### changeset 303:7963b76df6e1

¬odmax based HOD
author Shinji KONO Mon, 29 Jun 2020 17:56:06 +0900 304c271b3d47 2c111bfcc89a OD.agda 1 files changed, 162 insertions(+), 180 deletions(-) [+]
line wrap: on
line diff
```--- a/OD.agda	Sun Jun 28 18:09:04 2020 +0900
+++ b/OD.agda	Mon Jun 29 17:56:06 2020 +0900
@@ -67,10 +67,17 @@
--  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
+data One : Set n where
+  OneObj : One
+
+-- Ordinals in OD , the maximum
+Ords : OD
+Ords = record { def = λ x → One }
+
+record HOD : Set (suc n) where
field
-    hmax : {x : Ordinal } → x o< odmax
-    hdef : Ordinal  → Set n
+    od : OD
+    ¬odmax : ¬ (od ≡ Ords)

record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
field
@@ -88,34 +95,15 @@
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
+  od→ord : HOD  → Ordinal
+  ord→od : Ordinal  → HOD
+  c<→o<  :  {x y : HOD  }   → def (od y) ( od→ord x ) → od→ord x o< od→ord y
+  oiso   :  {x : HOD }      → ord→od ( od→ord x ) ≡ x
diso   :  {x : Ordinal } → od→ord ( ord→od x ) ≡ x
-  ==→o≡ : { x y : OD  } → (x == y) → x ≡ y
+  ==→o≡ : { x y : HOD  } → (od x == od 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
+  sup-od : ( HOD  → HOD ) →  HOD
+  sup-c< :  ( ψ : HOD  →  HOD ) → ∀ {x : HOD } → def (od ( sup-od ψ )) (od→ord ( ψ x ))
-- 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))
@@ -123,112 +111,103 @@
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
+-- 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)
+-- 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 }
+Ord : ( a : Ordinal  ) → HOD
+Ord  a = record { od = record { def = λ y → y o< a } ; ¬odmax = ? }

-od∅ : OD
+od∅ : HOD
od∅  = Ord o∅

+sup-o  :  ( HOD → Ordinal ) →  Ordinal
+sup-o  = ?
+sup-o< :  { ψ : HOD →  Ordinal } → ∀ {x : HOD } → ψ x  o<  sup-o ψ
+sup-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 )
+odef : HOD → Ordinal → Set n
+odef A x = def ( od A ) x

-_∋_ : ( a x : OD  ) → Set n
-_∋_  a x  = def a ( od→ord x )
+o<→c<→HOD=Ord : ( {x y : Ordinal  } → x o< y → odef (ord→od y) x ) → {x : HOD } →  x ≡ Ord (od→ord x)
+o<→c<→HOD=Ord o<→c< {x} = ==→o≡ ( record { eq→ = lemma1 ; eq← = lemma2 } ) where
+   lemma1 : {y : Ordinal} → odef x y → odef (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 → odef x k ) (sym diso) lt))
+   lemma2 : {y : Ordinal} → odef (Ord (od→ord x)) y → odef x y
+   lemma2 {y} lt = subst (λ k → odef k y ) oiso (o<→c< {y} {od→ord x} lt )

-_c<_ : ( x a : OD  ) → Set n
+_∋_ : ( a x : HOD  ) → Set n
+_∋_  a x  = odef a ( od→ord x )
+
+_c<_ : ( x a : HOD  ) → 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
+cseq : {n : Level} →  HOD  →  HOD
+cseq x = record { od = record { def = λ y → odef x (osuc y) } ; ¬odmax = ? } where

-sup-od : ( OD  → OD ) →  OD
-sup-od ψ = Ord ( sup-o ( λ x → od→ord (ψ x)) )
+odef-subst :  {Z : HOD } {X : Ordinal  }{z : HOD } {x : Ordinal  }→ odef Z X → Z ≡ z  →  X ≡ x  →  odef z x
+odef-subst df refl refl = df

-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 : {n : Level} {a x y : Ordinal  } → odef (Ord a) x → odef (Ord x) y → odef (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
+odef→o< :  {X : HOD } → {x : Ordinal } → odef X x → x o< od→ord X
+odef→o<  {X} {x} lt = o<-subst  {_} {_} {x} {od→ord X} ( c<→o< ( odef-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 : { x : HOD  } → { 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 : HOD  } → od (ord→od (od→ord x)) == od (ord→od (od→ord y))  →  od x == od 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 ))  }
+      eq→ = λ d →  lemma ( eq→  eq (odef-subst d (sym oiso) refl )) ;
+      eq← = λ d →  lemma ( eq←  eq (odef-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
+           lemma : {x : HOD  } {z : Ordinal } → odef (ord→od (od→ord x)) z → odef x z
+           lemma {x} {z} d = odef-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)
+=-iso :  {x y : HOD  } → (od x == od y) ≡ (od (ord→od (od→ord x)) == od y)
+=-iso  {_} {y} = cong ( λ k → od k == od y ) (sym oiso)

-ord→== : { x y : OD  } → od→ord x ≡  od→ord y →  x == y
+ord→== : { x y : HOD  } → od→ord x ≡  od→ord y →  od x == od 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 oy : Ordinal  ) → ox ≡ oy →  od (ord→od ox) == od (ord→od oy)
lemma ox ox  refl = ==-refl

-o≡→== : { x y : Ordinal  } → x ≡  y →  ord→od x == ord→od y
+o≡→== : { x y : Ordinal  } → x ≡  y →  od (ord→od x) == od (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
+     lemma0 :  {x : Ordinal} → odef (ord→od o∅) x → odef od∅ x
+     lemma0 {x} lt = o<-subst (c<→o<  {ord→od x} {ord→od o∅} (odef-subst  {ord→od o∅} {x} lt refl (sym diso)) ) diso diso
+     lemma1 :  {x : Ordinal} → odef od∅ x → odef (ord→od o∅) x
lemma1 {x} lt = ⊥-elim (¬x<0 lt)
-     lemma : ord→od o∅ == od∅
+     lemma : od (ord→od o∅) == od 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∅
+∅0 : record { def = λ x →  Lift n ⊥ } == od 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 : HOD  } → odef x (od→ord y ) → ¬ (  od x  == od 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 : HOD  }  → ¬ ( 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
+odef-iso : {A B : HOD } {x y : Ordinal } → x ≡ y  → (odef A y → odef B y)  → odef A x → odef B x
+odef-iso refl t = t

is-o∅ : ( x : Ordinal  ) → Dec ( x ≡ o∅  )
is-o∅ x with trio< x o∅
@@ -236,35 +215,35 @@
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))
+_,_ : HOD  → HOD  → HOD
+x , y = record { od = record { def = λ t → (t ≡ od→ord x ) ∨ ( t ≡ od→ord y ) } ; ¬odmax = ? }  --  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 )))))  }
+in-codomain : (X : HOD  ) → ( ψ : HOD  → HOD  ) → HOD
+in-codomain  X ψ = record { od = record { def = λ x → ¬ ( (y : Ordinal ) → ¬ ( odef X y ∧  ( x ≡ od→ord (ψ (ord→od y )))))  } ; ¬odmax = ? }

--- Power Set of X ( or constructible by λ y → def X (od→ord y )
+-- Power Set of X ( or constructible by λ y → odef 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
+ZFSubset : (A x : HOD  ) → HOD
+ZFSubset A x =  record { od = record { def = λ y → odef A y ∧  odef x y } ; ¬odmax = ? }  --   roughly x = A → Set

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

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

-record _⊆_ ( A B : OD   ) : Set (suc n) where
+record _⊆_ ( A B : HOD   ) : Set (suc n) where
field
-     incl : { x : OD } → A ∋ x →  B ∋ x
+     incl : { x : HOD } → 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 : HOD  } → ( {y : HOD } →  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 }
@@ -272,23 +251,23 @@

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 : HOD  → Set (suc n)}
+   → ( {x : HOD } → ({ y : HOD } →  x ∋ y → ψ y ) → ψ x )
+   → (x : HOD ) → 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 : HOD  ) → ( ne : ¬ (x == od∅ ) ) → (y : HOD ) → ¬ ( odef (minimal x ne) (od→ord y)) ∧ (odef 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
+HOD→ZF : ZF
+HOD→ZF   = record {
+    ZFSet = HOD
; _∋_ = _∋_
-    ; _≈_ = _==_
+    ; _≈_ = _=h=_
; ∅  = od∅
; _,_ = _,_
; Union = Union
@@ -298,18 +277,18 @@
; 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 }
+    ZFSet = HOD             -- is less than Ords because of maxod
+    Select : (X : HOD  ) → ((x : HOD  ) → Set n ) → HOD
+    Select X ψ = record { od = record { def = λ x →  ( odef X x ∧ ψ ( ord→od x )) } ; ¬odmax = ? }
+    Replace : HOD  → (HOD  → HOD  ) → HOD
+    Replace X ψ = record { od = record { def = λ x → (x o< sup-o  ( λ x → od→ord (ψ x))) ∧ odef (in-codomain X ψ) x } ; ¬odmax = ? }
_∩_ : ( 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 = record { od = record { def = λ x → odef A x ∧ odef B x }  ; ¬odmax = ? }
+    Union : HOD  → HOD
+    Union U = record { od = record { def = λ x → ¬ (∀ (u : Ordinal ) → ¬ ((odef U u) ∧ (odef (ord→od u) x)))  } ; ¬odmax = ? }
_∈_ : ( A B : ZFSet  ) → Set n
A ∈ B = B ∋ A
-    Power : OD  → OD
+    Power : HOD  → HOD
Power A = Replace (OPwr (Ord (od→ord A))) ( λ x → A ∩ x )
-- ｛_｝ : ZFSet → ZFSet
-- ｛ x ｝ = ( x ,  x )     -- it works but we don't use
@@ -319,12 +298,15 @@
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 }
+    infinite : HOD
+    infinite = record { od = record { def = λ x → infinite-d x } ; ¬odmax = ? }
+
+    _=h=_ : (x y : HOD) → Set n
+    x =h= y  = od x == od y

infixr  200 _∈_
-- infixr  230 _∩_ _∪_
-    isZF : IsZF (OD )  _∋_  _==_ od∅ _,_ Union Power Select Replace infinite
+    isZF : IsZF (HOD )  _∋_  _=h=_ od∅ _,_ Union Power Select Replace infinite
isZF = record {
isEquivalence  = record { refl = ==-refl ; sym = ==-sym; trans = ==-trans }
;   pair→  = pair→
@@ -345,15 +327,15 @@
-- ;   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  ) →  (x , y)  ∋ t  → ( t =h= x ) ∨ ( t =h= y )
+         pair→ x y t (case1 t≡x ) = case1 (subst₂ (λ j k → j =h= k ) oiso oiso (o≡→== t≡x ))
+         pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j =h= 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))
+         pair← : ( x y t : ZFSet  ) → ( t =h= x ) ∨ ( t =h= y ) →  (x , y)  ∋ t
+         pair← x y t (case1 t=h=x) = case1 (cong (λ k → od→ord k ) (==→o≡ t=h=x))
+         pair← x y t (case2 t=h=y) = case2 (cong (λ k → od→ord k ) (==→o≡ t=h=y))

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

o<→c< :  {x y : Ordinal } → x o< y → (Ord x) ⊆ (Ord y)
@@ -366,92 +348,92 @@
⊆→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 : HOD) → (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 )))
+              ; proj2 = subst ( λ k → odef k (od→ord z)) (sym oiso) (proj2 xx) } ))
+         union← :  (X z : HOD) (X∋z : Union X ∋ z) →  ¬  ( (u : HOD ) → ¬ ((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 }
+              lemma : {y : Ordinal} → odef X y ∧ odef (ord→od y) (od→ord z) → ¬ ((u : HOD) → ¬ (X ∋ u) ∧ (u ∋ z))
+              lemma {y} xx not = not (ord→od y) record { proj1 = subst ( λ k → odef 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 : HOD  → Set n} {x y : HOD } → 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 : HOD → Set n} {X y : HOD} → ((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))
+         replacement← : {ψ : HOD → HOD} (X x : HOD) →  X ∋ x → Replace X ψ ∋ ψ x
+         replacement← {ψ} X x lt = record { proj1 =  ? ; proj2 = lemma } where -- sup-c< ψ {x}
+             lemma : odef (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 : HOD → HOD} (X x : HOD) → (lt : Replace X ψ ∋ x) → ¬ ( (y : HOD) → ¬ (x =h= 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 :  ¬ ((y : Ordinal) → ¬ odef X y ∧ ((od→ord x) ≡ od→ord (ψ (ord→od y))))
+                    → ¬ ((y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= 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 ))
+                lemma3 : {y : Ordinal } → (od→ord x ≡ od→ord (ψ (ord→od  y))) → (ord→od (od→ord x) =h= ψ (ord→od y))
+                lemma3 {y} eq = subst (λ k  → ord→od (od→ord x) =h= k ) oiso (o≡→== eq )
+            lemma :  ( (y : HOD) → ¬ (x =h= ψ y)) → ( (y : Ordinal) → ¬ odef X y ∧ (ord→od (od→ord x) =h= ψ (ord→od y)) )
+            lemma not y not2 = not (ord→od y) (subst (λ k → k =h= ψ (ord→od y)) oiso  ( proj2 not2 ))

---
--- Power Set
---
-         ---    First consider ordinals in OD
+         ---    First consider ordinals in HOD
---
-         --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
+         --- ZFSubset A x =  record { def = λ y → odef A y ∧  odef x y }                   subset of A
--
--
-         ∩-≡ :  { a b : OD  } → ({x : OD  } → (a ∋ x → b ∋ x)) → a == ( b ∩ a )
+         ∩-≡ :  { a b : HOD  } → ({x : HOD  } → (a ∋ x → b ∋ x)) → a =h= ( 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  } ;
+                 proj1 = odef-subst  {_} {_} {b} {x} (inc (odef-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
+         -- we have t ∋ x → Ord a ∋ x means t is a subset of Ord a, that is ZFSubset (Ord a) t =h= 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}
+         ord-power← : (a : Ordinal ) (t : HOD) → ({x : HOD} → (t ∋ x → (Ord a) ∋ x)) → OPwr (Ord a) ∋ t
+         ord-power← a t t→A  = odef-subst  {_} {_} {OPwr (Ord a)} {od→ord t}
lemma refl (lemma1 lemma-eq )where
-              lemma-eq :  ZFSubset (Ord a) t == t
+              lemma-eq :  ZFSubset (Ord a) t =h= 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
+                 proj1 = odef-subst  {_} {_} {(Ord a)} {z}
+                    ( t→A (odef-subst  {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
+              lemma1 :  {a : Ordinal } { t : HOD }
+                 → (eq : ZFSubset (Ord a) t =h= 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
+         -- Every set in HOD 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 : HOD) → Power A ∋ t → {x : HOD} → 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 : ¬ ( (y : HOD) → ¬ (t =h=  (A ∩ y)))
lemma2 = replacement→ (OPwr (Ord (od→ord A))) t P∋t
-              lemma3 : (y : OD) → t == ( A ∩ y ) → ¬ ¬ (A ∋ x)
+              lemma3 : (y : HOD) → t =h= ( 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))
+              lemma4 : ¬ ((y : Ordinal) → ¬ (t =h= (A ∩ ord→od y)))
+              lemma4 not = lemma2 ( λ y not1 → not (od→ord y) (subst (λ k → t =h= ( A ∩ k )) (sym oiso) not1 ))
+              lemma5 : {y : Ordinal} → t =h= (A ∩ ord→od y) →  ¬ ¬  (odef 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 : HOD) → ({x : HOD} → (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 : HOD} → 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
@@ -466,14 +448,14 @@
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 :  odef (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  )))
+                  lemma6 = cong ( λ k → od→ord k ) (==→o≡ (subst (λ k → t =h= (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 : {x y : HOD} → 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
@@ -481,16 +463,16 @@
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
+         extensionality0 : {A B : HOD } → ((z : HOD) → (A ∋ z) ⇔ (B ∋ z)) → A =h= B
+         eq→ (extensionality0 {A} {B} eq ) {x} d = odef-iso  {A} {B} (sym diso) (proj1 (eq (ord→od x))) d
+         eq← (extensionality0 {A} {B} eq ) {x} d = odef-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)
+         extensionality : {A B w : HOD  } → ((z : HOD ) → (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
+         infinity∅ = odef-subst  {_} {_} {infinite} {od→ord (od∅ )} iφ refl lemma where
lemma : o∅ ≡ od→ord od∅
lemma =  let open ≡-Reasoning in begin
o∅
@@ -499,15 +481,15 @@
≡⟨ 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
+         infinity : (x : HOD) → infinite ∋ x → infinite ∋ Union (x , (x , x ))
+         infinity x lt = odef-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
+Union = ZF.Union HOD→ZF
+Power = ZF.Power HOD→ZF
+Select = ZF.Select HOD→ZF
+Replace = ZF.Replace HOD→ZF
+isZF = ZF.isZF  HOD→ZF```