open import Level
module HOD 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 HOD {n : Level}  : Set (suc n) where
  field
    def : (x : Ordinal {n} ) → Set n
    otrans : {x : Ordinal {n} } → def x → { y : Ordinal {n} } → y o< x → def y

open HOD
open import Data.Unit

open Ordinal
open _∧_

record _==_ {n : Level} ( a b :  HOD {n} ) : Set n where
  field
     eq→ : ∀ { x : Ordinal {n} } → def a x → def b x 
     eq← : ∀ { x : Ordinal {n} } → def b x → def a x 

id : {n : Level} {A : Set n} → A → A
id x = x

eq-refl : {n : Level} {  x :  HOD {n} } → x == x
eq-refl {n} {x} = record { eq→ = id ; eq← = id }

open  _==_ 

eq-sym : {n : Level} {  x y :  HOD {n} } → x == y → y == x
eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq }

eq-trans : {n : Level} {  x y z :  HOD {n} } → x == y → y == z → x == z
eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y  t) ; eq← = λ t → eq← x=y ( eq← y=z t) }

⇔→== : {n : Level} {  x y :  HOD {suc n} } → ( {z : Ordinal {suc n}} → def x z ⇔  def y z) → x == y 
eq→ ( ⇔→== {n} {x} {y}  eq ) {z} m = proj1 eq m 
eq← ( ⇔→== {n} {x} {y}  eq ) {z} m = proj2 eq m 

-- Ordinal in HOD ( and ZFSet )
Ord : { n : Level } → ( a : Ordinal {n} ) → HOD {n}
Ord {n} a = record { def = λ y → y o< a ; otrans = lemma }  where
   lemma : {x : Ordinal} → x o< a → {y : Ordinal} → y o< x → y o< a
   lemma {x}  x<a {y} y<x = ordtrans {n} {y} {x} {a} y<x x<a

od∅ : {n : Level} → HOD {n} 
od∅ {n} = Ord o∅ 

postulate      
  -- HOD can be iso to a subset of Ordinal ( by means of Godel Set )
  od→ord : {n : Level} → HOD {n} → Ordinal {n}
  ord→od : {n : Level} → Ordinal {n} → HOD {n} 
  c<→o<  : {n : Level} {x y : HOD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
  oiso   : {n : Level} {x : HOD {n}}     → ord→od ( od→ord x ) ≡ x
  diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
  ord-Ord :{n : Level} {x : Ordinal {n}} →  x ≡ od→ord (Ord x)   
  ==→o≡ : {n : Level} →  { x y : HOD {suc n} } → (x == y) → x ≡ y
  -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set
  -- o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
  -- supermum as Replacement Axiom
  sup-o  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
  sup-o< : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → ∀ {x : Ordinal {n}} →  ψ x  o<  sup-o ψ 
  -- contra-position of mimimulity of supermum required in Power Set Axiom
  sup-x  : {n : Level } → ( Ordinal {n} → Ordinal {n}) →  Ordinal {n}
  sup-lb : {n : Level } → { ψ : Ordinal {n} →  Ordinal {n}} → {z : Ordinal {n}}  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))
  -- sup-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( sup-o ψ ) 
  minimul : {n : Level } → (x : HOD {suc n} ) → ¬ (x == od∅ )→ HOD {suc n} 
  -- this should be ¬ (x == od∅ )→ ∃ ox → x ∋ Ord ox  ( minimum of x )
  x∋minimul : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
  minimul-1 : {n : Level } → (x : HOD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord  y) )
  -- we should prove this in agda, but simply put here
  ===-≡ : {n : Level} { x y : HOD {suc n}} → x == y  → x ≡ y

Ord-ord : {n : Level } {ox : Ordinal {suc n}} → Ord ox ≡ ord→od ox
Ord-ord {n} {px} = trans (sym oiso) (cong ( λ k → ord→od k ) (sym ord-Ord)) 

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

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

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

cseq : {n : Level} →  HOD {n} →  HOD {n}
cseq x = record { def = λ y → def x (osuc y) ; otrans = lemma } where
   lemma : {ox : Ordinal} → def x (osuc ox) → { oy : Ordinal}→ oy o< ox → def x (osuc oy)
   lemma {ox}  oox<Ox {oy} oy<ox  = otrans x  oox<Ox (osucc oy<ox )

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

o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x 
o<→c< {n} {x} {y} lt = subst ( λ k → k o< y ) ord-Ord lt 

sup-od : {n : Level } → ( HOD {n} → HOD {n}) →  HOD {n}
sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )

sup-c< : {n : Level } → ( ψ : HOD {n} →  HOD {n}) → ∀ {x : HOD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( sup-o ( λ x → od→ord (ψ (ord→od x ))) )} {od→ord ( ψ x )}
        lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where
    lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x)))
    lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso)  )

o<→o> : {n : Level} →  { x y : Ordinal {n} } →  (Ord x == Ord y) → x o< y → ⊥
o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with o<-subst (yx (case1 lt)) ord-Ord refl
... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
... | ()
o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with  o<-subst (yx (case2 lt)) ord-Ord refl
... | oyx with o<¬≡ refl (c<→o< {n} {Ord x} oyx )
... | ()

Ord==→≡ : {n : Level} { x y : Ordinal {suc n}} → Ord x == Ord y → x ≡ y
Ord==→≡ {n} {x} {y} eq with trio< x y
Ord==→≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq a )
Ord==→≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
Ord==→≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) c )


∅3 : {n : Level} →  { x : Ordinal {n}}  → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n}
∅3 {n} {x} = TransFinite {n} c2 c3 x where
   c0 : Nat →  Ordinal {n}  → Set n
   c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x))  → x ≡ o∅ {n}
   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 | ()

transitive : {n : Level } { z y x : HOD {suc n} } → z ∋ y → y ∋ x → z  ∋ x
transitive  {n} {z} {y} {x} z∋y x∋y  with  ordtrans ( c<→o< {suc n} {x} {y} x∋y ) (  c<→o< {suc n} {y} {z} z∋y ) 
... | t = otrans z z∋y (c<→o< {suc n} {x} {y} x∋y ) 

∅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)

ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y }
ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso

-- avoiding lv != Zero error
orefl : {n : Level} →  { x : HOD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
orefl refl = refl

==-iso : {n : Level} →  { x y : HOD {n} } → ord→od (od→ord x) == ord→od (od→ord y)  →  x == y
==-iso {n} {x} {y} eq = record {
      eq→ = λ d →  lemma ( eq→  eq (def-subst d (sym oiso) refl )) ;
      eq← = λ d →  lemma ( eq←  eq (def-subst d (sym oiso) refl ))  }
        where
           lemma : {x : HOD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z
           lemma {x} {z} d = def-subst d oiso refl

=-iso : {n : Level } {x y : HOD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y)
=-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso)

ord→== : {n : Level} →  { x y : HOD {n} } → od→ord x ≡  od→ord y →  x == y
ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where
   lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy →  (ord→od ox) == (ord→od oy)
   lemma ox ox  refl = eq-refl

o≡→== : {n : Level} →  { x y : Ordinal {n} } → x ≡  y →  ord→od x == ord→od y
o≡→== {n} {x} {.x} refl = eq-refl

>→¬< : {x y : Nat  } → (x < y ) → ¬ ( y < x )
>→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x

c≤-refl : {n : Level} →  ( x : HOD {n} ) → x c≤ x
c≤-refl x = case1 refl

∋→o< : {n : Level} →  { a x : HOD {suc n} } → a ∋ x → od→ord x o< od→ord a
∋→o< {n} {a} {x} lt = t where
         t : (od→ord x) o< (od→ord a)
         t = c<→o< {suc n} {x} {a} lt 

o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
    lemma :  o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
    lemma lt with o<→c< lt 
    lemma lt | t = o<¬≡ refl t
o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)

o<→¬c> : {n : Level} →  { x y : HOD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (y c< x )
o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where

o≡→¬c< : {n : Level} →  { x y : HOD {n} } →  (od→ord x ) ≡ ( od→ord y) →   ¬ x c< y
o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡  (orefl oeq ) (c<→o< lt) 

∅0 : {n : Level} →  record { def = λ x →  Lift n ⊥ ; otrans = λ () } == od∅ {n} 
eq→ ∅0 {w} (lift ())
eq← ∅0 {w} (case1 ())
eq← ∅0 {w} (case2 ())

∅< : {n : Level} →  { x y : HOD {n} } → def x (od→ord y ) → ¬ (  x  == od∅ {n} )
∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d
∅< {n} {x} {y} d eq | lift ()
       
∅6 : {n : Level} →  { x : HOD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
∅6 {n} {x} x∋x = o<¬≡ refl ( c<→o< {suc n} {x} {x} x∋x )

def-iso : {n : Level} {A B : HOD {n}} {x y : Ordinal {n}} → x ≡ y  → (def A y → def B y)  → def A x → def B x
def-iso refl t = t

is-o∅ : {n : Level} →  ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} )
is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl
is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () )
is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ())


-- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
-- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))

csuc :  {n : Level} →  HOD {suc n} → HOD {suc n}
csuc x = Ord ( osuc ( od→ord x ))

-- Power Set of X ( or constructible by λ y → def X (od→ord y )

ZFSubset : {n : Level} → (A x : HOD {suc n} ) → HOD {suc n}
ZFSubset A x =  record { def = λ y → def A y ∧  def x y ; otrans = lemma }   where
  lemma : {z : Ordinal} → def A z ∧ def x z → {y : Ordinal} → y o< z → def A y ∧ def x y
  lemma {z} d {y} y<z = record { proj1 = otrans A (proj1 d) y<z ; proj2 = otrans x (proj2 d) y<z }

Def :  {n : Level} → (A :  HOD {suc n}) → HOD {suc n}
Def {n} A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )  

OrdSubset : {n : Level} → (A x : Ordinal {suc n} ) → ZFSubset (Ord A) (Ord x) ≡ Ord ( minα A x )
OrdSubset {n} A x = ===-≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
  lemma1 :  {y : Ordinal} → def (ZFSubset (Ord A) (Ord x)) y → def (Ord (minα A x)) y
  lemma1 {y} s with trio< A x
  lemma1 {y} s | tri< a ¬b ¬c = proj1 s
  lemma1 {y} s | tri≈ ¬a refl ¬c = proj1 s
  lemma1 {y} s | tri> ¬a ¬b c = proj2 s
  lemma2 : {y : Ordinal} → def (Ord (minα A x)) y → def (ZFSubset (Ord A) (Ord x)) y
  lemma2 {y} lt with trio< A x
  lemma2 {y} lt | tri< a ¬b ¬c = record { proj1 = lt ; proj2 = ordtrans lt a }
  lemma2 {y} lt | tri≈ ¬a refl ¬c = record { proj1 = lt ; proj2 = lt }
  lemma2 {y} lt | tri> ¬a ¬b c = record { proj1 = ordtrans lt c ; proj2 = lt }

-- Constructible Set on α
-- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y <  od→ord x } 
-- L (Φ 0) = Φ
-- L (OSuc lv n) = { Def ( L n )  } 
-- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
L : {n : Level} → (α : Ordinal {suc n}) → HOD {suc n}
L {n}  record { lv = Zero ; ord = (Φ .0) } = od∅
L {n}  record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) 
L {n}  record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
    cseq ( Ord (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }))))

L00 :   {n : Level} → (ox : Ordinal {suc n}) → ox o<  sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x )))
L00 {n} ox =  o<-subst {suc n} {_} {_} {ox} {sup-o ( λ x → od→ord ( ZFSubset (Ord ox) (ord→od x )))}
  (sup-o< {suc n} {λ x → od→ord ( ZFSubset (Ord ox) (ord→od x ))} {ox} ) (lemma0 ox) refl where
      lemma1 : {n : Level } {ox z : Ordinal {suc n}} → ( def (Ord ox) z ∧ def (ord→od ox) z ) ⇔ def ( Ord ox ) z
      lemma1 {n} {ox} {z} = record { proj1 = proj1 ; proj2 = λ t → record { proj1 = t ; proj2 = subst (λ k → def k z ) Ord-ord t }}
      lemma0 :  {n : Level} → (ox : Ordinal {suc n}) →  od→ord (ZFSubset (Ord ox) (ord→od ox)) ≡ ox
      lemma0 {n} ox = trans (cong (λ k → od→ord k) (===-≡ (⇔→== lemma1) )) (sym ord-Ord)

-- L0 :  {n : Level} → (α : Ordinal {suc n}) → α o< β → L (osuc α) ∋ L α
-- L1 :  {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : HOD {suc n})  → L α ∋ x → L β ∋ x 

omega : { n : Level } → Ordinal {n}
omega = record { lv = Suc Zero ; ord = Φ 1 }

HOD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
HOD→ZF {n}  = record { 
    ZFSet = HOD {suc n}
    ; _∋_ = _∋_ 
    ; _≈_ = _==_ 
    ; ∅  = od∅
    ; _,_ = _,_
    ; Union = Union
    ; Power = Power
    ; Select = Select
    ; Replace = Replace
    ; infinite = Ord omega
    ; isZF = isZF 
 } where
    Replace : HOD {suc n} → (HOD {suc n} → HOD {suc n} ) → HOD {suc n}
    Replace X ψ = sup-od ψ
    Select : (X : HOD {suc n} ) → ((x : HOD {suc n} ) → Set (suc n) ) → HOD {suc n}
    Select X ψ = record { def = λ x → ((y : Ordinal {suc n} ) → X ∋ ord→od y  → ψ (ord→od y)) ∧ (X ∋ ord→od x )  ; otrans = lemma } where
       lemma :  {x : Ordinal} → ((y : Ordinal) → X ∋ ord→od y → ψ (ord→od y)) ∧ (X ∋ ord→od x) →
            {y : Ordinal} → y o< x → ((y₁ : Ordinal) → X ∋ ord→od y₁ → ψ (ord→od y₁)) ∧ (X ∋ ord→od y)
       lemma {x} select {y} y<x = record { proj1 = proj1 select ; proj2 = y<X } where
           y<X : X ∋ ord→od y
           y<X = otrans X (proj2 select) (o<-subst y<x (sym diso) (sym diso)  )
    _,_ : HOD {suc n} → HOD {suc n} → HOD {suc n}
    x , y = Ord (omax (od→ord x) (od→ord y))
    Union : HOD {suc n} → HOD {suc n}
    Union U = cseq U 
    -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x →  X ∋ x )
    ZFSet = HOD {suc n}
    _∈_ : ( A B : ZFSet  ) → Set (suc n)
    A ∈ B = B ∋ A
    _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set (suc n)
    _⊆_ A B {x} = A ∋ x →  B ∋ x
    _∩_ : ( A B : ZFSet  ) → ZFSet
    A ∩ B = Select (A , B) (  λ x → ( A ∋ x ) ∧ (B ∋ x) )
    Power : HOD {suc n} → HOD {suc n}
    Power A = Replace (Def (Ord (od→ord A))) ( λ x → A ∩ x )
    -- _∪_ : ( A B : ZFSet  ) → ZFSet
    -- A ∪ B = Select (A , B) (  λ x → (A ∋ x)  ∨ ( B ∋ x ) )
    ｛_｝ : ZFSet → ZFSet
    ｛ x ｝ = ( x ,  x )

    infixr  200 _∈_
    -- infixr  230 _∩_ _∪_
    infixr  220 _⊆_
    isZF : IsZF (HOD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace (Ord omega)
    isZF = record {
           isEquivalence  = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans }
       ;   pair  = pair
       ;   union-u = λ X z UX∋z → union-u {X} {z} UX∋z
       ;   union→ = union→
       ;   union← = union←
       ;   empty = empty
       ;   power→ = power→  
       ;   power← = power← 
       ;   extensionality = extensionality
       ;   minimul = minimul
       ;   regularity = regularity
       ;   infinity∅ = infinity∅
       ;   infinity = λ _ → infinity
       ;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
       ;   replacement = replacement
     } where

         pair : (A B : HOD {suc n} ) → ((A , B) ∋ A) ∧  ((A , B) ∋ B)
         proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B)
         proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B)

         empty : (x : HOD {suc n} ) → ¬  (od∅ ∋ x)
         empty x (case1 ())
         empty x (case2 ())

         ---
         --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
         --- Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )       Power X is a sup of all subset of A
         --
         --  if Power A ∋ t, from a propertiy of minimum sup there is osuc ZFSubset A ∋ t 
         --    then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x
         --    In case of later, ZFSubset A ∋ t and t ∋ x implies A ∋ x by transitivity
         --
         POrd : {a : Ordinal } {t : HOD} → Def (Ord a) ∋ t → Def (Ord a) ∋ Ord (od→ord t)
         POrd {a} {t} P∋t = o<→c< P∋t
         ord-power→ : (a : Ordinal ) ( t : HOD) → Def (Ord a) ∋ t → {x : HOD} → t ∋ x → Ord a ∋ x
         ord-power→ a t P∋t {x} t∋x with osuc-≡<  (sup-lb  (POrd P∋t))
         ... | case1 eq = proj1 (def-subst (Ltx t∋x) (sym (subst₂ (λ j k → j ≡ k ) oiso oiso ( cong (λ k → ord→od k) (sym eq) ))) refl )  where
              Ltx :   {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x
              Ltx {n} {x} {t} lt = c<→o< lt
         ... | case2 lt = otrans (Ord a) (proj1 (lemma1 lt) )
                   (c<→o< {suc n} {x} {Ord (od→ord t)} (Ltx t∋x) ) where
              sp =  sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x)))
              minsup :  HOD
              minsup =  ZFSubset (Ord a) ( ord→od ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))) 
              Ltx :   {n : Level} → {x t : HOD {suc n}} → t ∋ x → Ord (od→ord t) ∋ x
              Ltx {n} {x} {t} lt = c<→o< lt
              lemma1 : od→ord (Ord (od→ord t)) o< od→ord minsup → minsup ∋ Ord (od→ord t)
              lemma1 lt with OrdSubset a ( sup-x (λ x → od→ord ( ZFSubset (Ord a) (ord→od x))))
              ... | eq with subst ( λ k →  ZFSubset (Ord a) k ≡ Ord (minα a sp)) Ord-ord eq
              ... | eq1 = def-subst {suc n} {_} {_} {minsup} {od→ord (Ord (od→ord t))} (o<→c< lt) lemma2 (sym ord-Ord) where
                  lemma2 : Ord (od→ord (ZFSubset (Ord a) (ord→od sp))) ≡ minsup
                  lemma2 =  let open ≡-Reasoning in begin
                      Ord (od→ord (ZFSubset (Ord a) (ord→od sp)))
                    ≡⟨ cong (λ k → Ord (od→ord k)) eq1 ⟩
                      Ord (od→ord (Ord (minα a sp))) 
                    ≡⟨ cong (λ k → Ord (od→ord k)) Ord-ord  ⟩
                      Ord (od→ord (ord→od (minα a sp))) 
                    ≡⟨ cong (λ k → Ord k) diso ⟩
                      Ord (minα a sp)
                    ≡⟨ sym eq1 ⟩
                      minsup
                    ∎

         -- 
         -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t
         -- Power A is a sup of ZFSubset A t, so Power A ∋ t
         -- 
         ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → Def (Ord a) ∋ t
         ord-power← a t t→A  = def-subst {suc n} {_} {_} {Def (Ord a)} {od→ord t}
                 lemma refl (lemma1 lemma-eq )where
              lemma-eq :  ZFSubset (Ord a) t == t
              eq→ lemma-eq {z} w = proj2 w 
              eq← lemma-eq {z} w = record { proj2 = w  ;
                 proj1 = def-subst {suc n} {_} {_} {(Ord a)} {z}
                    ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
              lemma1 : {n : Level } {a : Ordinal {suc n}} { t : HOD {suc n}}
                 → (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
              lemma1 {n} {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (===-≡ eq ))
              lemma :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset (Ord a) (ord→od x)))
              lemma = sup-o<   

         power→ :  ( A t : HOD) → Power A ∋ t → {x : HOD} → t ∋ x → A ∋ x
         power→ = {!!}
         power← :  (A t : HOD) → ({x : HOD} → (t ∋ x → A ∋ x)) → Power A ∋ t
         power← = {!!}

         union-u : {X z : HOD {suc n}} → (U>z : Union X ∋ z ) → HOD {suc n}
         union-u {X} {z} U>z = Ord ( osuc ( od→ord z ) )
         union→ :  (X z u : HOD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
         union→ X z u xx with trio< ( od→ord u ) ( osuc ( od→ord z ))
         union→ X z u xx | tri< a ¬b ¬c with  osuc-< a (c<→o< (proj2 xx))
         union→ X z u xx | tri< a ¬b ¬c | ()
         union→ X z u xx | tri≈ ¬a b ¬c = def-subst {suc n} {_} {_} {X} {osuc (od→ord z)} (proj1 xx) refl b 
         union→ X z u xx | tri> ¬a ¬b c = otrans X (proj1 xx) c
         union← :  (X z : HOD) (X∋z : Union X ∋ z) → (X ∋  union-u {X} {z} X∋z ) ∧ (union-u {X} {z} X∋z ∋ z )
         union← X z X∋z = record { proj1 = lemma ; proj2 = <-osuc } where
             lemma : X ∋ union-u {X} {z} X∋z
             lemma = def-subst {suc n} {_} {_} {X} {od→ord (Ord (osuc ( od→ord z )))} X∋z refl ord-Ord

         ψiso :  {ψ : HOD {suc n} → Set (suc n)} {x y : HOD {suc n}} → ψ x → x ≡ y   → ψ y
         ψiso {ψ} t refl = t
         selection : {X : HOD } {ψ : (x : HOD ) → Set (suc n)} {y : HOD} → (((y₁ : HOD) → X ∋ y₁ → ψ y₁) ∧ (X ∋ y)) ⇔ (Select X ψ ∋ y)
         selection {X} {ψ} {y} =  record { proj1 = λ s → record {
             proj1 = λ y1 y1<X → proj1 s (ord→od y1) y1<X ; proj2 = subst (λ k → def X k ) (sym diso) (proj2 s) }
           ; proj2 = λ s → record { proj1 = λ y1 dy1 → subst (λ k → ψ k ) oiso ((proj1 s) (od→ord y1) (def-subst {suc n} {_} {_} {X} {_} dy1 refl (sym diso )))
                                  ; proj2 = def-subst {suc n} {_} {_} {X} {od→ord y} (proj2 s ) refl diso } } where

         replacement : {ψ : HOD → HOD} (X x : HOD) → Replace X ψ ∋ ψ x
         replacement {ψ} X x = sup-c< ψ {x}

         ∅-iso :  {x : HOD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 
         ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq  
         regularity :  (x : HOD) (not : ¬ (x == od∅)) →
            (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅)
         proj1 (regularity x not ) = x∋minimul x not
         proj2 (regularity x not ) = record { eq→ = lemma1 ; eq← = λ {y} d → lemma2 {y} d } where
             lemma1 : {x₁ : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁ → def od∅ x₁
             lemma1 {x₁} s = ⊥-elim  ( minimul-1 x not (ord→od x₁) lemma3 ) where
                 lemma3 : def (minimul x not) (od→ord (ord→od x₁)) ∧ def x (od→ord (ord→od x₁))
                 lemma3 = proj1 s x₁ (proj2 s)
             lemma2 : {x₁ : Ordinal} → def od∅ x₁ → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) x₁
             lemma2 {y} d = ⊥-elim (empty (ord→od y) (def-subst {suc n} {_} {_} {od∅} {od→ord (ord→od y)} d refl (sym diso) ))

         extensionality : {A B : HOD {suc n}} → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
         eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d  
         eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d  

         open  import  Relation.Binary.PropositionalEquality
         uxxx-ord : {x  : HOD {suc n}} → {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ⇔ ( y o< osuc (od→ord x) )
         uxxx-ord {x} {y} = subst (λ k → k ⇔ ( y o< osuc (od→ord x) )) (sym lemma) ( osuc2 y (od→ord x))  where
              lemma : {y : Ordinal {suc n}} → def (Union (x , (x , x))) y ≡ osuc y o< osuc (osuc (od→ord x)) 
              lemma {y} = let open ≡-Reasoning in begin
                   def (Union (x , (x , x))) y  
                ≡⟨⟩
                   def ( Ord ( omax (od→ord x) (od→ord (Ord (omax (od→ord x)  (od→ord x)  )) ))) ( osuc y )
                ≡⟨⟩
                   osuc y o<  omax (od→ord x) (od→ord (Ord (omax (od→ord x)  (od→ord x)  )) )
                ≡⟨ cong (λ k → osuc y o<  omax (od→ord x) k ) (sym ord-Ord)  ⟩
                   osuc y o<  omax (od→ord x) (omax (od→ord x)  (od→ord x)  ) 
                ≡⟨ cong (λ k → osuc y o<  k ) (omxxx  (od→ord x) )  ⟩
                   osuc y o< osuc (osuc (od→ord x))
                ∎ 
         infinite : HOD {suc n}
         infinite = Ord omega 
         infinity∅ : Ord omega  ∋ od∅ {suc n}
         infinity∅ = o<-subst (case1 (s≤s z≤n) ) ord-Ord refl
         infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
         infinity x lt = o<-subst ( lemma (od→ord x) lt ) eq refl where
              eq :  osuc (od→ord x) ≡ od→ord (Union (x , (x , x)))
              eq = let open ≡-Reasoning in begin
                    osuc (od→ord x)
                 ≡⟨ ord-Ord  ⟩
                    od→ord (Ord (osuc (od→ord x)))
                 ≡⟨ cong ( λ k → od→ord  k ) ( sym (==→o≡ ( ⇔→==  uxxx-ord ))) ⟩
                    od→ord (Union (x , (x , x)))
                 ∎
              lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega 
              lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n)
              lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n)
              lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ()))
              lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ()))
              lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2
              lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl
         -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set
         -- ∀ z [ ∀ x ( x ∈ z  → ¬ ( x ≈ ∅ ) )  ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y )  → x ∩ y ≈ ∅  ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ ｛ t ｝) ]
         record Choice (z : HOD {suc n}) : Set (suc (suc n)) where
             field
                 u : {x : HOD {suc n}} ( x∈z  : x ∈ z ) → HOD {suc n}
                 t : {x : HOD {suc n}} ( x∈z  : x ∈ z ) → (x : HOD {suc n} ) → HOD {suc n}
                 choice : { x : HOD {suc n} } → ( x∈z  : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝
         -- choice : {x :  HOD {suc n}} ( x ∈ z  → ¬ ( x ≈ ∅ ) ) →
         -- axiom-of-choice : { X : HOD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : HOD } → (A∈X : A ∈ X ) →  choice ¬x∅ A∈X ∈ A 
         -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}