (lx : Nat ) →  c0 lx (record { lv = Suc lx ; ord = ℵ lx } )  
   c1 lx not with not ( record { lv = lx ; ord = Φ lx } )
   ... | t with t (case1 ≤-refl )
   c1 lx not | t | ()
   c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } )
   c2 Zero not = refl
   c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } )
   ... | t with t (case1 ≤-refl )
   c2 (Suc lx) not | t | ()
   c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx  (record { lv = lx ; ord = x₁ })  → c0 lx (record { lv = lx ; ord = OSuc lx x₁ })
   c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } )
   ... | t with t (case2 Φ< )
   c3 lx (Φ .lx) d not | t | ()
   c3 lx (OSuc .lx x₁) d not with not (  record { lv = lx ; ord = OSuc lx x₁ } )
   ... | t with t (case2 (s< s<refl ) )
   c3 lx (OSuc .lx x₁) d not | t | ()
   c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) }  )
   ... | t with t (case2 (s< ℵΦ<   )) 
   c3 .(Suc lx) (ℵ lx) d not | t | ()

-- find : {n : Level} → ( x : Ordinal {n} ) → o∅ o< x → Ordinal {n}  
-- exists : {n : Level} → ( x : Ordinal {n} ) → (0<x : o∅ o< x ) → find x 0<x o< x

def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z  →  X ≡ x  →  def z x
def-subst df refl refl = df

transitive : {n : Level } { x y z : OD {n} } → y ∋ x → z ∋ y → z ∋ x
transitive  {n} {x} {y} {z} x∋y  z∋y with  ordtrans ( c<→o< {n} {x} {y} x∋y ) (  c<→o< {n} {y} {z} z∋y ) 
... | t = lemma0 (lemma t) where
   lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x )))
   lemma xo<z = o<→c< xo<z
   lemma0 :  def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) →  def z (od→ord x)
   lemma0 dz  = def-subst {n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso)

record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
  field
     mino : Ordinal {n}
     min<x :  mino o< x

ominimal : {n : Level} → (x : Ordinal {n} ) → o∅ o< x → Minimumo {n} x
ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case1 ())
ominimal {n} record { lv = Zero ; ord = (Φ .0) } (case2 ())
ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case1 ())
ominimal {n} record { lv = Zero ; ord = (OSuc .0 ord) } (case2 Φ<) = record { mino = record { lv = Zero ; ord = Φ 0 } ; min<x = case2 Φ< }
ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case1 (s≤s x)) = record { mino = record { lv = lv ; ord = Φ lv } ; min<x = case1 (s≤s ≤-refl)}
ominimal {n} record { lv = (Suc lv) ; ord = (Φ .(Suc lv)) } (case2 ())
ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case1 (s≤s x)) = record { mino = record { lv = (Suc lv) ; ord = ord } ; min<x = case2 s<refl}
ominimal {n} record { lv = (Suc lv) ; ord = (OSuc .(Suc lv) ord) } (case2 ())
ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case1 (s≤s z≤n)) = record { mino = record { lv = Suc lv ; ord = Φ (Suc lv) } ; min<x = case2 ℵΦ<  }
ominimal {n} record { lv = (Suc lv) ; ord = (ℵ .lv) } (case2 ())

∅4 : {n : Level} →  ( x : OD {n} )  →  x  ≡ od∅ {n}  → od→ord x ≡ o∅ {n}
∅4 {n} x refl =  ∅3 lemma1 where
  lemma0 :  (y : Ordinal {n}) → def ( od∅ {n} ) y →  ⊥
  lemma0 y (lift ())
  lemma1 : (y : Ordinal {n}) → y o< od→ord od∅ → ⊥
  lemma1 y y<o = lemma0 y ( def-subst {n} {ord→od (od→ord od∅ )} {od→ord (ord→od y)} (o<→c< y<o) oiso diso )

∅5 : {n : Level} →  ( x : Ordinal {n} )  → ¬ ( x  ≡ o∅ {n} ) → o∅ {n} o< x
∅5 {n} record { lv = Zero ; ord = (Φ .0) } not = ⊥-elim (not refl) 
∅5 {n} record { lv = Zero ; ord = (OSuc .0 ord) } not = case2 Φ<
∅5 {n} record { lv = (Suc lv) ; ord = ord } not = case1 (s≤s z≤n)

postulate extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality n (suc n)

∅6 : {n : Level } ( x : Ordinal {suc n}) → o∅ o< x → ¬ x ≡ o∅
∅6 {n} x lt eq with trio< {n} (o∅ {suc n}) x
∅6 {n} x lt refl | tri< a ¬b ¬c = ¬b refl
∅6 {n} x lt refl | tri≈ ¬a b ¬c =  ¬a lt
∅6 {n} x lt refl | tri> ¬a ¬b c = ¬b refl

∅8 : {n : Level} →  ( x : Ordinal {n} )  → ¬  x o< o∅ {n}
∅8 {n} x (case1 ())
∅8 {n} x (case2 ())

-- ∅10 : {n : Level} → (x : OD {n} ) → ¬ ( ( y : OD {n} ) → Lift (suc n) ( x ∋ y))   →  x ≡ od∅ 
-- ∅10 {n} x not = ?

open Ordinal

∋-subst : {n : Level} {X Y x y : OD {suc n} } → X ≡ x → Y ≡ y → X ∋ Y →  x ∋ y
∋-subst refl refl x = x

-- ∅77 : {n : Level} → (x : OD {suc n} ) →  ¬ ( ord→od (o∅ {suc n}) ∋ x )
-- ∅77 {n} x lt = {!!} where

∅7' : {n : Level} → ord→od (o∅ {n}) ≡ od∅ {n}
∅7' {n} = cong ( λ k → record { def = k }) ( ∅-base-def ) where

∅7 : {n : Level} →  ( x : OD {n} )   → od→ord x ≡ o∅ {n} →  x  ≡ od∅ {n}
∅7 {n} x eq = trans ( trans (sym oiso)( cong ( λ k → ord→od k ) eq )  )  ∅7' 

∅9 : {n : Level} → (x : OD {n} ) → ¬ x ≡ od∅ → o∅ o< od→ord x
∅9 x not = ∅5 ( od→ord x) lemma where
   lemma : ¬ od→ord x ≡ o∅
   lemma eq = not ( ∅7 x eq )

OD→ZF : {n : Level} → ZF {suc n} {suc n}
OD→ZF {n}  = record { 
    ZFSet = OD {n}
    ; _∋_ = λ a x → Lift (suc n) ( a ∋ x )
    ; _≈_ = _≡_ 
    ; ∅  = od∅
    ; _,_ = _,_
    ; Union = Union
    ; Power = Power
    ; Select = Select
    ; Replace = Replace
    ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero  }  }
    ; isZF = isZF 
 } where
    Replace : OD {n} → (OD {n} → OD {n} ) → OD {n}
    Replace X ψ = sup-od ψ
    Select : OD {n} → (OD {n} → Set (suc n) ) → OD {n}
    Select X ψ = record { def = λ x → select ( ord→od x ) } where
       select : OD {n} → Set n
       select x with ψ x
       ... | t =  Lift n ⊤
    _,_ : OD {n} → OD {n} → OD {n}
    x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) }
    Union : OD {n} → OD {n}
    Union x = record { def = λ y → {z : Ordinal {n}} → def x z  → def (ord→od z) y  }
    Power : OD {n} → OD {n}
    Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y )  }
    ZFSet = OD {n}
    _∈_ : ( A B : ZFSet  ) → Set n
    A ∈ B = B ∋ A
    _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set n
    _⊆_ A B {x} = A ∋ x →  B ∋ x
    _∩_ : ( A B : ZFSet  ) → ZFSet
    A ∩ B = Select (A , B) (  λ x → (Lift (suc n) ( A ∋ x )) ∧ (Lift (suc n) ( B ∋ x )  ))
    _∪_ : ( A B : ZFSet  ) → ZFSet
    A ∪ B = Select (A , B) (  λ x → (Lift (suc n) ( A ∋ x )) ∨ (Lift (suc n) ( B ∋ x )  ))
    infixr  200 _∈_
    infixr  230 _∩_ _∪_
    infixr  220 _⊆_
    isZF : IsZF (OD {n})  (λ a x → Lift (suc n) ( a ∋ x )) _≡_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero }  })
    isZF = record {
           isEquivalence  = record { refl = refl ; sym = sym ; trans = trans }
       ;   pair  = pair
       ;   union→ = {!!}
       ;   union← = {!!}
       ;   empty = empty
       ;   power→ = {!!}
       ;   power← = {!!}
       ;   extentionality = {!!}
       ;   minimul = minimul
       ;   regularity = {!!}
       ;   infinity∅ = {!!}
       ;   infinity = {!!}
       ;   selection = {!!}
       ;   replacement = {!!}
     } where
         open _∧_ 
         open Minimumo
         pair : (A B : OD {n} ) → Lift (suc n) ((A , B) ∋ A) ∧ Lift (suc n) ((A , B) ∋ B)
         proj1 (pair A B ) = lift ( case1 refl )
         proj2 (pair A B ) = lift ( case2 refl )
         empty : (x : OD {n} ) → ¬ Lift (suc n) (od∅ ∋ x)
         empty x (lift (lift ()))
         union→ : (X x y : OD {n} ) → Lift (suc n) (X ∋ x) → Lift (suc n) (x ∋ y) → Lift (suc n) (Union X ∋ y)
         union→ X x y (lift X∋x) (lift x∋y) = lift {!!}  where
            lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y
            lemma {z} X∋z = {!!}
         minord : (x : OD {n} ) → ¬ x ≡ od∅ → Minimumo (od→ord x)
         minord x not = ominimal (od→ord x) (∅9 x not)
         minimul : (x : OD {n} ) → ¬ x ≡ od∅ → OD {n} 
         minimul x  not = ord→od ( mino (minord x not))
         minimul<x : (x : OD {n} ) →  (not :  ¬ x ≡ od∅ ) → x ∋ minimul x not
         minimul<x x not = lemma0 (min<x (minord x not)) where
            lemma0 : mino (minord x not)  o< (od→ord x) → def x (od→ord (ord→od (mino (minord x not))))
            lemma0 m<x = def-subst {n} {ord→od (od→ord x)} {od→ord (ord→od (mino (minord x not)))} (o<→c< m<x) oiso refl
         regularity : (x : OD) → (not : ¬ x ≡ od∅ ) →
                Lift (suc n) (x ∋ minimul x not ) ∧
                (Select x (λ x₁ → Lift (suc n) ( minimul x not ∋ x₁) ∧ Lift (suc n) (x ∋ x₁)) ≡ od∅)
         proj1 ( regularity x non ) = lift ( minimul<x x non )
         proj2 ( regularity x non ) = cong ( λ k → record { def = k } ) ( extensionality ( λ y → lemma0 y) )   where
            lemma : ( z : Ordinal {n} ) → def (Select x (λ y → Lift (suc n) (minimul x non ∋ y) ∧ Lift (suc n) (x ∋ y))) z 
            lemma z with (minimul x non ∋ (ord→od z)) | x ∋ (ord→od z)
            ... | s | t = {!!}
            lemma0 : ( z : Ordinal {n} ) → def (Select x (λ y → Lift (suc n) (minimul x non ∋ y) ∧ Lift (suc n) (x ∋ y))) z ≡ Lift n ⊥ 
            lemma0 = {!!}