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ZF Set Theory in Agda
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 10 Jun 2019 09:53:45 +0900
parents 9829ba02877f (current diff) 52a82415dfc8 (diff)
children e022c0716936
files set-of-agda.agda
diffstat 5 files changed, 822 insertions(+), 151 deletions(-) [+]
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line diff
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+++ b/.hgtags	Mon Jun 10 09:53:45 2019 +0900
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+264784731a67c6781c7aa95f24feabe5c38629ea current
+92a11dc6425c89c9a19c6377571db0755c71492e current
+92a11dc6425c89c9a19c6377571db0755c71492e current
+b4742cf4ef978434d98a6f0a2f891a944dea5906 current
+b4742cf4ef978434d98a6f0a2f891a944dea5906 current
+a402881cc341fb6499f60bd0f55795dbef5efc70 current
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ordinal-definable.agda	Mon Jun 10 09:53:45 2019 +0900
@@ -0,0 +1,459 @@
+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 _ → Lift n ⊥ }
+
+
+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} 
+  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
+  -- 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 property of supermum required in Power Set Axiom
+  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-lb : {n : Level } → ( ψ : Ordinal {n} →  Ordinal {n}) → ( ∀ {x : Ordinal {n}} →  ψx  o<  z ) →  z o< osuc ( 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< ) refl (cong ( λ k → od→ord (ψ k) ) oiso)
+
+∅1 : {n : Level} →  ( x : OD {n} )  → ¬ ( x c< od∅ {n} )
+∅1 {n} x (lift ())
+
+∅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 : {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 x)
+   lemma xo<z = o<→c< xo<z
+   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
+     mino : Ordinal {n}
+     min<x :  mino o< x
+
+∅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
+
+>→¬< : {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 : 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 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 x} (o<→c< (case2 lt )) oiso refl )
+... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx )
+... | ()
+
+==→o≡ : {n : Level} →  { x y : Ordinal {suc n} } → ord→od x == ord→od y →  x ≡ y 
+==→o≡ {n} {x} {y} eq with trio< {n} x y
+==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c = ⊥-elim ( o<→o> eq (o<-subst a (sym ord-iso) (sym ord-iso )))
+==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b
+==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso )))
+
+≡-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 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} ))
+
+==→o≡1 : {n : Level} →  { x y : OD {suc n} } → x == y →  od→ord x ≡ od→ord y 
+==→o≡1 eq = ==→o≡ (subst₂ (λ k j → k == j ) (sym oiso) (sym oiso) eq )
+
+==-def-l : {n : Level } {x y : Ordinal {suc n} } { z : OD {suc n} }→ (ord→od x == ord→od y) → def z x → def z y
+==-def-l {n} {x} {y} {z} eq z>x = subst ( λ k → def z k ) (==→o≡ eq) z>x
+
+==-def-r : {n : Level } {x y : OD {suc n} } { z : Ordinal {suc n} }→ (x == y) → def x z → def y z
+==-def-r {n} {x} {y} {z} eq z>x = subst (λ k → def k z ) (subst₂ (λ j k → j ≡ k ) oiso oiso (cong (λ k → ord→od k) (==→o≡1 eq))) z>x  
+
+∋→o< : {n : Level} →  { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a
+∋→o< {n} {a} {x} lt = t where
+         t : (od→ord x) o< (od→ord a)
+         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 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}
+o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} ))
+o∅≡od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where
+    lemma :  o∅ {suc n } o< (od→ord (od∅ {suc n} )) → ⊥
+    lemma lt with def-subst (o<→c< lt) oiso refl
+    lemma lt | lift ()
+o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso
+o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c)
+
+o<→¬== : {n : Level} →  { x y : OD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (x == y )
+o<→¬== {n} {x} {y} lt eq = o<→o> eq lt
+
+o<→¬c> : {n : Level} →  { x y : OD {n} } → (od→ord x ) o< ( od→ord y) →  ¬ (y c< x )
+o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where
+
+o≡→¬c< : {n : Level} →  { x y : OD {n} } →  (od→ord x ) ≡ ( od→ord y) →   ¬ x c< y
+o≡→¬c< {n} {x} {y} oeq lt  = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) 
+
+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 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 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
+c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x
+c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y )
+c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y
+
+∅< : {n : Level} →  { x y : OD {n} } → def x (od→ord y ) → ¬ (  x  == od∅ {n} )
+∅< {n} {x} {y} d eq with eq→ eq 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} {x} {x} x∋x x∋x
+
+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-∋ : {n : Level} →  ( x y : OD {suc n} ) → Dec ( x ∋ y )
+is-∋ {n} x y with tri-c< x y
+is-∋ {n} x y | tri< a ¬b ¬c = no ¬c
+is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c
+is-∋ {n} x y | tri> ¬a ¬b c = yes c
+
+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 _∧_
+
+¬∅=→∅∈ :  {n : Level} →  { x : OD {suc n} } → ¬ (  x  == od∅ {suc n} ) → x ∋ od∅ {suc n} 
+¬∅=→∅∈  {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where
+     lemma : (ox : Ordinal {suc n}) →  ¬ (ord→od  ox  == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n}
+     lemma ox ne with is-o∅ ox
+     lemma ox ne | yes refl with ne ( ord→== lemma1 ) where
+         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< (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))
+
+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 }) )) }
+
+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→od ( record { lv = Suc Zero ; ord = Φ 1 } )
+    ; 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 ) )
+    infixr  200 _∈_
+    -- infixr  230 _∩_ _∪_
+    infixr  220 _⊆_
+    isZF : IsZF (OD {suc n})  _∋_  _==_ od∅ _,_ Union Power Select Replace (ord→od ( record { lv = Suc Zero ; ord = Φ 1} ))
+    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 _∧_ 
+         open Minimumo
+         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 ()
+         ---
+         --- ZFSubset A x =  record { def = λ y → def A y ∧  def x y }                   subset of A
+         --- 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 propertiy of minimum 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 = proj1 lemma-s where
+              minsup :  OD
+              minsup =  ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) 
+              lemma-t : csuc minsup ∋ t
+              lemma-t = o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) 
+              lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x)))))  ∋ x
+              lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso  )
+              lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl
+              lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst (o<→c< lt) oiso refl ) 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  = def-subst {suc n} {_} {_} {Power A} {od→ord t}
+                  ( o<→c< {suc n} {od→ord (ZFSubset A (ord→od (od→ord t)) )} {sup-o (λ x → od→ord (ZFSubset A (ord→od x)))}
+                      lemma ) refl lemma1 where
+              lemma-eq :  ZFSubset A t == t
+              eq→ lemma-eq {z} w = proj2 w 
+              eq← lemma-eq {z} w = record { proj2 = w  ;
+                 proj1 = def-subst {suc n} {_} {_} {A} {z} ( t→A (def-subst {suc n} {_} {_} {t} {od→ord (ord→od z)} w refl (sym diso) )) refl diso  }
+              lemma1 : od→ord (ZFSubset A (ord→od (od→ord t))) ≡ od→ord t
+              lemma1 = subst (λ k → od→ord (ZFSubset A k) ≡ od→ord t ) (sym oiso) (==→o≡1 (lemma-eq))
+              lemma :  od→ord (ZFSubset A (ord→od (od→ord t)) ) o< sup-o (λ x → od→ord (ZFSubset A (ord→od x)))
+              lemma = sup-o<   
+         union-lemma-u : {X z : OD {suc n}} → (U>z : Union X ∋ z ) → csuc 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 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))
+         union→ X y u xx | tri< a ¬b ¬c | ()
+         union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where
+             lemma : {oX ou ooy : Ordinal {suc n}} →  ou ≡ ooy  → ou o< oX   → ooy  o< oX
+             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 ∋ csuc z) ∧ (csuc z ∋ z )
+         union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc 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)
+         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  
+         minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} 
+         minimul x  not = od∅   
+         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 ) = ¬∅=→∅∈ not 
+         proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } 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 proj1 t
+            ... | x∈∅ = x∈∅
+         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  
+         xx-union : {x  : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) }
+         xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x))
+         xxx-union : {x  : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))}
+         xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where
+             lemma1 : {x  : OD {suc n}} → od→ord x o< od→ord (x , x)
+             lemma1 {x} = c<→o< ( proj1 (pair x x ) )
+             lemma2 : {x  : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x)
+             lemma2 = trans ( cong ( λ k →  od→ord k ) xx-union ) (sym ≡-def)
+             lemma : {x  : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x))
+             lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 )
+         uxxx-union : {x  : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) }
+         uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where
+             lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x))
+             lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def )
+         uxxx-2 : {x  : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) }
+         eq→ ( uxxx-2 {x} ) {m} lt = proj1 (osuc2 m (od→ord x)) lt
+         eq← ( uxxx-2 {x} ) {m} lt = proj2 (osuc2 m (od→ord x)) lt
+         uxxx-ord : {x  : OD {suc n}} → od→ord (Union (x , (x , x))) ≡ osuc (od→ord x)
+         uxxx-ord {x} = trans (cong (λ k →  od→ord k ) uxxx-union) (==→o≡ (subst₂ (λ j k → j == k ) (sym oiso) (sym od≡-def ) uxxx-2 )) 
+         omega = record { lv = Suc Zero ; ord = Φ 1 }
+         infinite : OD {suc n}
+         infinite = ord→od ( omega )
+         infinity∅ : ord→od ( omega ) ∋ od∅ {suc n}
+         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))
+              t  = ∋→o< {n} {infinite} {x} lt
+         infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x ))
+         infinite∋uxxx x lt = o<∋→ t 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 ) ( sym  (uxxx-ord {x} ) ) 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
+
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/ordinal.agda	Mon Jun 10 09:53:45 2019 +0900
@@ -0,0 +1,292 @@
+{-# OPTIONS --allow-unsolved-metas #-}
+open import Level
+module ordinal where
+
+open import zf
+
+open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 
+open import Data.Empty
+open import  Relation.Binary.PropositionalEquality
+
+data OrdinalD {n : Level} :  (lv : Nat) → Set n where
+   Φ : (lv : Nat) → OrdinalD  lv
+   OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
+
+record Ordinal {n : Level} : Set n where
+   field
+     lv : Nat
+     ord : OrdinalD {n} lv
+
+--
+--    Φ (Suc lv) < ℵ lv < OSuc (Suc lv) (ℵ lv) < OSuc ... < OSuc (Suc lv) (Φ (Suc lv)) < OSuc ...  < ℵ (Suc lv)
+--
+data _d<_ {n : Level} :   {lx ly : Nat} → OrdinalD {n} lx  →  OrdinalD {n} ly  → Set n where
+   Φ<  : {lx : Nat} → {x : OrdinalD {n} lx}  →  Φ lx d< OSuc lx x
+   s<  : {lx : Nat} → {x y : OrdinalD {n} lx}  →  x d< y  → OSuc  lx x d< OSuc  lx y
+
+open Ordinal
+
+_o<_ : {n : Level} ( x y : Ordinal ) → Set n
+_o<_ x y =  (lv x < lv y )  ∨ ( ord x d< ord y )
+
+s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x
+s<refl {n} {lv} {Φ lv}  = Φ<
+s<refl {n} {lv} {OSuc lv x}  = s< s<refl 
+
+trio<> : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  →  y d< x → x d< y → ⊥
+trio<>  {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t
+trio<> {n} {lx} {.(OSuc lx _)} {.(Φ lx)} Φ< ()
+
+d<→lv : {n : Level} {x y  : Ordinal {n}}   → ord x d< ord y → lv x ≡ lv y
+d<→lv Φ< = refl
+d<→lv (s< lt) = refl
+
+o<-subst : {n : Level } {Z X z x : Ordinal {n}}  → Z o< X → Z ≡ z  →  X ≡ x  →  z o< x
+o<-subst df refl refl = df
+
+open import Data.Nat.Properties 
+open import Data.Unit using ( ⊤ )
+open import Relation.Nullary
+
+open import Relation.Binary
+open import Relation.Binary.Core
+
+o∅ : {n : Level} → Ordinal {n}
+o∅  = record { lv = Zero ; ord = Φ Zero }
+
+open import Relation.Binary.HeterogeneousEquality using (_≅_;refl)
+
+ordinal-cong : {n : Level} {x y : Ordinal {n}}  →
+      lv x  ≡ lv y → ord x ≅ ord y →  x ≡ y
+ordinal-cong refl refl = refl
+
+ordinal-lv : {n : Level} {x y : Ordinal {n}}  → x ≡ y → lv x  ≡ lv y 
+ordinal-lv refl = refl
+
+ordinal-d : {n : Level} {x y : Ordinal {n}}  → x ≡ y → ord x  ≅ ord y 
+ordinal-d refl = refl
+
+≡→¬d< : {n : Level} →  {lv : Nat} → {x  : OrdinalD {n}  lv }  → x d< x → ⊥
+≡→¬d<  {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
+
+trio<≡ : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  → x ≡ y  → x d< y → ⊥
+trio<≡ refl = ≡→¬d<
+
+trio>≡ : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
+trio>≡ refl = ≡→¬d<
+
+triO : {n : Level} →  {lx ly : Nat} → OrdinalD {n} lx  →  OrdinalD {n} ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
+triO  {n} {lx} {ly} x y = <-cmp lx ly
+
+triOrdd : {n : Level} →  {lx : Nat}   → Trichotomous  _≡_ ( _d<_  {n} {lx} {lx} )
+triOrdd  {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
+triOrdd  {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
+triOrdd  {_} {lv} (OSuc lv x) (OSuc lv y) with triOrdd x y
+triOrdd  {_} {lv} (OSuc lv x) (OSuc lv y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) )  (  λ lt → trio<> lt (s< a) )
+triOrdd  {_} {lv} (OSuc lv x) (OSuc lv x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {_} {lv} (OSuc lv x) (OSuc lv y) | tri> ¬a ¬b c = tri> (  λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
+
+osuc : {n : Level} ( x : Ordinal {n} ) → Ordinal {n}
+osuc record { lv = lx ; ord = ox } = record { lv = lx ; ord = OSuc lx ox }
+
+<-osuc : {n : Level} { x : Ordinal {n} } → x o< osuc x
+<-osuc {n} {record { lv = lx ; ord = Φ .lx }} =  case2 Φ<
+<-osuc {n} {record { lv = lx ; ord = OSuc .lx ox }} = case2 ( s< s<refl )
+
+osuc-lveq : {n : Level} { x : Ordinal {n} } → lv x ≡ lv ( osuc x )
+osuc-lveq {n} = refl
+
+nat-<> : { x y : Nat } → x < y → y < x → ⊥
+nat-<>  (s≤s x<y) (s≤s y<x) = nat-<> x<y y<x
+
+nat-<≡ : { x : Nat } → x < x → ⊥
+nat-<≡  (s≤s lt) = nat-<≡ lt
+
+nat-≡< : { x y : Nat } → x ≡ y → x < y → ⊥
+nat-≡< refl lt = nat-<≡ lt
+
+¬a≤a : {la : Nat} → Suc la ≤ la → ⊥
+¬a≤a  (s≤s lt) = ¬a≤a  lt
+
+=→¬< : {x : Nat  } → ¬ ( x < x )
+=→¬< {Zero} ()
+=→¬< {Suc x} (s≤s lt) = =→¬< lt
+
+o<¬≡ : {n : Level } ( ox oy : Ordinal {n}) → ox ≡ oy  → ox o< oy  → ⊥
+o<¬≡ ox ox refl (case1 lt) =  =→¬< lt
+o<¬≡ ox ox refl (case2 (s< lt)) = trio<≡ refl lt
+
+¬x<0 : {n : Level} →  { x  : Ordinal {suc n} } → ¬ ( x o< o∅ {suc n} )
+¬x<0 {n} {x} (case1 ())
+¬x<0 {n} {x} (case2 ())
+
+o<> : {n : Level} →  {x y : Ordinal {n}  }  →  y o< x → x o< y → ⊥
+o<> {n} {x} {y} (case1 x₁) (case1 x₂) = nat-<>  x₁ x₂
+o<> {n} {x} {y} (case1 x₁) (case2 x₂) = nat-≡< (sym (d<→lv x₂)) x₁
+o<> {n} {x} {y} (case2 x₁) (case1 x₂) = nat-≡< (sym (d<→lv x₁)) x₂
+o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) (case2 ())
+o<> {n} {record { lv = lv₁ ; ord = .(OSuc lv₁ _) }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< y<x)) (case2 (s< x<y)) = 
+   o<> (case2 y<x) (case2 x<y)
+
+orddtrans : {n : Level} {lx : Nat} {x y z : OrdinalD {n}  lx }   → x d< y → y d< z → x d< z
+orddtrans {_} {lx} {.(Φ lx)} {.(OSuc lx _)} {.(OSuc lx _)} Φ< (s< y<z) = Φ< 
+orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
+
+osuc-≡< : {n : Level} { a x : Ordinal {n} } → x o< osuc a  →  (x ≡ a ) ∨ (x o< a)  
+osuc-≡< {n} {a} {x} (case1 lt) = case2 (case1 lt)
+osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case1 refl
+osuc-≡< {n} {record { lv = lv₁ ; ord = OSuc .lv₁ ord₁ }} {record { lv = .lv₁ ; ord = .(Φ lv₁) }} (case2 Φ<) = case2 (case2 Φ<)
+osuc-≡< {n} {record { lv = lv₁ ; ord = Φ .lv₁ }} {record { lv = .lv₁ ; ord = .(OSuc lv₁ _) }} (case2 (s< ()))
+osuc-≡< {n} {record { lv = la ; ord = OSuc la oa }} {record { lv = la ; ord = (OSuc la ox) }} (case2 (s< lt)) with
+      osuc-≡< {n} {record { lv = la ; ord = oa }} {record { lv = la ; ord = ox }} (case2 lt )
+... | case1 refl = case1 refl
+... | case2 (case2 x) = case2 (case2( s< x) )
+... | case2 (case1 x) = ⊥-elim (¬a≤a  x) where
+
+osuc-< : {n : Level} { x y : Ordinal {n} } → y o< osuc x  → x o< y → ⊥
+osuc-< {n} {x} {y} y<ox x<y with osuc-≡< y<ox
+osuc-< {n} {x} {x} y<ox (case1 x₁) | case1 refl = ⊥-elim (¬a≤a x₁)
+osuc-< {n} {x} {x} (case1 x₂) (case2 x₁) | case1 refl = ⊥-elim (¬a≤a x₂)
+osuc-< {n} {x} {x} (case2 x₂) (case2 x₁) | case1 refl = ≡→¬d<  x₁
+osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case1 x₁) = nat-<> x₂ x₁
+osuc-< {n} {x} {y} y<ox (case1 x₂) | case2 (case2 x₁) = nat-≡< (sym (d<→lv x₁)) x₂
+osuc-< {n} {x} {y} y<ox (case2 x<y) | case2 y<x = o<> (case2 x<y) y<x
+
+max : (x y : Nat) → Nat
+max Zero Zero = Zero
+max Zero (Suc x) = (Suc x)
+max (Suc x) Zero = (Suc x)
+max (Suc x) (Suc y) = Suc ( max x y )
+
+maxαd : {n : Level} → { lx : Nat } → OrdinalD {n} lx  →  OrdinalD  lx  →  OrdinalD  lx
+maxαd x y with triOrdd x y
+maxαd x y | tri< a ¬b ¬c = y
+maxαd x y | tri≈ ¬a b ¬c = x
+maxαd x y | tri> ¬a ¬b c = x
+
+_o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
+a o≤ b  = (a ≡ b)  ∨ ( a o< b )
+
+ordtrans : {n : Level} {x y z : Ordinal {n} }   → x o< y → y o< z → x o< z
+ordtrans {n} {x} {y} {z} (case1 x₁) (case1 x₂) = case1 ( <-trans x₁ x₂ )
+ordtrans {n} {x} {y} {z} (case1 x₁) (case2 x₂) with  d<→lv x₂
+... | refl = case1 x₁
+ordtrans {n} {x} {y} {z} (case2 x₁) (case1 x₂)  with  d<→lv x₁
+... | refl = case1 x₂
+ordtrans {n} {x} {y} {z} (case2 x₁) (case2 x₂) with d<→lv x₁ | d<→lv x₂
+... | refl | refl = case2 ( orddtrans x₁ x₂ )
+
+trio< : {n : Level } → Trichotomous {suc n} _≡_  _o<_ 
+trio< a b with <-cmp (lv a) (lv b)
+trio< a b | tri< a₁ ¬b ¬c = tri< (case1  a₁) (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) lemma1 where
+   lemma1 : ¬ (Suc (lv b) ≤ lv a) ∨ (ord b d< ord a)
+   lemma1 (case1 x) = ¬c x
+   lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) a₁  )
+trio< a b | tri> ¬a ¬b c = tri> lemma1 (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c) where
+   lemma1 : ¬ (Suc (lv a) ≤ lv b) ∨ (ord a d< ord b)
+   lemma1 (case1 x) = ¬a x
+   lemma1 (case2 x) = ⊥-elim (nat-≡< (sym ( d<→lv x )) c  )
+trio< a b | tri≈ ¬a refl ¬c with triOrdd ( ord a ) ( ord b )
+trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = tri< (case2 a) (λ refl → ¬b (lemma1 refl )) lemma2 where
+   lemma1 :  (record { lv = _ ; ord = x }) ≡ b →  x ≡ ord b
+   lemma1 refl = refl
+   lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< x)
+   lemma2 (case1 x) = ¬a x
+   lemma2 (case2 x) = trio<> x a
+trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> lemma2 (λ refl → ¬b (lemma1 refl )) (case2 c) where
+   lemma1 :  (record { lv = _ ; ord = x }) ≡ b →  x ≡ ord b
+   lemma1 refl = refl
+   lemma2 : ¬ (Suc (lv b) ≤ lv b) ∨ (x d< ord b)
+   lemma2 (case1 x) = ¬a x
+   lemma2 (case2 x) = trio<> x c
+trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ lemma1 refl lemma1 where
+   lemma1 : ¬ (Suc (lv b) ≤ lv b) ∨ (ord b d< ord b)
+   lemma1 (case1 x) = ¬a x
+   lemma1 (case2 x) = ≡→¬d< x
+
+maxα : {n : Level} →  Ordinal {n} →  Ordinal  → Ordinal
+maxα x y with <-cmp (lv x) (lv y)
+maxα x y | tri< a ¬b ¬c = x
+maxα x y | tri> ¬a ¬b c = y
+maxα x y | tri≈ ¬a refl ¬c = record { lv = lv x ; ord = maxαd (ord x) (ord y) }
+
+--
+--  max ( osuc x , osuc y )
+--
+
+omax : {n : Level} ( x y : Ordinal {suc n} ) → Ordinal {suc n}
+omax {n} x y with trio< x y
+omax {n} x y | tri< a ¬b ¬c = osuc y
+omax {n} x y | tri> ¬a ¬b c = osuc x
+omax {n} x y | tri≈ ¬a refl ¬c  = osuc x
+
+omax< : {n : Level} ( x y : Ordinal {suc n} ) → x o< y → osuc y ≡ omax x y
+omax< {n} x y lt with trio< x y
+omax< {n} x y lt | tri< a ¬b ¬c = refl
+omax< {n} x y lt | tri≈ ¬a b ¬c = ⊥-elim (¬a lt )
+omax< {n} x y lt | tri> ¬a ¬b c = ⊥-elim (¬a lt )
+
+omax≡ : {n : Level} ( x y : Ordinal {suc n} ) → x ≡ y → osuc y ≡ omax x y
+omax≡ {n} x y eq with trio< x y
+omax≡ {n} x y eq | tri< a ¬b ¬c = ⊥-elim (¬b eq )
+omax≡ {n} x y eq | tri≈ ¬a refl ¬c = refl
+omax≡ {n} x y eq | tri> ¬a ¬b c = ⊥-elim (¬b eq )
+
+omax-x : {n : Level} ( x y : Ordinal {suc n} ) → x o< omax x y
+omax-x {n} x y with trio< x y
+omax-x {n} x y | tri< a ¬b ¬c = ordtrans a <-osuc
+omax-x {n} x y | tri> ¬a ¬b c = <-osuc
+omax-x {n} x y | tri≈ ¬a refl ¬c = <-osuc
+
+omax-y : {n : Level} ( x y : Ordinal {suc n} ) → y o< omax x y
+omax-y {n} x y with  trio< x y
+omax-y {n} x y | tri< a ¬b ¬c = <-osuc
+omax-y {n} x y | tri> ¬a ¬b c = ordtrans c <-osuc
+omax-y {n} x y | tri≈ ¬a refl ¬c = <-osuc
+
+omxx : {n : Level} ( x : Ordinal {suc n} ) → omax x x ≡ osuc x
+omxx {n} x with  trio< x x
+omxx {n} x | tri< a ¬b ¬c = ⊥-elim (¬b refl )
+omxx {n} x | tri> ¬a ¬b c = ⊥-elim (¬b refl )
+omxx {n} x | tri≈ ¬a refl ¬c = refl
+
+omxxx : {n : Level} ( x : Ordinal {suc n} ) → omax x (omax x x ) ≡ osuc (osuc x)
+omxxx {n} x = trans ( cong ( λ k → omax x k ) (omxx x )) (sym ( omax< x (osuc x) <-osuc ))
+
+open _∧_
+
+osuc2 : {n : Level} ( x y : Ordinal {suc n} ) → ( osuc x o< osuc (osuc y )) ⇔ (x o< osuc y)
+proj1 (osuc2 {n} x y) (case1 lt) = case1 lt
+proj1 (osuc2 {n} x y) (case2 (s< lt)) = case2 lt
+proj2 (osuc2 {n} x y) (case1 lt) = case1 lt
+proj2 (osuc2 {n} x y) (case2 lt) with d<→lv lt
+... | refl = case2 (s< lt)
+
+-- omax≡ (omax x x ) (osuc x) (omxx x)
+
+OrdTrans : {n : Level} → Transitive {suc n} _o≤_
+OrdTrans (case1 refl) (case1 refl) = case1 refl
+OrdTrans (case1 refl) (case2 lt2) = case2 lt2
+OrdTrans (case2 lt1) (case1 refl) = case2 lt1
+OrdTrans (case2 x) (case2 y) = case2 (ordtrans x y)
+
+OrdPreorder : {n : Level} → Preorder (suc n) (suc n) (suc n)
+OrdPreorder {n} = record { Carrier = Ordinal
+   ; _≈_  = _≡_ 
+   ; _∼_   = _o≤_
+   ; isPreorder   = record {
+        isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
+        ; reflexive     = case1 
+        ; trans         = OrdTrans 
+     }
+ }
+
+TransFinite : {n : Level} → { ψ : Ordinal {n} → Set n }
+  → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ lx } ) )
+  → ( ∀ (lx : Nat ) → (x : OrdinalD lx )  → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc lx x } ) )
+  →  ∀ (x : Ordinal)  → ψ x
+TransFinite caseΦ caseOSuc record { lv = lv ; ord = (Φ (lv)) } = caseΦ lv
+TransFinite caseΦ caseOSuc record { lv = lx ; ord = (OSuc lx ox) } = 
+      caseOSuc lx ox (TransFinite caseΦ caseOSuc  record { lv = lx ; ord = ox })
+
--- a/set-of-agda.agda	Sat May 11 11:11:40 2019 +0900
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,104 +0,0 @@
-module set-of-agda where
-
-open import Level
-open import Data.Bool
-
--- infix  50 _∧_
-
--- record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
---    constructor _×_
---    field 
---       proj1 : A
---       proj2 : B
-
--- open _∧_
-
--- infix  50 _∨_
-
--- data  _∨_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
---    case1 : A → A ∨ B
---    case2 : B → A ∨ B
-
-data ZFSet {n : Level} : Set (suc (suc n)) where
-    elem : { A : Set  n } ( a : A ) → ZFSet 
-    ∅ : ZFSet  {n}
-    pair : {A B : Set n} (a : A ) (b : B ) → ZFSet 
-    union :  (A  : Set (suc n) ) → ZFSet  
-    -- repl :  ( ψ : ZFSet {n} →  Set zero )   → ZFSet 
-    infinite : ZFSet 
-    power : (A  : ZFSet {n})  → ZFSet 
-
-infix  60 _∋_ _∈_
-
-open import Relation.Binary.PropositionalEquality 
-
-data _∈_ {n : Level} : {A : Set n}  ( a : A )  ( Z : ZFSet {n} )  → Set (suc n) where
-    ∈-elm : {A : Set n } {a : A} →  a ∈ (elem a)   
-    ∈-pair-1 : {A : Set n } {B : Set n} {b : B} {a : A}  → a ∈  (pair a b)   
-    ∈-pair-2 : {A : Set n } {B : Set n} {b : A} {a : B}  → b ∈  (pair a b)   
---    ∈-union : {Z : Set (suc n)}  {A : Z }  → {a : {!!} } → a ∈ (union Z)  
---    ∈-repl : {A : Set n } { B : Set n}  → { ψ : B → A } → { b : B } → ψ b ∈  {!!} -- (repl {!!}) 
-    -- ∈-infinite-1 : ∅ ∈ infinite 
---    ∈-infinite : {A : Set n} {a : A} → _∈_  infinite {A} a 
-    ∈-power : {A B : Set n} {Z : ZFSet {n}} {a : A → B } → a ∈  (power Z)  
-
--- _∈_  : {n : Level} { A : ZFSet {n} } → {B : Set n} →  (a : B ) →  Set n  → Bool
--- _∈_ {_} {A} a _  = A ∋ a
-
-infix  40 _⇔_
-
--- _⇔_ : {n : Level} (A  B : Set n)  → Set  n 
--- A ⇔ B = ( ∀ {x : A } → x ∈ B  ) ∧  ( ∀ {x : B } → x ∈ A )
-
--- Axiom of extentionality
--- ∀ x ∀ y [ ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w ) ]
-
--- set-extentionality : {n : Level } {x y : Set n }  → { z : x } → ( z ∈ x ⇔ z ∈ y ) → ∀ (w : Set (suc n))  → ( x ∈ w ⇔ y ∈ w ) 
--- proj1 (set-extentionality {n} {x} {y} {z} (z∈x→z∈y × z∈x←z∈y) w) {elem .x} = elem ( elem x )
--- proj2 (set-extentionality {n} {x} {y} {z} (z∈x→z∈y × z∈x←z∈y) w) {elem .y} = elem ( elem y )
-
-
-open import Relation.Nullary
-open import Data.Empty
-
--- data ∅  : Set where
-
-infix  50 _∩_
-
--- record _∩_ {n m : Level} (A : Set n) ( B : Set m)  : Set (n ⊔ m) where
---    field
---       inL : {x : A } → x ∈ A
---       inR : {x : B } → x ∈ B
-
--- open _∩_
-
--- lemma : {n m : Level} (A : Set n) ( B : Set m) → (a : A ) → a ∈ (A ∩ B) 
--- lemma A B a A∩B = inL A∩B
-
--- Axiom of regularity 
--- ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
-
--- set-regularity : {n : Level } → ( x : Set n) → ( ¬ ( x ⇔ ∅ ) ) → { y : x } → ( y ∩ x  ⇔ ∅ )
--- set-regularity = {!!}
-
---  Axiom of pairing
--- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z).
-
--- pair : {n m : Level}  ( x : Set n ) ( y : Set m ) → Set (n ⊔ m)
--- pair x y = {!!} -- ( x × y )
-
--- Axiom of Union
--- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t  ∈ x))
-
--- axiom of infinity
--- ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
-
--- axiom of replacement
--- ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
-
--- axiom of power set
--- ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) ) 
-
-
-
-
--- a/zf.agda	Sat May 11 11:11:40 2019 +0900
+++ b/zf.agda	Mon Jun 10 09:53:45 2019 +0900
@@ -2,77 +2,94 @@
 
 open import Level
 
+data Bool : Set where
+   true : Bool
+   false : Bool
 
 record  _∧_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
    field 
       proj1 : A
       proj2 : B
 
-open _∧_
-
-
 data  _∨_  {n m : Level} (A  : Set n) ( B : Set m ) : Set (n ⊔ m) where
    case1 : A → A ∨ B
    case2 : B → A ∨ B
 
-open import Relation.Binary.PropositionalEquality 
+_⇔_ : {n : Level } → ( A B : Set n )  → Set  n
+_⇔_ A B =  ( A → B ) ∧ ( B → A )
 
-_⇔_ : {n : Level } → ( A B : Set n )  → Set  n
-_⇔_ A B =  ( A → B ) ∧ ( B  → A )
+open import Relation.Nullary
+open import Relation.Binary
 
 infixr  130 _∧_
 infixr  140 _∨_
 infixr  150 _⇔_
 
-open import Data.Empty
-open import Relation.Nullary
+record IsZF {n m : Level }
+     (ZFSet : Set n)
+     (_∋_ : ( A x : ZFSet  ) → Set m)
+     (_≈_ : Rel ZFSet m)
+     (∅  : ZFSet)
+     (_,_ : ( A B : ZFSet  ) → ZFSet)
+     (Union : ( A : ZFSet  ) → ZFSet)
+     (Power : ( A : ZFSet  ) → ZFSet)
+     (Select : ZFSet → ( ZFSet → Set m ) → ZFSet )
+     (Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet )
+     (infinite : ZFSet)
+       : Set (suc (n ⊔ m)) where
+  field
+     isEquivalence : IsEquivalence {n} {m} {ZFSet} _≈_ 
+     -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
+     pair : ( A B : ZFSet  ) →  ( (A , B)  ∋ A ) ∧ ( (A , B)  ∋ B  )
+     -- ∀ x ∃ y ∀ z (z ∈ y ⇔ ∃ u ∈ x  ∧ (z ∈ u))
+     union-u : ( X z : ZFSet  ) → Union X ∋ z → ZFSet
+     union→ : ( X z u : ZFSet ) → ( X ∋ u ) ∧ (u ∋ z ) → Union X ∋ z
+     union← : ( X z : ZFSet ) → (X∋z : Union X ∋ z ) → (X ∋ union-u X z X∋z)  ∧ (union-u X z X∋z ∋ z )
+  _∈_ : ( A B : ZFSet  ) → Set m
+  A ∈ B = B ∋ A
+  _⊆_ : ( A B : ZFSet  ) → ∀{ x : ZFSet } →  Set m
+  _⊆_ A B {x} = A ∋ x →  B ∋ x
+  _∩_ : ( A B : ZFSet  ) → ZFSet
+  A ∩ B = Select A (  λ x → ( A ∋ x ) ∧ ( B ∋ x )  )
+  _∪_ : ( A B : ZFSet  ) → ZFSet
+  A ∪ B = Union (A , B) 
+  ｛_｝ : ZFSet → ZFSet
+  ｛ x ｝ = ( x ,  x )
+  infixr  200 _∈_
+  infixr  230 _∩_ _∪_
+  infixr  220 _⊆_ 
+  field
+     empty :  ∀( x : ZFSet  ) → ¬ ( ∅ ∋ x )
+     -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
+     power→ : ∀( A t : ZFSet  ) → Power A ∋ t → ∀ {x}  →  _⊆_ t A {x} 
+     power← : ∀( A t : ZFSet  ) → ( ∀ {x}  →  _⊆_ t A {x})  → Power A ∋ t 
+     -- extensionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
+     extensionality :  { A B : ZFSet  } → ( (z : ZFSet) → ( A ∋ z ) ⇔ (B ∋ z)  ) → A ≈ B
+     -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
+     minimul : (x : ZFSet ) → ¬ (x ≈ ∅) → ZFSet 
+     regularity : ∀( x : ZFSet  ) → (not : ¬ (x ≈ ∅)) → (  minimul x not  ∈ x ∧  (  minimul x not  ∩ x  ≈ ∅ ) )
+     -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
+     infinity∅ :  ∅ ∈ infinite
+     infinity :  ∀( X x : ZFSet  ) → x ∈ infinite →  ( x ∪ ｛ x ｝) ∈ infinite 
+     selection : { ψ : ZFSet → Set m } → ∀ { X y : ZFSet  } →  ( ( y ∈ X ) ∧ ψ y ) ⇔ (y ∈  Select X ψ ) 
+     -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
+     replacement : {ψ : ZFSet → ZFSet} → ∀ ( X x : ZFSet  ) →  ( ψ x ∈  Replace X ψ )  
 
-record ZF (n m : Level ) : Set (suc (n ⊔ m)) where
-  coinductive
+record ZF {n m : Level } : Set (suc (n ⊔ m)) where
+  infixr  210 _,_
+  infixl  200 _∋_ 
+  infixr  220 _≈_
   field
      ZFSet : Set n
      _∋_ : ( A x : ZFSet  ) → Set m 
      _≈_ : ( A B : ZFSet  ) → Set m
   -- ZF Set constructor
      ∅  : ZFSet
-     _×_ : ( A B : ZFSet  ) → ZFSet
+     _,_ : ( A B : ZFSet  ) → ZFSet
      Union : ( A : ZFSet  ) → ZFSet
      Power : ( A : ZFSet  ) → ZFSet
-     Restrict : ( ZFSet → Set m ) → ZFSet
-  infixl  200 _∋_ 
-  infixr  210 _×_
-  infixr  220 _≈_
-  field
-     -- ∀ x ∀ y ∃ z(x ∈ z ∧ y ∈ z)
-     pair : ( A B : ZFSet  ) →  A × B  ∋ A  ∧ A × B  ∋ B 
-     -- ∀ X ∃ A∀ t(t ∈ A ⇔ ∃ x ∈ X(t  ∈ x))
-     union→ : ( X x y : ZFSet  ) → X ∋ x  → x ∋ y → Union X  ∋ y
-     union← : ( X x y : ZFSet  ) → Union X  ∋ y → X ∋ x  → x ∋ y 
-  _∈_ : ( A B : ZFSet  ) → Set m
-  A ∈ B = B ∋ A
-  _⊆_ : ( A B : ZFSet  ) → { x : ZFSet } → { A∋x : Set m } → Set m
-  _⊆_ A B {x} {A∋x} = B ∋ x
-  _∩_ : ( A B : ZFSet  ) → ZFSet
-  A ∩ B = Restrict ( λ x → ( A ∋ x ) ∧ ( B ∋ x ) )
-  _∪_ : ( A B : ZFSet  ) → ZFSet
-  A ∪ B = Restrict ( λ x → ( A ∋ x ) ∨ ( B ∋ x ) )
-  infixr  200 _∈_
-  infixr  230 _∩_ _∪_
-  infixr  220 _⊆_ 
-  field
-     empty :  ∀( x : ZFSet  ) → ¬ ( ∅ ∋ x )
-     -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ t ⊆ X ) )
-     power→ : ( A t : ZFSet  ) → Power A ∋ t → ∀ {x} {y} →  _⊆_ t A {x} {y}
-     power← : ( A t : ZFSet  ) → ∀ {x} {y} →  _⊆_ t A {x} {y} → Power A ∋ t 
-     -- extentionality : ∀ z ( z ∈ x ⇔ z ∈ y ) ⇒ ∀ w ( x ∈ w ⇔ y ∈ w )
-     extentionality :  ( A B z  : ZFSet  ) → A ∋ z ⇔ B ∋ z → A ≈ B
-     -- regularity : ∀ x ( x ≠ ∅ → ∃ y ∈ x ( y ∩ x = ∅ ) )
-     -- smaller : ZFSet → ZFSet
-     -- regularity : ( x : ZFSet  ) → ¬ (x ≈ ∅) → smaller x  ∩ x  ≈ ∅ 
-     -- infinity : ∃ A ( ∅ ∈ A ∧ ∀ x ∈ A ( x ∪ { x } ∈ A ) )
+     Select : ZFSet → ( ZFSet → Set m ) → ZFSet
+     Replace : ZFSet → ( ZFSet → ZFSet ) → ZFSet
      infinite : ZFSet
-     infinity∅ :  ∅ ∈ infinite
-     infinity :  ( x : ZFSet  ) → x ∈ infinite →  ( x ∪ Restrict ( λ y → x ≈ y )) ∈ infinite 
-     -- replacement : ∀ x ∀ y ∀ z ( ( ψ ( x , y ) ∧ ψ ( x , z ) ) → y = z ) → ∀ X ∃ A ∀ y ( y ∈ A ↔ ∃ x ∈ X ψ ( x , y ) )
-     replacement : ( ψ :  ZFSet → Set m ) → ( y : ZFSet  ) →  y  ∈  Restrict ψ  → ψ y
+     isZF : IsZF ZFSet _∋_ _≈_ ∅ _,_ Union Power Select Replace infinite