### changeset 297:be6670af87fa

maxod try
author Shinji KONO Mon, 22 Jun 2020 16:43:31 +0900 42f89e5efb00 3795ffb127d0 OD.agda 1 files changed, 69 insertions(+), 61 deletions(-) [+]
line wrap: on
line diff
```--- a/OD.agda	Mon Jun 15 18:15:48 2020 +0900
+++ b/OD.agda	Mon Jun 22 16:43:31 2020 +0900
@@ -70,15 +70,18 @@
record ODAxiom : Set (suc n) where
-- OD can be iso to a subset of Ordinal ( by means of Godel Set )
field
+  maxod : Ordinal
od→ord : OD  → Ordinal
-  ord→od : Ordinal  → OD
+  ord→od : (x : Ordinal ) → x o< maxod  → OD
+  o<max : {x : OD } → od→ord x o< maxod
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
+  oiso   :  {x : OD }      → ord→od ( od→ord x ) o<max ≡ x
+  diso   :  {x : Ordinal } → (lt : x o< maxod)   → od→ord ( ord→od x lt ) ≡ 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
+  sup-<od : { ψ : OD →  OD } → ∀ {x : OD } → sup-o (λ x → od→ord (ψ x)) o< maxod
-- 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))
@@ -93,8 +96,8 @@
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

-- Ordinal in OD ( and ZFSet ) Transitive Set
Ord : ( a : Ordinal  ) → OD
@@ -104,12 +107,12 @@
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 )
+-- 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 )
@@ -129,50 +132,55 @@
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)  )
+    lemma : od→ord (ψ (ord→od (od→ord x) o<max )) o< sup-o (λ x → od→ord (ψ x))
+    lemma = subst₂ (λ j k → j o< k ) refl (diso (sup-<od {ψ} {x}) ) (o<-subst (sup-o< ) refl (sym (diso sup-<od)))

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
+-- 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 lemma)))) (diso lemma) (diso o<max) where
+--     lemma : x o< maxod
+--     lemma = subst (λ k → k o< maxod ) (diso {!!} ) (otrans o<max ( c<→o< lt ))

-- 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 : OD  } → ord→od (od→ord x) o<max == ord→od (od→ord y) o<max →  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 : OD  } {z : Ordinal } → def (ord→od (od→ord x) o<max ) 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 :  {x y : OD  } → (x == y) ≡ (ord→od (od→ord x) o<max == y)
=-iso  {_} {y} = cong ( λ k → k == y ) (sym oiso)

+<-irr : {x y z : Ordinal } → x ≡ y → (x o< z) ≡ (y o< z)
+<-irr refl = refl
+
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
+ord→==  {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq) o<max o<max ) where
+    lemma : ( ox oy : Ordinal  ) → ox ≡ oy → (x<m : ox o< maxod) (y<m : oy o< maxod)  →  (ord→od ox x<m ) == (ord→od oy y<m )
+    lemma ox ox  refl x<m y<m = subst (λ  k → ord→od ox x<m == ord→od ox k) {!!} ==-refl

-o≡→== : { x y : Ordinal  } → x ≡  y →  ord→od x == ord→od y
+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∅ : 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} → 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 : 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∅ ) )
+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 ())
@@ -201,7 +209,7 @@
-- 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 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 )

@@ -234,10 +242,10 @@
→ ( {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
+     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 → {!!} )
@@ -258,13 +266,13 @@
} 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 )) }
+    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)))  }
+    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
@@ -275,7 +283,7 @@
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-d  (od→ord ( Union (ord→od x {!!} , (ord→od x {!!} , ord→od x {!!} ) ) ))

infinite : OD
infinite = record { def = λ x → infinite-d x }
@@ -321,16 +329,16 @@
⊆→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 ))
+         ⊆→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 }
+              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
@@ -345,13 +353,13 @@
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 :  ¬ ((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 ))
+                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
@@ -364,7 +372,7 @@
∩-≡ :  { 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  } ;
+                 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
@@ -379,11 +387,11 @@
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  }
+                    ( 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
+                 → (eq : ZFSubset (Ord a) t == t)  → od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max )) ≡ 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 :  od→ord (ZFSubset (Ord a) (ord→od (od→ord t) o<max ) ) o< sup-o  (λ x → od→ord (ZFSubset (Ord a) x))
lemma = sup-o<

--
@@ -401,10 +409,10 @@
lemma2 = replacement→ (Def (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 : ¬ ((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
+              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
@@ -413,9 +421,9 @@
lemma0 {x} t∋x = c<→o< (t→A t∋x)
lemma3 : Def (Ord a) ∋ t
lemma3 = ord-power← a t lemma0
-              lemma4 :  (A ∩ ord→od (od→ord t)) ≡ t
+              lemma4 :  (A ∩ ord→od (od→ord t) {!!} ) ≡ t
lemma4 = let open ≡-Reasoning in begin
-                    A ∩ ord→od (od→ord t)
+                    A ∩ ord→od (od→ord t) {!!}
≡⟨ cong (λ k → A ∩ k) oiso ⟩
A ∩ t
≡⟨ sym (==→o≡ ( ∩-≡ t→A )) ⟩
@@ -426,7 +434,7 @@
lemma4 (sup-o<  {λ x → od→ord (A ∩ x)}  )
lemma2 :  def (in-codomain (Def (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 : 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))
@@ -440,8 +448,8 @@
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
+         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
@@ -452,14 +460,14 @@
lemma : o∅ ≡ od→ord od∅
lemma =  let open ≡-Reasoning in begin
o∅
-                 ≡⟨ sym diso ⟩
-                    od→ord ( ord→od 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))))
+               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
```