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 ) 

open import  Relation.Binary.PropositionalEquality

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

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

data _o<_  :  {lx ly : Nat} → OridinalD  lx  →  OridinalD  ly  → Set n where
   l< : {lx ly : Nat }  → {x : OridinalD  lx } →  {y : OridinalD  ly } → lx < ly → x o< y
   Φ<  : {lx : Nat} → {x : OridinalD  lx}  →  Φ  {lx} o< OSuc  {lx} x
   s<  : {lx : Nat} → {x y : OridinalD  lx}  →  x o< y  → OSuc  {lx} x o< OSuc  {lx} y
   ℵΦ< : {lx : Nat} → {x : OridinalD  (Suc lx) } →  Φ  {Suc lx} o< (ℵ lx) 
   ℵ<  : {lx : Nat} → {x : OridinalD  (Suc lx) } →  OSuc  {Suc lx} x o< (ℵ lx) 

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

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


≡→¬< : { x y : Nat } → x ≡ y → x < y → ⊥
≡→¬< {Zero} {Zero} refl ()
≡→¬< {Suc x} {.(Suc x)} refl (s≤s t) = ≡→¬< {x} {x} refl t

x≤x : { x : Nat } → x ≤ x
x≤x {Zero} = z≤n
x≤x {Suc x} =  s≤s ( x≤x  )

x<>y : { x y : Nat } → x > y → x < y → ⊥
x<>y {.(Suc _)} {.(Suc _)} (s≤s lt) (s≤s lt1) = x<>y lt lt1

triO> : {lx ly : Nat} {x  : OridinalD  lx } { y : OridinalD  ly }  →  ly < lx → x o< y → ⊥
triO>  {lx} {ly} {x} {y} y<x xo<y with <-cmp  lx ly
triO>  {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c =  ¬c y<x 
triO>  {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c =  ¬c y<x 
triO>  {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c =  ¬a x₁ 
triO>  {lx} {ly} {Φ} {OSuc _} y<x Φ< | tri> ¬a ¬b c =  ¬b refl 
triO>  {lx} {ly} {OSuc px} {OSuc py} y<x (s< w) | tri> ¬a ¬b c =  triO> y<x w
triO>  {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
triO>  {lx} {ly} {(OSuc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c =  ¬b refl

≡→¬o< : {lv : Nat} → {x  : OridinalD  lv }  → x o< x → ⊥
≡→¬o<  {lx} {x} (l< y) = ≡→¬< refl y
≡→¬o<  {lx} {OSuc y} (s< t) = ≡→¬o< t

trio<> : {lx : Nat} {x  : OridinalD  lx } { y : OridinalD  lx }  →  y o< x → x o< y → ⊥
trio<>  {lx} {x} {y} (l< lt) _ = ≡→¬< refl lt
trio<>  {lx} {x} {y} _ (l< lt)  = ≡→¬< refl lt
trio<>  {lx} {.(OSuc _)} {.(OSuc _)} (s< s) (s< t) = 
    trio<> s t

trio<≡ : {lx : Nat} {x  : OridinalD  lx } { y : OridinalD  lx }  → x ≡ y  → x o< y → ⊥
trio<≡ refl = ≡→¬o<

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

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

triOonSameLevel : {lx : Nat}   → Trichotomous  _≡_ ( _o<_  {lx} {lx} )
triOonSameLevel  {lv} Φ Φ = tri≈ ≡→¬o< refl ≡→¬o<
triOonSameLevel  {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬o< refl ≡→¬o<
triOonSameLevel  {lv} Φ (OSuc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
triOonSameLevel  {.(Suc lv)} Φ (ℵ lv) = tri<  (ℵΦ<  {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ<  {lv} {Φ} )) )
triOonSameLevel  {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ<  {lv} {Φ} ) ) (λ ()) (ℵΦ<  {lv} {Φ} )
triOonSameLevel  {Suc lv} (ℵ lv) (OSuc y) = tri> ( λ lt → trio<> lt (ℵ<  {lv} {y} )  ) (λ ()) (ℵ<  {lv} {y} )
triOonSameLevel  {lv} (OSuc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
triOonSameLevel  {.(Suc lv)} (OSuc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
triOonSameLevel  {lv} (OSuc x) (OSuc y) with triOonSameLevel x y
triOonSameLevel  {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) )
triOonSameLevel  {lv} (OSuc x) (OSuc x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬o< refl ≡→¬o<
triOonSameLevel  {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)

<→≤ : {lx ly : Nat} → lx < ly → (Suc lx ≤ ly)
<→≤ {Zero} {Suc ly} (s≤s lt) = s≤s z≤n
<→≤ {Suc lx} {Zero} ()
<→≤ {Suc lx} {Suc ly} (s≤s lt) = s≤s (<→≤ lt) 

orddtrans : {lx ly lz : Nat} {x  : OridinalD  lx } { y : OridinalD  ly } { z : OridinalD  lz } → x o< y → y o< z → x o< z 
orddtrans {lx} {ly} {lz} x<y y<z with <-cmp lx ly  | <-cmp ly lz
orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = l< ( <-trans a a₁ )
orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri≈ ¬a refl ¬c₁ = l< a
orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = l< {!!} -- ⊥-elim ( ¬a c )
orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = l< {!!}
orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri≈ ¬a₁ refl ¬c = l< {!!}
orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = {!!}
orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = l< a
orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = l< {!!}
orddtrans {lx} {lx} {lx} x<y y<z | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = orddtrans1  x<y y<z where
  orddtrans1 : {lx : Nat} {x y z : OridinalD  lx }   → x o< y → y o< z → x o< z
  orddtrans1 = {!!}

  

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 )

--  use cannot use OridinalD  (Data.Nat_⊔_ lx  ly), I don't know why

maxα> : { lx ly : Nat } → OridinalD  lx  →  OridinalD  ly  → lx > ly  → OridinalD  lx
maxα> x y _ = x

maxα= : { lx : Nat } → OridinalD  lx  →  OridinalD  lx  →  OridinalD  lx
maxα= x y with triOonSameLevel x y
maxα= x y | tri< a ¬b ¬c = y
maxα= x y | tri≈ ¬a b ¬c = x
maxα= x y | tri> ¬a ¬b c = x

OrdTrans : Transitive (λ ( a b : Ordinal ) → (a ≡ b) ∨ (Ordinal.lv a < Ordinal.lv b) ∨ (Ordinal.ord a o< Ordinal.ord b) )
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)) = case2 {!!}
OrdTrans (case2 (case2 x)) (case2 (case1 y)) = case2 {!!}
OrdTrans (case2 (case2 x)) (case2 (case2 y)) = case2 {!!}

OrdPreorder : Preorder n n n
OrdPreorder = record { Carrier = Ordinal
   ; _≈_  = _≡_ 
   ; _∼_   = λ a b → (a ≡ b)  ∨ (Ordinal.lv a < Ordinal.lv b)  ∨ (Ordinal.ord a o< Ordinal.ord b )  
   ; isPreorder   = record {
        isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
        ; reflexive     = case1 
        ; trans         = OrdTrans
     }
 }

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

data Constructible {lv : Nat} ( α : OridinalD  lv )  :  Set (suc n) where
    fsub : ( ψ : OridinalD  lv → Set n ) → Constructible  α
    xself : OridinalD  lv → Constructible  α

record ConstructibleSet  : Set (suc n) where
  field
    level : Nat
    α : OridinalD  level 
    constructible : Constructible α

open ConstructibleSet

data _c∋_  : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' } →
        Constructible  {lv} α → Constructible  {lv'} α' → Set n where
    c> : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' }
        (ta : Constructible  {lv} α ) ( tx : Constructible  {lv'} α' ) → α' o< α →  ta c∋ tx
    xself-fsub  : {lv : Nat} {α : OridinalD  lv } 
         (ta : OridinalD  lv ) ( ψ : OridinalD  lv → Set n ) → _c∋_  {_} {_} {α} {α} (xself ta ) ( fsub ψ)  
    fsub-fsub : {lv lv' : Nat} {α : OridinalD  lv } 
          ( ψ : OridinalD  lv → Set n ) ( ψ₁ : OridinalD  lv → Set n ) →
         ( ∀ ( x :  OridinalD  lv ) → ψ x →  ψ₁ x ) →  _c∋_  {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) 

_∋_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set n 
a ∋ x  = constructible a c∋ constructible 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 = {!!}

data _c≈_  : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' } →
        Constructible  {lv} α → Constructible  {lv'} α' → Set n where
    crefl :  {lv : Nat} {α : OridinalD  lv } → _c≈_  {_} {_} {α} {α} (xself α ) (xself α )
    feq :  {lv : Nat} {α : OridinalD  lv }
          → ( ψ : OridinalD  lv → Set n ) ( ψ₁ : OridinalD  lv → Set n ) 
          → (∀ ( x :  OridinalD  lv ) → ψ x  ⇔ ψ₁ x ) → _c≈_  {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)

_≈_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set n 
a ≈ x  = constructible a c≈ constructible x

ConstructibleSet→ZF : ZF {suc n} 
ConstructibleSet→ZF   = record { 
    ZFSet = ConstructibleSet 
    ; _∋_ = _∋_
    ; _≈_ = _≈_ 
    ; ∅  = record { level = Zero ; α = Φ ; constructible = xself Φ }
    ; _×_ = {!!}
    ; Union = {!!}
    ; Power = {!!}
    ; Select = {!!}
    ; Replace = {!!}
    ; infinite = {!!}
    ; isZF = {!!}
 }