--- a/ordinal-definable.agda	Sat Jun 08 13:18:10 2019 +0900
+++ b/ordinal-definable.agda	Sat Jun 08 17:33:09 2019 +0900
@@ -5,13 +5,10 @@
 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
 
@@ -26,16 +23,6 @@
 
 open Ordinal
 
-postulate      
-  od→ord : {n : Level} → OD {n} → Ordinal {n}
-  ord→od : {n : Level} → Ordinal {n} → OD {n} 
-
-_∋_ : { n : Level } → ( a x : OD {n} ) → Set n
-_∋_ {n} a x  = def a ( od→ord x )
-
-_c<_ : { n : Level } → ( x a : OD {n} ) → Set n
-x c< a = a ∋ x 
-
 record _==_ {n : Level} ( a b :  OD {n} ) : Set n where
   field
      eq→ : ∀ { x : Ordinal {n} } → def a x → def b x 
@@ -55,19 +42,37 @@
 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) }
 
-_c≤_ : {n : Level} →  OD {n} →  OD {n} → Set (suc n)
-a c≤ b  = (a ≡ b)  ∨ ( b ∋ a )
-
 od∅ : {n : Level} → OD {n} 
 od∅ {n} = record { def = λ _ → Lift n ⊥ }
 
 postulate      
-  c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord y
-  o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y
-  oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x
-  diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
-  sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
-  sup-c< : {n : Level } → ( ψ : OD {n} →  OD {n}) → ∀ {x : OD {n}} → ψ x  c< sup-od ψ
+  od→ord : {n : Level} → OD {n} → Ordinal {n}
+  ord→od : {n : Level} → Ordinal {n} → OD {n} 
+  c<→o<  : {n : Level} {x y : OD {n} }      → def y ( od→ord x ) → od→ord x o< od→ord y
+  o<→c<  : {n : Level} {x y : Ordinal {n} } → x o< y             → def (ord→od y) x 
+  oiso   : {n : Level} {x : OD {n}}      → ord→od ( od→ord x ) ≡ x
+  diso   : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x
+  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 
+
+_∋_ : { n : Level } → ( a x : OD {n} ) → Set n
+_∋_ {n} a x  = def a ( od→ord x )
+
+_c<_ : { n : Level } → ( x a : OD {n} ) → Set n
+x c< a = a ∋ 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
+
+sup-od : {n : Level } → ( OD {n} → OD {n}) →  OD {n}
+sup-od ψ = ord→od ( 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} {_} {_} {sup-od ψ} {od→ord ( ψ x )}
+        ( o<→c< ( sup-o< ( λ y → od→ord (ψ (ord→od y ))) {od→ord x } )) refl (cong ( λ k → od→ord (ψ k) ) oiso)
 
 ∅1 : {n : Level} →  ( x : OD {n} )  → ¬ ( x c< od∅ {n} )
 ∅1 {n} x (lift ())
@@ -89,16 +94,13 @@
    ... | t with t (case2 (s< s<refl ) )
    c3 lx (OSuc .lx x₁) d not | t | ()
 
-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
-
 transitive : {n : Level } { z y x : OD {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 = lemma0 (lemma t) where
-   lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x )))
+   lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x)
    lemma xo<z = o<→c< xo<z
-   lemma0 :  def ( ord→od ( od→ord z )) ( od→ord ( ord→od ( od→ord x ))) →  def z (od→ord x)
-   lemma0 dz  = def-subst {suc n} { ord→od ( od→ord z )} { od→ord ( ord→od ( od→ord x))} dz (oiso) (diso)
+   lemma0 :  def ( ord→od ( od→ord z )) ( od→ord x) →  def z (od→ord x)
+   lemma0 dz  = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso)  refl
 
 record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where
   field
@@ -144,11 +146,11 @@
 
 o<→o> : {n : Level} →  { x y : OD {n} } →  (x == y) → (od→ord x ) o< ( od→ord y) → ⊥
 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with
-     yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case1 lt )) oiso diso )
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case1 lt )) oiso refl )
 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
 ... | ()
 o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with
-     yx (def-subst {n} {ord→od (od→ord y)} {od→ord (ord→od (od→ord x))} (o<→c< (case2 lt )) oiso diso )
+     yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} (o<→c< (case2 lt )) oiso refl )
 ... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
 ... | ()
 
@@ -161,10 +163,10 @@
 ≡-def : {n : Level} →  { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } )
 ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where
     lemma :  ord→od x == record { def = λ z → z o< x }
-    eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso diso t where 
-         t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
-         t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso))
-    eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl diso
+    eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where 
+        t : (od→ord ( ord→od w)) o< (od→ord (ord→od x))
+        t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso))
+    eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} ( o<→c< {suc n} {_} {_} z ) refl refl
 
 od≡-def : {n : Level} →  { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } 
 od≡-def {n} {x} = subst (λ k  → ord→od x ≡ k ) oiso (cong ( λ k → ord→od k ) (≡-def {n} {x} ))
@@ -175,8 +177,8 @@
          t = c<→o< {suc n} {x} {a} lt 
 
 o<∋→ : {n : Level} →  { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x 
-o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso diso t  where
-         t : def (ord→od (od→ord a)) (od→ord (ord→od (od→ord x)))
+o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t  where
+         t : def (ord→od (od→ord a)) (od→ord x)
          t = o<→c< {suc n} {od→ord x} {od→ord a} lt 
 
 o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n}
@@ -199,9 +201,9 @@
 
 tri-c< : {n : Level} →  Trichotomous _==_ (_c<_ {suc n})
 tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) 
-tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso diso) (o<→¬== a) ( o<→¬c> a )
+tri-c< {n} x y | tri< a ¬b ¬c = tri< (def-subst (o<→c< a) oiso refl) (o<→¬== a) ( o<→¬c> a )
 tri-c< {n} x y | tri≈ ¬a b ¬c = tri≈ (o≡→¬c< b) (ord→== b) (o≡→¬c< (sym b))
-tri-c< {n} x y | tri> ¬a ¬b c = tri>  ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso diso)
+tri-c< {n} x y | tri> ¬a ¬b c = tri>  ( o<→¬c> c) (λ eq → o<→¬== c (eq-sym eq ) ) (def-subst (o<→c< c) oiso refl)
 
 c<> : {n : Level } { x y : OD {suc n}} → x c<  y  → y c< x  →  ⊥
 c<> {n} {x} {y} x<y y<x with tri-c< x y
@@ -240,13 +242,13 @@
          lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅
          lemma1 = cong ( λ k → od→ord k ) o∅≡od∅
      lemma o∅ ne | yes refl | ()
-     lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (∅5 ¬p))  
+     lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ (o<→c< (subst (λ k → k o< ox ) (sym diso) (∅5 ¬p)) )  
 
 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) 
 -- postulate f-extensionality : { n : Level}  → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n))
 
-Def : OD {suc n} → OD {suc n}
-Def X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → def X x → def (ord→od y) x  }
+Def :  {n : Level} →  OD {suc n} → OD {suc n}
+Def {n} X = record { def = λ y → ∀ (x : Ordinal {suc n} ) → def X x → def (ord→od y) x  }
 
 
 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
@@ -313,15 +315,15 @@
          empty x ()
          --- Power X = record { def = λ t → ∀ (x : Ordinal {suc n} ) → def (ord→od t) x  →  def X x }
          power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x
-         power→ A t P∋t {x} t∋x = ?
+         power→ A t P∋t {x} t∋x = {!!}
          power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t
-         power← A t t→A z _ = ?
+         power← A t t→A z _ = {!!}
          union-u : (X z : OD {suc n}) → Union X ∋ z → OD {suc n}
          union-u X z U>z = ord→od ( osuc ( od→ord z ) )
          union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → union-u X z U>z  ∋ z
          union-lemma-u {X} {z} U>z = lemma <-osuc where
              lemma : {oz ooz : Ordinal {suc n}} → oz o< ooz → def (ord→od ooz) oz
-             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} (o<→c< lt) refl diso
+             lemma {oz} {ooz} lt = def-subst {suc n} {ord→od  ooz} (o<→c< lt) refl refl
          union→ :  (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z
          union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y ))
          union→ X y u xx | tri< a ¬b ¬c with  osuc-< a (c<→o< (proj2 xx))
@@ -331,7 +333,7 @@
              lemma refl lt = lt
          union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) 
          union← :  (X z : OD) (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 = def-subst {suc n} (o<→c< X∋z) oiso refl ; proj2 = union-lemma-u X∋z } 
+         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (union-u X z X∋z)} (o<→c< X∋z) oiso (sym diso) ; proj2 = union-lemma-u X∋z } 
          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)
@@ -378,8 +380,8 @@
          infinite : OD {suc n}
          infinite = ord→od ( omega )
          infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
-         infinity∅ = def-subst {suc n} {_} {od→ord (ord→od o∅)} {infinite} {od→ord od∅}
-              (o<→c< ( case1 (s≤s z≤n )))  refl (cong (λ k →  od→ord k) o∅≡od∅ )
+         infinity∅ = def-subst {suc n} {_} {o∅} {infinite} {od→ord od∅}
+              (o<→c< ( case1 (s≤s z≤n )))  refl (subst ( λ k → ( k ≡ od→ord od∅ )) diso (cong (λ k →  od→ord k) o∅≡od∅ ))
          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))