module constructible-set  where

open import Level
open import zf

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

open import  Relation.Binary.PropositionalEquality

data Ordinal {n : Level } : (lv : Nat) → Set n where
   Φ : {lv : Nat} → Ordinal {n} lv
   T-suc : {lv : Nat} → Ordinal {n} lv → Ordinal lv
   ℵ_ :  (lv : Nat) → Ordinal (Suc lv)

data _o<_ {n : Level } :  {lx ly : Nat} → Ordinal {n} lx  →  Ordinal {n} ly  → Set n where
   l< : {lx ly : Nat }  → {x : Ordinal {n} lx } →  {y : Ordinal {n} ly } → lx < ly → x o< y
   Φ<  : {lx : Nat} → {x : Ordinal {n} lx}  →  Φ {n} {lx} o< T-suc {n} {lx} x
   s<  : {lx : Nat} → {x y : Ordinal {n} lx}  →  x o< y  → T-suc {n} {lx} x o< T-suc {n} {lx} y
   ℵΦ< : {lx : Nat} → {x : Ordinal {n} (Suc lx) } →  Φ {n} {Suc lx} o< (ℵ lx) 
   ℵ<  : {lx : Nat} → {x : Ordinal {n} (Suc lx) } →  T-suc {n} {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


nat< : { x y : Nat } → x ≡ y → x < y → ⊥
nat< {Zero} {Zero} refl ()
nat< {Suc x} {.(Suc x)} refl (s≤s t) = nat< {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> : {n : Level } → {lx ly : Nat} {x  : Ordinal {n} lx } { y : Ordinal {n} ly }  →  ly < lx → x o< y → ⊥
triO> {n} {lx} {ly} {x} {y} y<x xo<y with <-cmp  lx ly
triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c =  ¬c y<x 
triO> {n} {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c =  ¬c y<x 
triO> {n} {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c =  ¬a x₁ 
triO> {n} {lx} {ly} {Φ} {T-suc _} y<x Φ< | tri> ¬a ¬b c =  ¬b refl 
triO> {n} {lx} {ly} {T-suc px} {T-suc py} y<x (s< w) | tri> ¬a ¬b c =  triO> y<x w
triO> {n} {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
triO> {n} {lx} {ly} {(T-suc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c =  ¬b refl

trio! : {n : Level } → {lv : Nat} → {x  : Ordinal {n} lv }  → x o< x → ⊥
trio! {n} {lx} {x} (l< y) = nat< refl y
trio! {n} {lx} {T-suc y} (s< t) = trio! t

trio<> : {n : Level } → {lx : Nat} {x  : Ordinal {n} lx } { y : Ordinal {n} lx }  →  y o< x → x o< y → ⊥
trio<> {n} {lx} {x} {y} (l< lt) _ = nat< refl lt
trio<> {n} {lx} {x} {y} _ (l< lt)  = nat< refl lt
trio<> {n} {lx} {.(T-suc _)} {.(T-suc _)} (s< s) (s< t) = 
    trio<> s t

trio<≡ : {n : Level } → {lx : Nat} {x  : Ordinal {n} lx } { y : Ordinal {n} lx }  → x ≡ y  → x o< y → ⊥
trio<≡ refl = trio!

trio>≡ : {n : Level } → {lx : Nat} {x  : Ordinal {n} lx } { y : Ordinal {n} lx }  → x ≡ y  → y o< x → ⊥
trio>≡ refl = trio!

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

triOonSameLevel : {n : Level } → {lx : Nat}   → Trichotomous  _≡_ ( _o<_ {n} {lx} {lx} )
triOonSameLevel {n} {lv} Φ Φ = tri≈ trio! refl trio!
triOonSameLevel {n} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ trio! refl trio!
triOonSameLevel {n} {lv} Φ (T-suc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
triOonSameLevel {n} {.(Suc lv)} Φ (ℵ lv) = tri<  (ℵΦ< {n} {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {n} {lv} {Φ} )) )
triOonSameLevel {n} {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ< {n} {lv} {Φ} ) ) (λ ()) (ℵΦ< {n} {lv} {Φ} )
triOonSameLevel {n} {Suc lv} (ℵ lv) (T-suc y) = tri> ( λ lt → trio<> lt (ℵ< {n} {lv} {y} )  ) (λ ()) (ℵ< {n} {lv} {y} )
triOonSameLevel {n} {lv} (T-suc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
triOonSameLevel {n} {.(Suc lv)} (T-suc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
triOonSameLevel {n} {lv} (T-suc x) (T-suc y) with triOonSameLevel x y
triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) )  (  λ lt → trio<> lt (s< a) )
triOonSameLevel {n} {lv} (T-suc x) (T-suc x) | tri≈ ¬a refl ¬c = tri≈ trio! refl trio!
triOonSameLevel {n} {lv} (T-suc x) (T-suc y) | tri> ¬a ¬b c = tri> (  λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)


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 )

maxα> : {n : Level } → { lx ly : Nat } → Ordinal {n} lx  →  Ordinal {n} ly  → lx > ly  → Ordinal {n} lx
maxα> x y _ = x

maxα= : {n : Level } → { lx : Nat } → Ordinal {n} lx  →  Ordinal {n} lx  →  Ordinal {n} 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

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

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

record ConstructibleSet {n : Level } : Set (suc n) where
  field
    level : Nat
    α : Ordinal {n} level 
    constructible : Constructible α

open ConstructibleSet

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

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

transitiveness : {n : Level} → (a b c : ConstructibleSet {n}) → a ∋ b → b ∋ c → a ∋ c
transitiveness = {!!}

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

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

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