### view OD.agda @ 302:304c271b3d47

HOD and reduction mapping of Ordinals
author Shinji KONO Sun, 28 Jun 2020 18:09:04 +0900 b012a915bbb5 7963b76df6e1
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open import Level
open import Ordinals
module OD {n : Level } (O : Ordinals {n} ) 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
open import Data.Nat.Properties
open import Data.Empty
open import Relation.Nullary
open import Relation.Binary
open import Relation.Binary.Core

open import logic
open import nat

open inOrdinal O

-- Ordinal Definable Set

record OD : Set (suc n ) where
field
def : (x : Ordinal  ) → Set n

open OD

open _∧_
open _∨_
open Bool

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

id : {A : Set n} → A → A
id x = x

==-refl :  {  x :  OD  } → x == x
==-refl  {x} = record { eq→ = id ; eq← = id }

open  _==_

==-trans : { x y z : OD } →  x == y →  y == z →  x ==  z
==-trans x=y y=z  = record { eq→ = λ {m} t → eq→ y=z (eq→ x=y t) ; eq← =  λ {m} t → eq← x=y (eq← y=z t) }

==-sym : { x y  : OD } →  x == y →  y == x
==-sym x=y = record { eq→ = λ {m} t → eq← x=y t ; eq← =  λ {m} t → eq→ x=y t }

⇔→== :  {  x y :  OD  } → ( {z : Ordinal } → def x z ⇔  def y z) → x == y
eq→ ( ⇔→==  {x} {y}  eq ) {z} m = proj1 eq m
eq← ( ⇔→==  {x} {y}  eq ) {z} m = proj2 eq m

-- next assumptions are our axiom
--  In classical Set Theory, HOD is used, as a subset of OD,
--     HOD = { x | TC x ⊆ OD }
--  where TC x is a transitive clusure of x, i.e. Union of all elemnts of all subset of x.
--  This is not possible because we don't have V yet.
--  We simply assume V=OD here.
--
--  We also assumes ODs are isomorphic to Ordinals, which is ususally proved by Goedel number tricks.
--  ODs have an ovbious maximum, but Ordinals are not. This means, od→ord is not an on-to mapping.
--
--  ==→o≡ is necessary to prove axiom of extensionality.
--
--  In classical Set Theory, sup is defined by Uion. Since we are working on constructive logic,
--  we need explict assumption on sup.

record HOD (odmax : Ordinal) : Set (suc n) where
field
hmax : {x : Ordinal } → x o< odmax
hdef : Ordinal  → Set n

record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
field
os→ : (x : Ordinal) → x o< maxordinal → Ordinal
os← : Ordinal → Ordinal
os←limit : (x : Ordinal) → os← x o< maxordinal
os-iso← : {x : Ordinal} →  os→  (os← x) (os←limit x) ≡ x
os-iso→ : {x : Ordinal} → (lt : x o< maxordinal ) →  os← (os→ x lt) ≡ x

open HOD

-- HOD→OD : {x : Ordinal} → HOD x → OD
-- HOD→OD hod = record { def = hdef {!!} }

record ODAxiom : Set (suc n) where
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )
field
od→ord : OD  → Ordinal
ord→od : Ordinal  → OD
c<→o<  :  {x y : OD  }   → def y ( od→ord x ) → od→ord x o< od→ord y
oiso   :  {x : OD }      → ord→od ( od→ord x ) ≡ x
diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
==→o≡ : { x y : OD  } → (x == y) → x ≡ y
-- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
sup-o  :  ( OD → Ordinal ) →  Ordinal
sup-o< :  { ψ : OD →  Ordinal } → ∀ {x : OD } → ψ x  o<  sup-o ψ
-- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
-- sup-x  : {n : Level } → ( OD → Ordinal ) →  Ordinal
-- sup-lb : {n : Level } → { ψ : OD →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))

record HODAxiom : Set (suc n) where
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )
field
mod : Ordinal
mod-limit :  ¬ ((y : Ordinal) → mod ≡ osuc y)
os : OrdinalSubset mod
od→ord : HOD mod → Ordinal
ord→od : Ordinal  → HOD mod
c<→o<  :  {x y : HOD mod }   → hdef y (od→ord x) → od→ord x o< od→ord y
oiso   :  {x : HOD mod }      → ord→od ( od→ord x ) ≡ x
diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
==→o≡ : { x y : OD  } → (x == y) → x ≡ y
-- supermum as Replacement Axiom ( corresponding Ordinal of OD has maximum )
sup-o  :  ( HOD mod → Ordinal ) →  Ordinal
sup-o< :  { ψ : HOD mod →  Ordinal } → ∀ {x : HOD mod } → ψ x  o<  sup-o ψ
-- contra-position of mimimulity of supermum required in Power Set Axiom which we don't use
-- sup-x  : {n : Level } → ( OD → Ordinal ) →  Ordinal
-- sup-lb : {n : Level } → { ψ : OD →  Ordinal } → {z : Ordinal }  →  z o< sup-o ψ → z o< osuc (ψ (sup-x ψ))

postulate  odAxiom : ODAxiom
open ODAxiom odAxiom

data One : Set n where
OneObj : One

-- Ordinals in OD , the maximum
Ords : OD
Ords = record { def = λ x → One }

maxod : {x : OD} → od→ord x o< od→ord Ords
maxod {x} = c<→o< OneObj

-- we have to avoid this contradiction

bad-bad = osuc-< <-osuc (c<→o< { record { def = λ x → One }} OneObj)

-- Ordinal in OD ( and ZFSet ) Transitive Set
Ord : ( a : Ordinal  ) → OD
Ord  a = record { def = λ y → y o< a }

od∅ : OD
od∅  = Ord o∅

o<→c<→OD=Ord : ( {x y : Ordinal  } → x o< y → def (ord→od y) x ) → {x : OD } →  x ≡ Ord (od→ord x)
o<→c<→OD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
lemma1 : {y : Ordinal} → def x y → def (Ord (od→ord x)) y
lemma1 {y} lt = subst ( λ k → k o< od→ord x ) diso (c<→o< {ord→od y} {x} (subst (λ k → def x k ) (sym diso) lt))
lemma2 : {y : Ordinal} → def (Ord (od→ord x)) y → def x y
lemma2 {y} lt = subst (λ k → def k y ) oiso (o<→c< {y} {od→ord x} lt )

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

_c<_ : ( x a : OD  ) → Set n
x c< a = a ∋ x

cseq : {n : Level} →  OD  →  OD
cseq x = record { def = λ y → def x (osuc y) } where

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

sup-od : ( OD  → OD ) →  OD
sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )

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

otrans : {n : Level} {a x y : Ordinal  } → def (Ord a) x → def (Ord x) y → def (Ord a) y
otrans x<a y<x = ordtrans y<x x<a

def→o< :  {X : OD } → {x : Ordinal } → def X x → x o< od→ord X
def→o<  {X} {x} lt = o<-subst  {_} {_} {x} {od→ord X} ( c<→o< ( def-subst  {X} {x}  lt (sym oiso) (sym diso) )) diso diso

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

==-iso : { x y : OD  } → ord→od (od→ord x) == ord→od (od→ord y)  →  x == y
==-iso  {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  } {z : Ordinal } → def (ord→od (od→ord x)) z → def x z
lemma {x} {z} d = def-subst d oiso refl

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

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

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

o∅≡od∅ : ord→od (o∅ ) ≡ od∅
o∅≡od∅  = ==→o≡ lemma where
lemma0 :  {x : Ordinal} → def (ord→od o∅) x → def od∅ x
lemma0 {x} lt = o<-subst (c<→o<  {ord→od x} {ord→od o∅} (def-subst  {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
lemma1 :  {x : Ordinal} → def od∅ x → def (ord→od o∅) x
lemma1 {x} lt = ⊥-elim (¬x<0 lt)
lemma : ord→od o∅ == od∅
lemma = record { eq→ = lemma0 ; eq← = lemma1 }

ord-od∅ : od→ord (od∅ ) ≡ o∅
ord-od∅  = sym ( subst (λ k → k ≡  od→ord (od∅ ) ) diso (cong ( λ k → od→ord k ) o∅≡od∅ ) )

∅0 : record { def = λ x →  Lift n ⊥ } == od∅
eq→ ∅0 {w} (lift ())
eq← ∅0 {w} lt = lift (¬x<0 lt)

∅< : { x y : OD  } → def x (od→ord y ) → ¬ (  x  == od∅  )
∅<  {x} {y} d eq with eq→ (==-trans eq (==-sym ∅0) ) d
∅<  {x} {y} d eq | lift ()

∅6 : { x : OD  }  → ¬ ( x ∋ x )    --  no Russel paradox
∅6  {x} x∋x = o<¬≡ refl ( c<→o<  {x} {x} x∋x )

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

is-o∅ : ( x : Ordinal  ) → Dec ( x ≡ o∅  )
is-o∅ x with trio< x o∅
is-o∅ x | tri< a ¬b ¬c = no ¬b
is-o∅ x | tri≈ ¬a b ¬c = yes b
is-o∅ x | tri> ¬a ¬b c = no ¬b

_,_ : OD  → OD  → OD
x , y = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } --  Ord (omax (od→ord x) (od→ord y))

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

in-codomain : (X : OD  ) → ( ψ : OD  → OD  ) → OD
in-codomain  X ψ = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( def X y ∧  ( x ≡ od→ord (ψ (ord→od y )))))  }

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

ZFSubset : (A x : OD  ) → OD
ZFSubset A x =  record { def = λ y → def A y ∧  def x y }  --   roughly x = A → Set

OPwr :  (A :  OD ) → OD
OPwr  A = Ord ( sup-o  ( λ x → od→ord ( ZFSubset A x) ) )

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

record _⊆_ ( A B : OD   ) : Set (suc n) where
field
incl : { x : OD } → A ∋ x →  B ∋ x

open _⊆_

infixr  220 _⊆_

subset-lemma : {A x : OD  } → ( {y : OD } →  x ∋ y → ZFSubset A x ∋  y ) ⇔  ( x ⊆ A  )
subset-lemma  {A} {x} = record {
proj1 = λ lt  → record { incl = λ x∋z → proj1 (lt x∋z)  }
; proj2 = λ x⊆A lt → record { proj1 = incl x⊆A lt ; proj2 = lt }
}

open import Data.Unit

ε-induction : { ψ : OD  → Set (suc n)}
→ ( {x : OD } → ({ y : OD } →  x ∋ y → ψ y ) → ψ x )
→ (x : OD ) → ψ x
ε-induction {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc )  where
induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (ord→od oy)) → ψ (ord→od ox)
induction ox prev = ind  ( λ {y} lt → subst (λ k → ψ k ) oiso (prev (od→ord y) (o<-subst (c<→o< lt) refl diso )))
ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (ord→od oy)
ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (ord→od oy)} induction oy

-- minimal-2 : (x : OD  ) → ( ne : ¬ (x == od∅ ) ) → (y : OD ) → ¬ ( def (minimal x ne) (od→ord y)) ∧ (def x (od→ord  y) )
-- minimal-2 x ne y = contra-position ( ε-induction (λ {z} ind → {!!} ) x ) ( λ p → {!!} )

OD→ZF : ZF
OD→ZF   = record {
ZFSet = OD
; _∋_ = _∋_
; _≈_ = _==_
; ∅  = od∅
; _,_ = _,_
; Union = Union
; Power = Power
; Select = Select
; Replace = Replace
; infinite = infinite
; isZF = isZF
} where
ZFSet = OD             -- is less than Ords because of maxod
Select : (X : OD  ) → ((x : OD  ) → Set n ) → OD
Select X ψ = record { def = λ x →  ( def X x ∧ ψ ( ord→od x )) }
Replace : OD  → (OD  → OD  ) → OD
Replace X ψ = record { def = λ x → (x o< sup-o  ( λ x → od→ord (ψ x))) ∧ def (in-codomain X ψ) x }
_∩_ : ( A B : ZFSet  ) → ZFSet
A ∩ B = record { def = λ x → def A x ∧ def B x }
Union : OD  → OD
Union U = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((def U u) ∧ (def (ord→od u) x)))  }
_∈_ : ( A B : ZFSet  ) → Set n
A ∈ B = B ∋ A
Power : OD  → OD
Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
-- ｛_｝ : ZFSet → ZFSet
-- ｛ x ｝ = ( x ,  x )     -- it works but we don't use

data infinite-d  : ( x : Ordinal  ) → Set n where
iφ :  infinite-d o∅
isuc : {x : Ordinal  } →   infinite-d  x  →
infinite-d  (od→ord ( Union (ord→od x , (ord→od x , ord→od x ) ) ))

infinite : OD
infinite = record { def = λ x → infinite-d x }

infixr  200 _∈_
-- infixr  230 _∩_ _∪_
isZF : IsZF (OD )  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
isZF = record {
isEquivalence  = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
;   pair→  = pair→
;   pair←  = pair←
;   union→ = union→
;   union← = union←
;   empty = empty
;   power→ = power→
;   power← = power←
;   extensionality = λ {A} {B} {w} → extensionality {A} {B} {w}
;   ε-induction = ε-induction
;   infinity∅ = infinity∅
;   infinity = infinity
;   selection = λ {X} {ψ} {y} → selection {X} {ψ} {y}
;   replacement← = replacement←
;   replacement→ = replacement→
-- ;   choice-func = choice-func
-- ;   choice = choice
} where

pair→ : ( x y t : ZFSet  ) →  (x , y)  ∋ t  → ( t == x ) ∨ ( t == y )
pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡x ))
pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y ))

pair← : ( x y t : ZFSet  ) → ( t == x ) ∨ ( t == y ) →  (x , y)  ∋ t
pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x))
pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y))

empty : (x : OD  ) → ¬  (od∅ ∋ x)
empty x = ¬x<0

o<→c< :  {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
o<→c< lt = record { incl = λ z → ordtrans z lt }

⊆→o< :  {x y : Ordinal } → (Ord x) ⊆ (Ord y) →  x o< osuc y
⊆→o< {x} {y}  lt with trio< x y
⊆→o< {x} {y}  lt | tri< a ¬b ¬c = ordtrans a <-osuc
⊆→o< {x} {y}  lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
⊆→o< {x} {y}  lt | tri> ¬a ¬b c with (incl lt)  (o<-subst c (sym diso) refl )
... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))

union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
union→ X z u xx not = ⊥-elim ( not (od→ord u) ( record { proj1 = proj1 xx
; proj2 = subst ( λ k → def k (od→ord z)) (sym oiso) (proj2 xx) } ))
union← :  (X z : OD) (X∋z : Union X ∋ z) →  ¬  ( (u : OD ) → ¬ ((X ∋  u) ∧ (u ∋ z )))
union← X z UX∋z =  FExists _ lemma UX∋z where
lemma : {y : Ordinal} → def X y ∧ def (ord→od y) (od→ord z) → ¬ ((u : OD) → ¬ (X ∋ u) ∧ (u ∋ z))
lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → def X k ) (sym diso) (proj1 xx ) ; proj2 = proj2 xx }

ψiso :  {ψ : OD  → Set n} {x y : OD } → ψ x → x ≡ y   → ψ y
ψiso {ψ} t refl = t
selection : {ψ : OD → Set 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) →  X ∋ x → Replace X ψ ∋ ψ x
replacement← {ψ} X x lt = record { proj1 =  sup-c< ψ {x} ; proj2 = lemma } where
lemma : def (in-codomain X ψ) (od→ord (ψ x))
lemma not = ⊥-elim ( not ( od→ord x ) (record { proj1 = lt ; proj2 = cong (λ k → od→ord (ψ k)) (sym oiso)} ))
replacement→ : {ψ : OD → OD} (X x : OD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : OD) → ¬ (x == ψ y))
replacement→ {ψ} X x lt = contra-position lemma (lemma2 (proj2 lt)) where
lemma2 :  ¬ ((y : Ordinal) → ¬ def X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
→ ¬ ((y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)))
lemma2 not not2  = not ( λ y d →  not2 y (record { proj1 = proj1 d ; proj2 = lemma3 (proj2 d)})) where
lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od  y))) → (ord→od (od→ord x) == ψ (ord→od y))
lemma3 {y} eq = subst (λ k  → ord→od (od→ord x) == k ) oiso (o≡→== eq )
lemma :  ( (y : OD) → ¬ (x == ψ y)) → ( (y : Ordinal) → ¬ def X y ∧ (ord→od (od→ord x) == ψ (ord→od y)) )
lemma not y not2 = not (ord→od y) (subst (λ k → k == ψ (ord→od y)) oiso  ( proj2 not2 ))

---
--- Power Set
---
---    First consider ordinals in OD
---
--- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
--
--
∩-≡ :  { a b : OD  } → ({x : OD  } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
∩-≡ {a} {b} inc = record {
eq→ = λ {x} x<a → record { proj2 = x<a ;
proj1 = def-subst  {_} {_} {b} {x} (inc (def-subst  {_} {_} {a} {_} x<a refl (sym diso) )) refl diso  } ;
eq← = λ {x} x<a∩b → proj2 x<a∩b }
--
-- Transitive Set case
-- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t == t
-- OPwr (Ord a) is a sup of ZFSubset (Ord a) t, so OPwr (Ord a) ∋ t
-- OPwr  A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
--
ord-power← : (a : Ordinal ) (t : OD) → ({x : OD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
ord-power← a t t→A  = def-subst  {_} {_} {OPwr (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  {_} {_} {(Ord a)} {z}
( t→A (def-subst  {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
lemma1 :  {a : Ordinal } { t : OD }
→ (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t))) ≡ od→ord t
lemma1  {a} {t} eq = subst (λ k → od→ord (ZFSubset (Ord a) k) ≡ od→ord t ) (sym oiso) (cong (λ k → od→ord k ) (==→o≡ eq ))
lemma :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t)) ) o< sup-o  (λ x → od→ord (ZFSubset (Ord a) x))
lemma = sup-o<

--
-- Every set in OD is a subset of Ordinals, so make OPwr (Ord (od→ord A)) first
-- then replace of all elements of the Power set by A ∩ y
--
-- Power A = Replace (OPwr (Ord (od→ord A))) ( λ y → A ∩ y )

-- we have oly double negation form because of the replacement axiom
--
power→ :  ( A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → ¬ ¬ (A ∋ x)
power→ A t P∋t {x} t∋x = FExists _ lemma5 lemma4 where
a = od→ord A
lemma2 : ¬ ( (y : OD) → ¬ (t ==  (A ∩ y)))
lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t
lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
lemma3 y eq not = not (proj1 (eq→ eq t∋x))
lemma4 : ¬ ((y : Ordinal) → ¬ (t == (A ∩ ord→od y)))
lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t == ( A ∩ k )) (sym oiso) not1 ))
lemma5 : {y : Ordinal} → t == (A ∩ ord→od y) →  ¬ ¬  (def A (od→ord x))
lemma5 {y} eq not = (lemma3 (ord→od y) eq) not

power← :  (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
power← A t t→A = record { proj1 = lemma1 ; proj2 = lemma2 } where
a = od→ord A
lemma0 : {x : OD} → t ∋ x → Ord a ∋ x
lemma0 {x} t∋x = c<→o< (t→A t∋x)
lemma3 : OPwr (Ord a) ∋ t
lemma3 = ord-power← a t lemma0
lemma4 :  (A ∩ ord→od (od→ord t)) ≡ t
lemma4 = let open ≡-Reasoning in begin
A ∩ ord→od (od→ord t)
≡⟨ cong (λ k → A ∩ k) oiso ⟩
A ∩ t
≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
t
∎
lemma1 : od→ord t o< sup-o  (λ x → od→ord (A ∩ x))
lemma1 = subst (λ k → od→ord k o< sup-o   (λ x → od→ord (A ∩ x)))
lemma4 (sup-o<  {λ x → od→ord (A ∩ x)}  )
lemma2 :  def (in-codomain (OPwr (Ord (od→ord A))) (_∩_ A)) (od→ord t)
lemma2 not = ⊥-elim ( not (od→ord t) (record { proj1 = lemma3 ; proj2 = lemma6 }) ) where
lemma6 : od→ord t ≡ od→ord (A ∩ ord→od (od→ord t))
lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t == (A ∩ k)) (sym oiso) ( ∩-≡ t→A  )))

ord⊆power : (a : Ordinal) → (Ord (osuc a)) ⊆ (Power (Ord a))
ord⊆power a = record { incl = λ {x} lt →  power← (Ord a) x (lemma lt) } where
lemma : {x y : OD} → od→ord x o< osuc a → x ∋ y →  Ord a ∋ y
lemma lt y<x with osuc-≡< lt
lemma lt y<x | case1 refl = c<→o< y<x
lemma lt y<x | case2 x<a = ordtrans (c<→o< y<x) x<a

continuum-hyphotheis : (a : Ordinal) → Set (suc n)
continuum-hyphotheis a = Power (Ord a) ⊆ Ord (osuc a)

extensionality0 : {A B : OD } → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B
eq→ (extensionality0 {A} {B} eq ) {x} d = def-iso  {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
eq← (extensionality0 {A} {B} eq ) {x} d = def-iso  {B} {A} (sym diso) (proj2 (eq (ord→od x))) d

extensionality : {A B w : OD  } → ((z : OD ) → (A ∋ z) ⇔ (B ∋ z)) → (w ∋ A) ⇔ (w ∋ B)
proj1 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) ( ==→o≡ (extensionality0 {A} {B} eq) ) d
proj2 (extensionality {A} {B} {w} eq ) d = subst (λ k → w ∋ k) (sym ( ==→o≡ (extensionality0 {A} {B} eq) )) d

infinity∅ : infinite  ∋ od∅
infinity∅ = def-subst  {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
lemma : o∅ ≡ od→ord od∅
lemma =  let open ≡-Reasoning in begin
o∅
≡⟨ sym diso ⟩
od→ord ( ord→od o∅ )
≡⟨ cong ( λ k → od→ord k ) o∅≡od∅ ⟩
od→ord od∅
∎
infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
infinity x lt = def-subst  {_} {_} {infinite} {od→ord (Union (x , (x , x )))} ( isuc {od→ord x} lt ) refl lemma where
lemma :  od→ord (Union (ord→od (od→ord x) , (ord→od (od→ord x) , ord→od (od→ord x))))
≡ od→ord (Union (x , (x , x)))
lemma = cong (λ k → od→ord (Union ( k , ( k , k ) ))) oiso

Union = ZF.Union OD→ZF
Power = ZF.Power OD→ZF
Select = ZF.Select OD→ZF
Replace = ZF.Replace OD→ZF
isZF = ZF.isZF  OD→ZF
```