open import Level
module constructible-set (n : Level) where

open import zf

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

open import  Relation.Binary.PropositionalEquality

data OrdinalD  : (lv : Nat) → Set n where
   Φ : {lv : Nat} → OrdinalD  lv
   OSuc : {lv : Nat} → OrdinalD  lv → OrdinalD lv
   ℵ_ :  (lv : Nat) → OrdinalD (Suc lv)

record Ordinal : Set n where
   field
     lv : Nat
     ord : OrdinalD lv

data _d<_  :  {lx ly : Nat} → OrdinalD  lx  →  OrdinalD  ly  → Set n where
   Φ<  : {lx : Nat} → {x : OrdinalD  lx}  →  Φ  {lx} d< OSuc  {lx} x
   s<  : {lx : Nat} → {x y : OrdinalD  lx}  →  x d< y  → OSuc  {lx} x d< OSuc  {lx} y
   ℵΦ< : {lx : Nat} → {x : OrdinalD  (Suc lx) } →  Φ  {Suc lx} d< (ℵ lx) 
   ℵ<  : {lx : Nat} → {x : OrdinalD  (Suc lx) } →  OSuc  {Suc lx} x d< (ℵ lx) 

open Ordinal

_o<_ : ( x y : Ordinal ) → Set n
_o<_ x y =  (lv x < lv y )  ∨ ( ord x d< ord y )

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

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

o∅ : Ordinal
o∅  = record { lv = Zero ; ord = Φ }


≡→¬d< : {lv : Nat} → {x  : OrdinalD  lv }  → x d< x → ⊥
≡→¬d<  {lx} {OSuc y} (s< t) = ≡→¬d< t

trio<> : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  →  y d< x → x d< y → ⊥
trio<>  {lx} {.(OSuc _)} {.(OSuc _)} (s< s) (s< t) = 
    trio<> s t

trio<≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → x d< y → ⊥
trio<≡ refl = ≡→¬d<

trio>≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
trio>≡ refl = ≡→¬d<

triO : {lx ly : Nat} → OrdinalD  lx  →  OrdinalD  ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
triO  {lx} {ly} x y = <-cmp lx ly

triOrdd : {lx : Nat}   → Trichotomous  _≡_ ( _d<_  {lx} {lx} )
triOrdd  {lv} Φ Φ = tri≈ ≡→¬d< refl ≡→¬d<
triOrdd  {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d<
triOrdd  {lv} Φ (OSuc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
triOrdd  {.(Suc lv)} Φ (ℵ lv) = tri<  (ℵΦ<  {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ<  {lv} {Φ} )) )
triOrdd  {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ<  {lv} {Φ} ) ) (λ ()) (ℵΦ<  {lv} {Φ} )
triOrdd  {Suc lv} (ℵ lv) (OSuc y) = tri> ( λ lt → trio<> lt (ℵ<  {lv} {y} )  ) (λ ()) (ℵ<  {lv} {y} )
triOrdd  {lv} (OSuc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
triOrdd  {.(Suc lv)} (OSuc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
triOrdd  {lv} (OSuc x) (OSuc y) with triOrdd x y
triOrdd  {lv} (OSuc x) (OSuc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) )  (  λ lt → trio<> lt (s< a) )
triOrdd  {lv} (OSuc x) (OSuc x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d<
triOrdd  {lv} (OSuc x) (OSuc y) | tri> ¬a ¬b c = tri> (  λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)

d<→lv :  {x y  : Ordinal }   → ord x d< ord y → lv x ≡ lv y
d<→lv Φ< = refl
d<→lv (s< lt) = refl
d<→lv ℵΦ< = refl
d<→lv ℵ< = refl

orddtrans : {lx : Nat} {x y z : OrdinalD  lx }   → x d< y → y d< z → x d< z
orddtrans {lx} {.Φ} {.(OSuc _)} {.(OSuc _)} Φ< (s< y<z) = Φ< 
orddtrans {Suc lx} {Φ {Suc lx}} {OSuc y} {ℵ lx} Φ< ℵ< = ℵΦ< {lx} {y}
orddtrans {lx} {.(OSuc _)} {.(OSuc _)} {.(OSuc _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
orddtrans {.(Suc _)} {.(OSuc _)} {.(OSuc _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ<
orddtrans {.(Suc _)} {.Φ} {.(ℵ _)} {z} ℵΦ< ()
orddtrans {.(Suc _)} {.(OSuc _)} {.(ℵ _)} {z} ℵ< ()

max : (x y : Nat) → Nat
max Zero Zero = Zero
max Zero (Suc x) = (Suc x)
max (Suc x) Zero = (Suc x)
max (Suc x) (Suc y) = Suc ( max x y )

maxαd : { lx : Nat } → OrdinalD  lx  →  OrdinalD  lx  →  OrdinalD  lx
maxαd x y with triOrdd x y
maxαd x y | tri< a ¬b ¬c = y
maxαd x y | tri≈ ¬a b ¬c = x
maxαd x y | tri> ¬a ¬b c = x

maxα :  Ordinal →  Ordinal  → Ordinal
maxα x y with <-cmp (lv x) (lv y)
maxα x y | tri< a ¬b ¬c = x
maxα x y | tri> ¬a ¬b c = y
maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) }

_o≤_ : Ordinal → Ordinal → Set n
a o≤ b  = (a ≡ b)  ∨ ( a o< b )

trio< : Trichotomous  _≡_  _o<_ 
trio< a b with <-cmp (lv a) (lv b)
trio< a b | tri< a₁ ¬b ¬c = tri< (case1  a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) {!!}
trio< a b | tri> ¬a ¬b c = tri> {!!} (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c)
trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b )
trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b {!!} )  {!!}
trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ {!!} refl {!!}
trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> {!!} {!!} (case2 c)

OrdTrans : Transitive _o≤_
OrdTrans (case1 refl) (case1 refl) = case1 refl
OrdTrans (case1 refl) (case2 lt2) = case2 lt2
OrdTrans (case2 lt1) (case1 refl) = case2 lt1
OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) )
OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y
OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x )
OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x
OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y)
OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y
OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y ))

OrdPreorder : Preorder n n n
OrdPreorder = record { Carrier = Ordinal
   ; _≈_  = _≡_ 
   ; _∼_   = _o≤_
   ; isPreorder   = record {
        isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
        ; reflexive     = case1 
        ; trans         = OrdTrans
     }
 }

TransFinite : ( ψ : Ordinal  → Set n ) 
  → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) 
  → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ } ) )
  → ( ∀ (lx : Nat ) → (x : OrdinalD lx )  → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc x } ) )
  →  ∀ (x : Ordinal)  → ψ x
TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ } = caseΦ lv
TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc ord₁ } = caseOSuc lv ord₁
    ( TransFinite ψ caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } ))
TransFinite ψ caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁

-- X' = { x ∈ X |  ψ  x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '

record ConstructibleSet  : Set (suc (suc n)) where
  field
    α : Ordinal
    constructible : Ordinal  → Set (suc n)

open ConstructibleSet

_∋_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set (suc n)
a ∋ x  = ( α x o< α a ) ∧ constructible a ( α x )

c∅ : ConstructibleSet
c∅  = record {α = o∅ ; constructible = λ x → Lift (suc n) ⊥ }

record SupR {n m : Level} {S : Set n} ( _≤_ : S → S → Set m  ) (ψ : S → S ) (X : S) : Set (n ⊔ m)  where
  field
    sup : S
    smax  : ∀ { x : S } → x ≤ X  → ψ x ≤ sup 
    suniq : {max : S} → ( ∀ { x :  S } → x ≤ X  → ψ x ≤ max ) → max ≤ sup 

open SupR

_⊆_ : ( A B : ConstructibleSet  ) → ∀{ x : ConstructibleSet } →  Set (suc n)
_⊆_ A B {x} = A ∋ x →  B ∋ x

suptraverse : (X : ConstructibleSet ) ( max : ConstructibleSet) ( ψ : ConstructibleSet  → ConstructibleSet ) → ConstructibleSet
suptraverse X max ψ  = {!!} 

Sup : (ψ : ConstructibleSet → ConstructibleSet )  → (X : ConstructibleSet)  → SupR (λ x a → (α a ≡ α x) ∨ (a ∋ x)) ψ X
sup (Sup ψ X ) = suptraverse X c∅ ψ 
smax (Sup ψ X ) = {!!} -- TransFinite {!!} {!!} {!!} {!!} {!!} 
suniq (Sup ψ X ) = {!!}
     

-- transitiveness : (a b c : ConstructibleSet ) → a ∋ b → b ∋ c → a ∋ c
-- transitiveness  a b c a∋b b∋c with constructible a c∋ constructible b | constructible b c∋ constructible c
-- ... | t1 | t2 = {!!}

open import Data.Unit
open SupR

ConstructibleSet→ZF : ZF {suc (suc n)} {suc (suc n)}
ConstructibleSet→ZF   = record { 
    ZFSet = ConstructibleSet 
    ; _∋_ = λ a b → Lift (suc (suc n)) ( a ∋ b )
    ; _≈_ = _≡_ 
    ; ∅  = c∅ 
    ; _,_ = _,_
    ; Union = Union
    ; Power = {!!}
    ; Select = Select
    ; Replace = Replace
    ; infinite = {!!}
    ; isZF = {!!}
 } where
    conv : (ConstructibleSet → Set (suc (suc n))) → ConstructibleSet → Set (suc n)
    conv ψ x with ψ x
    ... | t =  Lift ( suc n ) ⊤
    Select : (X : ConstructibleSet) → (ConstructibleSet → Set (suc (suc n))) → ConstructibleSet
    Select X ψ = record { α = α X ; constructible = λ x → (conv ψ) (record { α = x ; constructible = λ x → constructible X x }  ) }
    Replace : (X : ConstructibleSet) → (ConstructibleSet → ConstructibleSet) → ConstructibleSet
    Replace X ψ  = record { α = α (sup (Sup ψ X))  ; constructible = λ x → {!!}  }
    _,_ : ConstructibleSet → ConstructibleSet → ConstructibleSet
    a , b  = record { α = maxα (α a) (α b) ; constructible = λ x → {!!} }
    Union : ConstructibleSet → ConstructibleSet
    Union a = {!!}