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

open Ordinal

record _==_ {n : Level} ( a b :  OD {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 :  OD {n} } → x == x
eq-refl {n} {x} = record { eq→ = id ; eq← = id }

open  _==_ 

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

eq-trans : {n : Level} {  x y z :  OD {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) }

od∅ : {n : Level} → OD {n} 
od∅ {n} = record { def = λ x → x o< o∅ }

postulate      
  -- OD can be iso to a subset of Ordinal ( by means of Godel Set )
  od→ord : {n : Level} → OD {n} → Ordinal {n}
  ord→od : {n : Level} → Ordinal {n} → OD {n} 
  oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
  diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ 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 
  -- a contra-position of minimality of supermum 
  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-min : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → {z : Ordinal {n}}  →  ψ z  o<  z  →   sup-o ψ  o< osuc z
  minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
  x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) )
  minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord  y) )


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

Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n}
Ord {n} a = record { def = λ y → y o< a } 

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

postulate      
   c<→o< : { n : Level } → { x a : Ordinal {n} } → Ord a  ∋ Ord x →  x o<  a
   o<→c< : { n : Level } → { x a : Ordinal {n} } → x o< a → Ord a ∋ Ord x 

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


∅∨ : { n : Level } → { x y : Ordinal {suc n} } → ( Ord {suc n} x == Ord y ) ∨ ( ¬ ( Ord x == Ord y ) )
∅∨ {n} {x} {y} with trio< x y
∅∨ {n} {x} {y} | tri< a ¬b ¬c = case2 ( λ eq → ¬b ( ==→o≡ eq ) )
∅∨ {n} {x} {y} | tri≈ ¬a refl ¬c = case1 ( record { eq→ =  id ; eq← = id } ) 
∅∨ {n} {x} {y} | tri> ¬a ¬b c = case2 ( λ eq → ¬b ( ==→o≡ eq ) )

-- ¬x∋x' : { n : Level } → { x  : Ordinal {n} } → ¬ ( record { def = λ y → y o< x } ∋ record { def = λ y → y o< x } )
-- ¬x∋x' {n} {record { lv = Zero ; ord = ord }} (case1 ())
-- ¬x∋x' {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = lx ; ord = Φ lx }} (case1 {!!}) 
-- ¬x∋x' {n} {record { lv = Suc lx ; ord = OSuc (Suc lx) ox }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = Suc lx ; ord = ox}}  (case1 {!!}) 
-- ¬x∋x' {n} {record { lv = lv ; ord = Φ (lv) }} (case2 ())
-- ¬x∋x' {n} {record { lv = lv ; ord = OSuc (lv) ox }} (case2 x) = 
--    ¬x∋x' {n} {record { lv = lv ; ord = ox }} (case2 {!!}) 

-- ¬x∋x : { n : Level } → { x  : OD {n} } → ¬ x ∋ x
-- ¬x∋x = {!!}

oc-lemma : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → def (record { def = λ y → y o< oa }) oa → ⊥
oc-lemma {n} {x} {oa} lt = o<¬≡ oa oa refl lt

-- oc-lemma1 : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → od→ord (record { def = λ y → y o< oa }) o< oa → ⊥
-- oc-lemma1 {n} {x} {oa} lt = ¬x∋x' {n}   lt -- lt : def (record { def = λ y → y o< oa }) (record { def = λ y → y o< oa })

-- this one cannot be proved because if we have this OD and Ordinal has one to one corespondence
-- oc-lemma2 : { n : Level } → { x a : OD {n} } → { oa : Ordinal {n} } → oa o< od→ord (record { def = λ y → y o< oa }) → ⊥
--   this is not allowed in our case. ( avoid one-to-one of Ord and OD )
-- Ord=ord→od : {n : Level} →  { x : Ordinal {n} } → Ord x ≡  ord→od x

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

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

o<-def : {n : Level } {x y : Ordinal {n} } → x o< y  →  def (record { def = λ x → x o< y }) x
o<-def x<y = x<y

def-o< : {n : Level } {x y : Ordinal {n} } → def (record { def = λ x → x o< y }) x → x o< y    
def-o< x<y = x<y

sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
sup-od ψ = record { def = λ y → y o< ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) }

sup-c< : {n : Level } → ( ψ : OD {n} →  OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x ))
sup-c< {n} ψ {x} = def-subst {n} {_} {_} {record { def = λ y → y o< ( 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)  )

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

∅1 : {n : Level} →  ( x : Ordinal {n} )  → ¬ ( x c< o∅ {n} )
∅1 {n} record { lv = Zero ; ord = (Φ .0) } (case1 ())
∅1 {n} record { lv = Zero ; ord = (Φ .0) } (case2 ())
∅1 {n} record { lv = Zero ; ord = (OSuc .0 ox) } (case1 ())
∅1 {n} record { lv = Zero ; ord = (OSuc .0 ox) } (case2 ())
∅1 {n} record { lv = (Suc lx) ; ord = (Φ .(Suc lx)) } (case1 ())
∅1 {n} record { lv = (Suc lx) ; ord = (Φ .(Suc lx)) } (case2 ())
∅1 {n} record { lv = (Suc lx) ; ord = (OSuc .(Suc lx) ox) } (case1 ())
∅1 {n} record { lv = (Suc lx) ; ord = (OSuc .(Suc lx) ox) } (case2 ())

∅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-Ord : {n : Level } { z y x : Ordinal {suc n} } → Ord z ∋ Ord y → Ord y ∋ Ord x → Ord z  ∋ Ord x
transitive-Ord {n} {z} {y} {x} z∋y x∋y  =  o<→c< ( ordtrans ( c<→o< {suc n} {x} {y} x∋y ) (  c<→o< {suc n} {y} {z} z∋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 : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y
orefl refl = refl

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

ord→== : {n : Level} →  { x y : OD {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

O≡→== : {n : Level} →  { x y : Ordinal {n} } → x ≡  y →  Ord x == Ord 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 : OD {n} ) → x c≤ x
c≤-refl x = case1 refl

o<→o> : {n : Level} →  { x y : Ordinal {suc n} } →  (Ord x == Ord y) → x o< y → ⊥
o<→o> {n} {x} {y} eq lt with   ==→o≡ {n} eq 
... | refl = o<¬≡ _ _ refl lt

==-def-r : {n : Level } {x y z : Ordinal {suc n} } → (Ord x == Ord y) → def (Ord x) z → def (Ord y) z
==-def-r {n} {x} {y} {z} eq z>x = eq→ eq z>x  

o<→¬== : {n : Level} →  { x y : Ordinal {suc n} } → x o< y →  ¬ (Ord x == Ord y )
o<→¬== {n} {x} {y} lt eq = o<→o> {n} eq lt

∅< : {n : Level} →  { x y : OD {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 : OD {suc n} }  → ¬ ( x ∋ x )    --  no Russel paradox
-- ∅6 {n} {x} x∋x = c<> {n} {{!!}} {{!!}} {!!} {!!}

def-iso : {n : Level} {A B : OD {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 _∧_

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


¬∅=→∅∈ :  {n : Level} →  { x : Ordinal {suc n} } → ¬ (  Ord x  == od∅ {suc n} ) → Ord x ∋ od∅ {suc n} 
¬∅=→∅∈  {n} {x} ne with is-o∅ x
¬∅=→∅∈ {n} {x} ne | yes refl = ⊥-elim ( ne (eq-sym (eq-refl) ))
¬∅=→∅∈ {n} {x} ne | no ¬p = o<-subst (∅5 ¬p) ord-od∅ refl


-- 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} →  OD {suc n} → OD {suc n}
csuc x = ord→od ( osuc ( od→ord x ))

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

ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n}
ZFSubset A x =  record { def = λ y → def A y ∧  def x y }  

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

-- Constructible Set on α
L : {n : Level} → (α : Ordinal {suc n}) → OD {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 )
       record { def = λ y → osuc y o< (od→ord (L {n}  (record { lv = lx ; ord = Φ lx }) )) }

Ordsuc : {n : Level} → Ordinal {suc n} → OD {suc n}
Ordsuc x = record { def = λ y → y o< osuc x  }
OrdSubset : {n : Level} →(A x : Ordinal {suc n} ) → OD {suc n}
OrdSubset A x =  record { def = λ y → (y o< A) ∧ (y  o< x ) }  
OrdDef :  {n : Level} → (A :  Ordinal {suc n}) → OD {suc n}
OrdDef A = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( OrdSubset A x))) }

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

OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
OD→ZF {n}  = record { 
    ZFSet = OD {suc n}
    ; _∋_ = _∋_ 
    ; _≈_ = _==_ 
    ; ∅  = od∅
    ; _,_ = _,_
    ; Union = Union
    ; Power = Power
    ; Select = Select
    ; Replace = Replace
    ; infinite = Ord omega
    ; isZF = isZF 
 } where
    Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n}
    Replace X ψ = sup-od ψ
    Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n}
    Select X ψ = record { def = λ x →  ( def X  x ∧  ψ ( ord→od x )) } 
    _,_ : OD {suc n} → OD {suc n} → OD {suc n}
    x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) }
    Union : OD {suc n} → OD {suc n}
    Union U = record { def = λ y → osuc y o< (od→ord U) }
    -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x →  X ∋ x )
    Power : OD {suc n} → OD {suc n}
    Power A = Def A
    ZFSet = OD {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) )
    -- _∪_ : ( 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 (OD {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 _ z _ → csuc z
       ;   union→ = union→
       ;   union← = union←
       ;   empty = empty
       ;   power→ = power→
       ;   power← = power← 
       ;   extensionality = extensionality
       ;   minimul = minimul
       ;   regularity = regularity
       ;   infinity∅ = infinity∅
       ;   infinity = λ _ → infinity
       ;   selection = λ {ψ} {X} {y} → selection {ψ} {X} {y}
       ;   replacement = replacement
     } where
         open _∧_ 
         pair : (A B : OD {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 : OD {suc n} ) → ¬  (od∅ ∋ x)
         empty x (case1 ())
         empty x (case2 ())

         power→Ord : (A t : Ordinal) → OrdDef {suc n} A ∋ (Ord t) → {x : OD} → Ord t ∋ x → Ord A ∋ x
         power→Ord A t P∋t {x} t∋x = proj1 lemma-s where
              minsup :  OD
              minsup =  OrdSubset A ( sup-x (λ x → od→ord ( OrdSubset A x)))
              lemma-t : Ord (sup-x (λ x → od→ord ( OrdSubset A x))) ∋ Ord t
              lemma-t = {!!} -- sup-lb  P∋t = (od→ord (OrdSubset A (ord→od (sup-x (λ x₁ → od→ord (OrdSubset A (ord→od x₁)))))))
              lemma-s : OrdSubset A ( sup-x (λ x → od→ord ( OrdSubset A x)))  ∋ x
              lemma-s with osuc-≡< {suc n} ( o<-subst (c<→o< {!!} ) refl diso  )
              lemma-s | case1 eq = {!!}
              lemma-s | case2 lt = {!!}
         power←Ord : (A t : Ordinal) → ({x : OD} → (Ord t ∋ x → Ord A ∋ x)) → OrdDef {suc n} A ∋ Ord t
         power←Ord = {!!}
         ---
         ---   ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
         ---   Def {n} A    = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )))) }
         ---      = 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 minimulity of 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 ZFSubset A ∋ x by transitivity
         --
         power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x
         power→ A t P∋t {x} t∋x = {!!}
         -- 
         -- 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
         -- 
         power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
         power← A t t→A  = {!!}
         union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
         union→ = {!!}
         union← :  (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z )
         union← = {!!}
         ψiso :  {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y   → ψ y
         ψiso {ψ} t refl = t
         selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y)
         selection {ψ} {X} {y} = record {
              proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso)  }
            ; proj2 = λ select → record { proj1 = proj1 select  ; proj2 =  ψiso {ψ} (proj2 select) oiso  }
           }
         replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x
         replacement {ψ} X x = sup-c< ψ {x}
         ∅-iso :  {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) 
         ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq  
         regularity :  (x : OD) (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→ = reg ; eq← = {!!} } where
            reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y
            reg {y} t  with minimul-1 x not (ord→od y) (proj2 t ) 
            ... | t1 = {!!}
         extensionality : {A B : OD {suc n}} → ((z : OD) → (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  
         infinite : OD {suc n}
         infinite = Ord omega 
         infinity∅ : Ord omega  ∋ od∅ {suc n}
         infinity∅ = {!!}
         infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega
         infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where
              t  : od→ord x o< od→ord (ord→od (omega))
              t  = {!!}
         infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x ))
         infinite∋uxxx x lt = {!!} where
              t  :  od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega))
              t  = subst (λ k →  od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) {!!} lt ) 
         infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
         infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt ))   where
              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 : OD {suc n}) : Set (suc (suc n)) where
             field
                 u : {x : OD {suc n}} ( x∈z  : x ∈ z ) → OD {suc n}
                 t : {x : OD {suc n}} ( x∈z  : x ∈ z ) → (x : OD {suc n} ) → OD {suc n}
                 choice : { x : OD {suc n} } → ( x∈z  : x ∈ z ) → ( u x∈z ∩ x) == ｛ t x∈z x ｝
         -- choice : {x :  OD {suc n}} ( x ∈ z  → ¬ ( x ≈ ∅ ) ) →
         -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) →  choice ¬x∅ A∈X ∈ A 
         -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}