# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1557809539 -32400
# Node ID 6a668c6086a5639d01db82586553fc3be6d0391b
# Parent  ac362cc8b10fa003ed18d64c0061ec112a73da7b
clean up

diff -r ac362cc8b10f -r 6a668c6086a5 constructible-set.agda
--- a/constructible-set.agda	Tue May 14 12:53:52 2019 +0900
+++ b/constructible-set.agda	Tue May 14 13:52:19 2019 +0900
@@ -7,22 +7,26 @@
 
 open import  Relation.Binary.PropositionalEquality
 
-data OridinalD  : (lv : Nat) → Set n where
-   Φ : {lv : Nat} → OridinalD  lv
-   OSuc : {lv : Nat} → OridinalD  lv → OridinalD lv
-   ℵ_ :  (lv : Nat) → OridinalD (Suc lv)
+data OrdinalD  : (lv : Nat) → Set n where
+   Φ : {lv : Nat} → OrdinalD  lv
+   OSuc : {lv : Nat} → OrdinalD  lv → OrdinalD lv
+   ℵ_ :  (lv : Nat) → OrdinalD (Suc lv)
 
 record Ordinal : Set n where
    field
      lv : Nat
-     ord : OridinalD lv
+     ord : OrdinalD lv
 
-data _o<_  :  {lx ly : Nat} → OridinalD  lx  →  OridinalD  ly  → Set n where
-   l< : {lx ly : Nat }  → {x : OridinalD  lx } →  {y : OridinalD  ly } → lx < ly → x o< y
-   Φ<  : {lx : Nat} → {x : OridinalD  lx}  →  Φ  {lx} o< OSuc  {lx} x
-   s<  : {lx : Nat} → {x y : OridinalD  lx}  →  x o< y  → OSuc  {lx} x o< OSuc  {lx} y
-   ℵΦ< : {lx : Nat} → {x : OridinalD  (Suc lx) } →  Φ  {Suc lx} o< (ℵ lx) 
-   ℵ<  : {lx : Nat} → {x : OridinalD  (Suc lx) } →  OSuc  {Suc lx} x o< (ℵ lx) 
+data _d<_  :  {lx ly : Nat} → OrdinalD  lx  →  OrdinalD  ly  → Set n where
+   Φ<  : {lx : Nat} → {x : OrdinalD  lx}  →  Φ  {lx} d< OSuc  {lx} x
+   s<  : {lx : Nat} → {x y : OrdinalD  lx}  →  x d< y  → OSuc  {lx} x d< OSuc  {lx} y
+   ℵΦ< : {lx : Nat} → {x : OrdinalD  (Suc lx) } →  Φ  {Suc lx} d< (ℵ lx) 
+   ℵ<  : {lx : Nat} → {x : OrdinalD  (Suc lx) } →  OSuc  {Suc lx} x d< (ℵ lx) 
+
+open Ordinal
+
+_o<_ : ( x y : Ordinal ) → Set n
+_o<_ x y =  (lv x < lv y )  ∨ ( ord x d< ord y )
 
 open import Data.Nat.Properties 
 open import Data.Empty
@@ -31,81 +35,49 @@
 open import Relation.Binary
 open import Relation.Binary.Core
 
-
-≡→¬< : { x y : Nat } → x ≡ y → x < y → ⊥
-≡→¬< {Zero} {Zero} refl ()
-≡→¬< {Suc x} {.(Suc x)} refl (s≤s t) = ≡→¬< {x} {x} refl t
-
-x≤x : { x : Nat } → x ≤ x
-x≤x {Zero} = z≤n
-x≤x {Suc x} =  s≤s ( x≤x  )
-
-x<>y : { x y : Nat } → x > y → x < y → ⊥
-x<>y {.(Suc _)} {.(Suc _)} (s≤s lt) (s≤s lt1) = x<>y lt lt1
+≡→¬d< : {lv : Nat} → {x  : OrdinalD  lv }  → x d< x → ⊥
+≡→¬d<  {lx} {OSuc y} (s< t) = ≡→¬d< t
 
-triO> : {lx ly : Nat} {x  : OridinalD  lx } { y : OridinalD  ly }  →  ly < lx → x o< y → ⊥
-triO>  {lx} {ly} {x} {y} y<x xo<y with <-cmp  lx ly
-triO>  {lx} {ly} {x} {y} y<x xo<y | tri< a ¬b ¬c =  ¬c y<x 
-triO>  {lx} {ly} {x} {y} y<x xo<y | tri≈ ¬a b ¬c =  ¬c y<x 
-triO>  {lx} {ly} {x} {y} y<x (l< x₁) | tri> ¬a ¬b c =  ¬a x₁ 
-triO>  {lx} {ly} {Φ} {OSuc _} y<x Φ< | tri> ¬a ¬b c =  ¬b refl 
-triO>  {lx} {ly} {OSuc px} {OSuc py} y<x (s< w) | tri> ¬a ¬b c =  triO> y<x w
-triO>  {lx} {ly} {Φ {u}} {ℵ w} y<x ℵΦ< | tri> ¬a ¬b c = ¬b refl
-triO>  {lx} {ly} {(OSuc _)} {ℵ u} y<x ℵ< | tri> ¬a ¬b c =  ¬b refl
-
-≡→¬o< : {lv : Nat} → {x  : OridinalD  lv }  → x o< x → ⊥
-≡→¬o<  {lx} {x} (l< y) = ≡→¬< refl y
-≡→¬o<  {lx} {OSuc y} (s< t) = ≡→¬o< t
-
-trio<> : {lx : Nat} {x  : OridinalD  lx } { y : OridinalD  lx }  →  y o< x → x o< y → ⊥
-trio<>  {lx} {x} {y} (l< lt) _ = ≡→¬< refl lt
-trio<>  {lx} {x} {y} _ (l< lt)  = ≡→¬< refl lt
+trio<> : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  →  y d< x → x d< y → ⊥
 trio<>  {lx} {.(OSuc _)} {.(OSuc _)} (s< s) (s< t) = 
     trio<> s t
 
-trio<≡ : {lx : Nat} {x  : OridinalD  lx } { y : OridinalD  lx }  → x ≡ y  → x o< y → ⊥
-trio<≡ refl = ≡→¬o<
+trio<≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → x d< y → ⊥
+trio<≡ refl = ≡→¬d<
 
-trio>≡ : {lx : Nat} {x  : OridinalD  lx } { y : OridinalD  lx }  → x ≡ y  → y o< x → ⊥
-trio>≡ refl = ≡→¬o<
+trio>≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
+trio>≡ refl = ≡→¬d<
 
-triO : {lx ly : Nat} → OridinalD  lx  →  OridinalD  ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
+triO : {lx ly : Nat} → OrdinalD  lx  →  OrdinalD  ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
 triO  {lx} {ly} x y = <-cmp lx ly
 
-triOonSameLevel : {lx : Nat}   → Trichotomous  _≡_ ( _o<_  {lx} {lx} )
-triOonSameLevel  {lv} Φ Φ = tri≈ ≡→¬o< refl ≡→¬o<
-triOonSameLevel  {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬o< refl ≡→¬o<
-triOonSameLevel  {lv} Φ (OSuc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
-triOonSameLevel  {.(Suc lv)} Φ (ℵ lv) = tri<  (ℵΦ<  {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ<  {lv} {Φ} )) )
-triOonSameLevel  {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ<  {lv} {Φ} ) ) (λ ()) (ℵΦ<  {lv} {Φ} )
-triOonSameLevel  {Suc lv} (ℵ lv) (OSuc y) = tri> ( λ lt → trio<> lt (ℵ<  {lv} {y} )  ) (λ ()) (ℵ<  {lv} {y} )
-triOonSameLevel  {lv} (OSuc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
-triOonSameLevel  {.(Suc lv)} (OSuc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
-triOonSameLevel  {lv} (OSuc x) (OSuc y) with triOonSameLevel x y
-triOonSameLevel  {lv} (OSuc x) (OSuc y) | tri< a ¬b ¬c = tri< (s< a) (λ tx=ty → trio<≡ tx=ty (s< a) )  (  λ lt → trio<> lt (s< a) )
-triOonSameLevel  {lv} (OSuc x) (OSuc x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬o< refl ≡→¬o<
-triOonSameLevel  {lv} (OSuc x) (OSuc y) | tri> ¬a ¬b c = tri> (  λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
+triOrdd : {lx : Nat}   → Trichotomous  _≡_ ( _d<_  {lx} {lx} )
+triOrdd  {lv} Φ Φ = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {lv} Φ (OSuc y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
+triOrdd  {.(Suc lv)} Φ (ℵ lv) = tri<  (ℵΦ<  {lv} {Φ} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ<  {lv} {Φ} )) )
+triOrdd  {Suc lv} (ℵ lv) Φ = tri> ( λ lt → trio<> lt (ℵΦ<  {lv} {Φ} ) ) (λ ()) (ℵΦ<  {lv} {Φ} )
+triOrdd  {Suc lv} (ℵ lv) (OSuc y) = tri> ( λ lt → trio<> lt (ℵ<  {lv} {y} )  ) (λ ()) (ℵ<  {lv} {y} )
+triOrdd  {lv} (OSuc x) Φ = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
+triOrdd  {.(Suc lv)} (OSuc x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
+triOrdd  {lv} (OSuc x) (OSuc y) with triOrdd x y
+triOrdd  {lv} (OSuc x) (OSuc 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 x) (OSuc x) | tri≈ ¬a refl ¬c = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {lv} (OSuc x) (OSuc y) | tri> ¬a ¬b c = tri> (  λ lt → trio<> lt (s< c) ) (λ tx=ty → trio>≡ tx=ty (s< c) ) (s< c)
 
-<→≤ : {lx ly : Nat} → lx < ly → (Suc lx ≤ ly)
-<→≤ {Zero} {Suc ly} (s≤s lt) = s≤s z≤n
-<→≤ {Suc lx} {Zero} ()
-<→≤ {Suc lx} {Suc ly} (s≤s lt) = s≤s (<→≤ lt) 
+d<→lv :  {x y  : Ordinal }   → ord x d< ord y → lv x ≡ lv y
+d<→lv Φ< = refl
+d<→lv (s< lt) = refl
+d<→lv ℵΦ< = refl
+d<→lv ℵ< = refl
 
-orddtrans : {lx ly lz : Nat} {x  : OridinalD  lx } { y : OridinalD  ly } { z : OridinalD  lz } → x o< y → y o< z → x o< z 
-orddtrans {lx} {ly} {lz} x<y y<z with <-cmp lx ly  | <-cmp ly lz
-orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri< a₁ ¬b₁ ¬c₁ = l< ( <-trans a a₁ )
-orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri≈ ¬a refl ¬c₁ = l< a
-orddtrans {lx} {ly} {lz} x<y y<z | tri< a ¬b ¬c | tri> ¬a ¬b₁ c = l< {!!} -- ⊥-elim ( ¬a c )
-orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri< a ¬b₁ ¬c = l< {!!}
-orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri≈ ¬a₁ refl ¬c = l< {!!}
-orddtrans {lx} {ly} {lz} x<y y<z | tri> ¬a ¬b c | tri> ¬a₁ ¬b₁ c₁ = {!!}
-orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri< a ¬b ¬c₁ = l< a
-orddtrans {lx} {ly} {lz} x<y y<z | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = l< {!!}
-orddtrans {lx} {lx} {lx} x<y y<z | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = orddtrans1  x<y y<z where
-  orddtrans1 : {lx : Nat} {x y z : OridinalD  lx }   → x o< y → y o< z → x o< z
-  orddtrans1 = {!!}
-
-  
+orddtrans : {lx : Nat} {x y z : OrdinalD  lx }   → x d< y → y d< z → x d< z
+orddtrans {lx} {.Φ} {.(OSuc _)} {.(OSuc _)} Φ< (s< y<z) = Φ< 
+orddtrans {Suc lx} {Φ {Suc lx}} {OSuc y} {ℵ lx} Φ< ℵ< = ℵΦ< {lx} {y}
+orddtrans {lx} {.(OSuc _)} {.(OSuc _)} {.(OSuc _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
+orddtrans {.(Suc _)} {.(OSuc _)} {.(OSuc _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ<
+orddtrans {.(Suc _)} {.Φ} {.(ℵ _)} {z} ℵΦ< ()
+orddtrans {.(Suc _)} {.(OSuc _)} {.(ℵ _)} {z} ℵ< ()
 
 max : (x y : Nat) → Nat
 max Zero Zero = Zero
@@ -113,30 +85,34 @@
 max (Suc x) Zero = (Suc x)
 max (Suc x) (Suc y) = Suc ( max x y )
 
---  use cannot use OridinalD  (Data.Nat_⊔_ lx  ly), I don't know why
-
-maxα> : { lx ly : Nat } → OridinalD  lx  →  OridinalD  ly  → lx > ly  → OridinalD  lx
-maxα> x y _ = x
+maxαd : { lx : Nat } → OrdinalD  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
 
-maxα= : { lx : Nat } → OridinalD  lx  →  OridinalD  lx  →  OridinalD  lx
-maxα= x y with triOonSameLevel x y
-maxα= x y | tri< a ¬b ¬c = y
-maxα= x y | tri≈ ¬a b ¬c = x
-maxα= x y | tri> ¬a ¬b c = x
+maxα :  Ordinal →  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) }
 
-OrdTrans : Transitive (λ ( a b : Ordinal ) → (a ≡ b) ∨ (Ordinal.lv a < Ordinal.lv b) ∨ (Ordinal.ord a o< Ordinal.ord b) )
+OrdTrans : Transitive (λ ( a b : Ordinal ) → (a ≡ b) ∨ (Ordinal.lv a < Ordinal.lv b) ∨ (Ordinal.ord a d< Ordinal.ord b) )
 OrdTrans (case1 refl) (case1 refl) = case1 refl
 OrdTrans (case1 refl) (case2 lt2) = case2 lt2
 OrdTrans (case2 lt1) (case1 refl) = case2 lt1
-OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 ( case1 ( <-trans x y ) )
-OrdTrans (case2 (case1 x)) (case2 (case2 y)) = case2 {!!}
-OrdTrans (case2 (case2 x)) (case2 (case1 y)) = case2 {!!}
-OrdTrans (case2 (case2 x)) (case2 (case2 y)) = case2 {!!}
+OrdTrans (case2 (case1 x)) (case2 (case1 y)) = case2 (case1 ( <-trans x y ) )
+OrdTrans (case2 (case1 x)) (case2 (case2 y)) with d<→lv y
+OrdTrans (case2 (case1 x)) (case2 (case2 y)) | refl = case2 (case1 x )
+OrdTrans (case2 (case2 x)) (case2 (case1 y)) with d<→lv x
+OrdTrans (case2 (case2 x)) (case2 (case1 y)) | refl = case2 (case1 y)
+OrdTrans (case2 (case2 x)) (case2 (case2 y)) with d<→lv x | d<→lv y
+OrdTrans (case2 (case2 x)) (case2 (case2 y)) | refl | refl = case2 (case2 (orddtrans x y ))
 
 OrdPreorder : Preorder n n n
 OrdPreorder = record { Carrier = Ordinal
    ; _≈_  = _≡_ 
-   ; _∼_   = λ a b → (a ≡ b)  ∨ (Ordinal.lv a < Ordinal.lv b)  ∨ (Ordinal.ord a o< Ordinal.ord b )  
+   ; _∼_   = λ a b → (a ≡ b)  ∨ ( a o< b )  
    ; isPreorder   = record {
         isEquivalence = record { refl = refl  ; sym = sym ; trans = trans }
         ; reflexive     = case1 
@@ -146,41 +122,40 @@
 
 -- X' = { x ∈ X |  ψ  x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
 
-data Constructible {lv : Nat} ( α : OridinalD  lv )  :  Set (suc n) where
-    fsub : ( ψ : OridinalD  lv → Set n ) → Constructible  α
-    xself : OridinalD  lv → Constructible  α
+data Constructible ( α : Ordinal  )  :  Set (suc n) where
+    fsub : ( ψ : Ordinal  → Set n ) → Constructible  α
+    xself : Ordinal → Constructible  α
 
 record ConstructibleSet  : Set (suc n) where
   field
-    level : Nat
-    α : OridinalD  level 
+    α : Ordinal
     constructible : Constructible α
 
 open ConstructibleSet
 
-data _c∋_  : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' } →
-        Constructible  {lv} α → Constructible  {lv'} α' → Set n where
-    c> : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' }
-        (ta : Constructible  {lv} α ) ( tx : Constructible  {lv'} α' ) → α' o< α →  ta c∋ tx
-    xself-fsub  : {lv : Nat} {α : OridinalD  lv } 
-         (ta : OridinalD  lv ) ( ψ : OridinalD  lv → Set n ) → _c∋_  {_} {_} {α} {α} (xself ta ) ( fsub ψ)  
-    fsub-fsub : {lv lv' : Nat} {α : OridinalD  lv } 
-          ( ψ : OridinalD  lv → Set n ) ( ψ₁ : OridinalD  lv → Set n ) →
-         ( ∀ ( x :  OridinalD  lv ) → ψ x →  ψ₁ x ) →  _c∋_  {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁) 
+data _c∋_  : {α α' : Ordinal  }  →
+        Constructible  α → Constructible   α' → Set n where
+    c> :  {α α' : Ordinal }
+        (ta : Constructible  α ) ( tx : Constructible   α' ) → α' o< α →  ta c∋ tx
+    xself-fsub  :  {α : Ordinal  } 
+         (ta : Ordinal ) ( ψ : Ordinal  → Set n ) → _c∋_  {α} {α} (xself ta ) ( fsub ψ)  
+    fsub-fsub :  {α : Ordinal   } 
+          ( ψ : Ordinal   → Set n ) ( ψ₁ : Ordinal   → Set n ) →
+         ( ∀ ( x :  Ordinal  ) → ψ x →  ψ₁ x ) →  _c∋_  {α} {α} ( fsub ψ ) ( fsub ψ₁) 
 
 _∋_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set n 
 a ∋ x  = constructible a c∋ constructible x
 
-transitiveness : (a b c : ConstructibleSet ) → a ∋ b → b ∋ c → a ∋ c
-transitiveness  a b c a∋b b∋c with constructible a c∋ constructible b | constructible b c∋ constructible c
-... | t1 | t2 = {!!}
+-- transitiveness : (a b c : ConstructibleSet ) → a ∋ b → b ∋ c → a ∋ c
+-- transitiveness  a b c a∋b b∋c with constructible a c∋ constructible b | constructible b c∋ constructible c
+-- ... | t1 | t2 = {!!}
 
-data _c≈_  : {lv lv' : Nat} {α : OridinalD  lv } {α' : OridinalD  lv' } →
-        Constructible  {lv} α → Constructible  {lv'} α' → Set n where
-    crefl :  {lv : Nat} {α : OridinalD  lv } → _c≈_  {_} {_} {α} {α} (xself α ) (xself α )
-    feq :  {lv : Nat} {α : OridinalD  lv }
-          → ( ψ : OridinalD  lv → Set n ) ( ψ₁ : OridinalD  lv → Set n ) 
-          → (∀ ( x :  OridinalD  lv ) → ψ x  ⇔ ψ₁ x ) → _c≈_  {_} {_} {α} {α} ( fsub ψ ) ( fsub ψ₁)
+data _c≈_  :  {α α' : Ordinal}  →
+        Constructible  α → Constructible   α' → Set n where
+    crefl :  {α : Ordinal  } → _c≈_  {α} {α} (xself α ) (xself α )
+    feq :  {lv : Nat} {α : Ordinal }
+          → ( ψ : Ordinal  → Set n ) ( ψ₁ : Ordinal → Set n ) 
+          → (∀ ( x :  Ordinal ) → ψ x  ⇔ ψ₁ x ) → _c≈_    {α} {α} ( fsub ψ ) ( fsub ψ₁)
 
 _≈_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set n 
 a ≈ x  = constructible a c≈ constructible x
@@ -190,7 +165,7 @@
     ZFSet = ConstructibleSet 
     ; _∋_ = _∋_
     ; _≈_ = _≈_ 
-    ; ∅  = record { level = Zero ; α = Φ ; constructible = xself Φ }
+    ; ∅  = record {  α = record {lv = Zero ; ord = Φ } ; constructible = xself ( record {lv = Zero ; ord = Φ }) }
     ; _×_ = {!!}
     ; Union = {!!}
     ; Power = {!!}