Mercurial > hg > Members > kono > Proof > ZF-in-agda

changeset 24:3186bbee99dd

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separte level
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 18 May 2019 16:03:10 +0900
parents 7293a151d949
children 0f3d98e97593
files constructible-set.agda
diffstat 1 files changed, 113 insertions(+), 90 deletions(-) [+]
line wrap: on
line diff
--- a/constructible-set.agda	Sat May 18 08:29:08 2019 +0900
+++ b/constructible-set.agda	Sat May 18 16:03:10 2019 +0900
@@ -1,5 +1,5 @@
 open import Level
-module constructible-set (n : Level) where
+module constructible-set where
 
 open import zf
 
@@ -7,25 +7,25 @@
 
 open import  Relation.Binary.PropositionalEquality
 
-data OrdinalD  : (lv : Nat) → Set n where
-   Φ : {lv : Nat} → OrdinalD  lv
-   OSuc : {lv : Nat} → OrdinalD  lv → OrdinalD lv
+data OrdinalD {n : Level} :  (lv : Nat) → Set n where
+   Φ : (lv : Nat) → OrdinalD  lv
+   OSuc : (lv : Nat) → OrdinalD {n} lv → OrdinalD lv
    ℵ_ :  (lv : Nat) → OrdinalD (Suc lv)
 
-record Ordinal : Set n where
+record Ordinal {n : Level} : Set n where
    field
      lv : Nat
-     ord : OrdinalD lv
+     ord : OrdinalD {n} lv
 
-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) 
+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
+   ℵΦ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } →  Φ  (Suc lx) d< (ℵ lx) 
+   ℵ<  : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } →  OSuc  (Suc lx) x d< (ℵ lx) 
 
 open Ordinal
 
-_o<_ : ( x y : Ordinal ) → Set n
+_o<_ : {n : Level} ( x y : Ordinal ) → Set (suc n)
 _o<_ x y =  (lv x < lv y )  ∨ ( ord x d< ord y )
 
 open import Data.Nat.Properties 
@@ -35,53 +35,53 @@
 open import Relation.Binary
 open import Relation.Binary.Core
 
-o∅ : Ordinal
-o∅  = record { lv = Zero ; ord = Φ }
+o∅ : {n : Level} → Ordinal {n}
+o∅  = record { lv = Zero ; ord = Φ Zero }
 
 
-≡→¬d< : {lv : Nat} → {x  : OrdinalD  lv }  → x d< x → ⊥
-≡→¬d<  {lx} {OSuc y} (s< t) = ≡→¬d< t
+≡→¬d< : {n : Level} →  {lv : Nat} → {x  : OrdinalD {n}  lv }  → x d< x → ⊥
+≡→¬d<  {n} {lx} {OSuc lx y} (s< t) = ≡→¬d< t
 
-trio<> : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  →  y d< x → x d< y → ⊥
-trio<>  {lx} {.(OSuc _)} {.(OSuc _)} (s< s) (s< t) = 
+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<≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → x d< y → ⊥
+trio<≡ : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  → x ≡ y  → x d< y → ⊥
 trio<≡ refl = ≡→¬d<
 
-trio>≡ : {lx : Nat} {x  : OrdinalD  lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
+trio>≡ : {n : Level} →  {lx : Nat} {x  : OrdinalD {n} lx } { y : OrdinalD  lx }  → x ≡ y  → y d< x → ⊥
 trio>≡ refl = ≡→¬d<
 
-triO : {lx ly : Nat} → OrdinalD  lx  →  OrdinalD  ly  → Tri (lx < ly) ( lx ≡ ly ) ( lx > ly )
-triO  {lx} {ly} x y = <-cmp lx ly
+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 : {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)
+triOrdd : {n : Level} →  {lx : Nat}   → Trichotomous  _≡_ ( _d<_  {n} {lx} {lx} )
+triOrdd  {_} {lv} (Φ lv) (Φ lv) = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {_} {Suc lv} (ℵ lv) (ℵ lv) = tri≈ ≡→¬d< refl ≡→¬d<
+triOrdd  {_} {lv} (Φ lv) (OSuc lv y) = tri< Φ< (λ ()) ( λ lt → trio<> lt Φ< )
+triOrdd  {_} {.(Suc lv)} (Φ (Suc lv)) (ℵ lv) = tri<  (ℵΦ<  {_} {lv} {Φ (Suc lv)} ) (λ ()) ( λ lt → trio<> lt ((ℵΦ< {_} {lv} {Φ (Suc lv)} )) )
+triOrdd  {_} {Suc lv} (ℵ lv) (Φ (Suc lv)) = tri> ( λ lt → trio<> lt (ℵΦ< {_} {lv} {Φ (Suc lv)} ) ) (λ ()) (ℵΦ<  {_} {lv} {Φ (Suc lv)} )
+triOrdd  {_} {Suc lv} (ℵ lv) (OSuc (Suc lv) y) = tri> ( λ lt → trio<> lt (ℵ< {_} {lv} {y} )  ) (λ ()) (ℵ< {_} {lv} {y} )
+triOrdd  {_} {lv} (OSuc lv x) (Φ lv) = tri> (λ lt → trio<> lt Φ<) (λ ()) Φ<
+triOrdd  {_} {.(Suc lv)} (OSuc (Suc lv) x) (ℵ lv) = tri< ℵ< (λ ()) (λ lt → trio<> lt ℵ< )
+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)
 
-d<→lv :  {x y  : Ordinal }   → ord x d< ord y → lv x ≡ lv y
+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
 d<→lv ℵΦ< = refl
 d<→lv ℵ< = refl
 
-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< ()
+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 {_} {Suc lx} {Φ (Suc lx)} {OSuc (Suc lx) y} {ℵ lx} Φ< ℵ< = ℵΦ< {_} {lx} {y}
+orddtrans {_} {lx} {.(OSuc lx _)} {.(OSuc lx _)} {.(OSuc lx _)} (s< x<y) (s< y<z) = s< ( orddtrans x<y y<z )
+orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) _)} {.(ℵ _)} (s< x<y) ℵ< = ℵ<
+orddtrans {_} {Suc lx} {Φ (Suc lx)} {.(ℵ _)} {z} ℵΦ< ()
+orddtrans {_} {Suc lx} {OSuc (Suc lx) _} {.(ℵ _)} {z} ℵ< ()
 
 max : (x y : Nat) → Nat
 max Zero Zero = Zero
@@ -89,31 +89,52 @@
 max (Suc x) Zero = (Suc x)
 max (Suc x) (Suc y) = Suc ( max x y )
 
-maxαd : { lx : Nat } → OrdinalD  lx  →  OrdinalD  lx  →  OrdinalD  lx
+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
 
-maxα :  Ordinal →  Ordinal  → Ordinal
+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) }
 
-_o≤_ : Ordinal → Ordinal → Set n
+_o≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
 a o≤ b  = (a ≡ b)  ∨ ( a o< b )
 
-trio< : Trichotomous  _≡_  _o<_ 
+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 ) ) {!!}
-trio< a b | tri> ¬a ¬b c = tri> {!!} (λ refl → ¬b (cong ( λ x → lv x ) refl ) ) (case1 c)
+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) with d<→lv x
+   lemma1 (case2 x) | refl = ¬b refl
+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) with d<→lv x
+   lemma1 (case2 x) | refl = ¬b refl
 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 {!!} )  {!!}
-trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri≈ ¬a₁ refl ¬c₁ = tri≈ {!!} refl {!!}
-trio< record { lv = .(lv b) ; ord = x } b | tri≈ ¬a refl ¬c | tri> ¬a₁ ¬b c = tri> {!!} {!!} (case2 c)
+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
 
-OrdTrans : Transitive _o≤_
+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
@@ -125,41 +146,41 @@
 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
+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
+        ; trans         = OrdTrans 
      }
  }
 
-TransFinite : ( ψ : Ordinal  → Set n ) 
+TransFinite : {n : Level} → ( ψ : Ordinal {n} → Set n ) 
   → ( ∀ (lx : Nat ) → ψ ( record { lv = Suc lx ; ord = ℵ lx } )) 
-  → ( ∀ (lx : Nat ) → ψ ( record { lv = lx ; ord = Φ } ) )
-  → ( ∀ (lx : Nat ) → (x : OrdinalD lx )  → ψ ( record { lv = lx ; ord = x } ) → ψ ( record { lv = lx ; ord = OSuc x } ) )
+  → ( ∀ (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 ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ } = caseΦ lv
-TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc ord₁ } = caseOSuc lv ord₁
+TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = Φ lv } = caseΦ lv
+TransFinite ψ caseℵ caseΦ caseOSuc record { lv = lv ; ord = OSuc lv ord₁ } = caseOSuc lv ord₁
     ( TransFinite ψ caseℵ caseΦ caseOSuc (record { lv = lv ; ord = ord₁ } ))
 TransFinite ψ caseℵ caseΦ caseOSuc record { lv = Suc lv₁ ; ord = ℵ lv₁ } = caseℵ lv₁
 
 -- X' = { x ∈ X |  ψ  x } ∪ X , Mα = ( ∪ (β < α) Mβ ) '
 
-record ConstructibleSet  : Set (suc (suc n)) where
+record ConstructibleSet {n : Level} : Set (suc n) where
   field
-    α : Ordinal
-    constructible : Ordinal  → Set (suc n)
+    α : Ordinal {suc n}
+    constructible : Ordinal {suc n} → Set n
 
 open ConstructibleSet
 
-_∋_  : (ConstructibleSet ) → (ConstructibleSet  ) → Set (suc n)
+_∋_  : {n : Level} →  (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set (suc n)
 a ∋ x  = ( α x o< α a ) ∧ constructible a ( α x )
 
-c∅ : ConstructibleSet
-c∅  = record {α = o∅ ; constructible = λ x → Lift (suc n) ⊥ }
+c∅ : {n : Level} → ConstructibleSet
+c∅ {n} = record {α = o∅ ; constructible = λ x → Lift n ⊥ }
 
 record SupR {n m : Level} {S : Set n} ( _≤_ : S → S → Set m  ) (ψ : S → S ) (X : S) : Set (n ⊔ m)  where
   field
@@ -169,47 +190,49 @@
 
 open SupR
 
-_⊆_ : ( A B : ConstructibleSet  ) → ∀{ x : ConstructibleSet } →  Set (suc n)
+_⊆_ : {n : Level} → ( A B : ConstructibleSet  ) → ∀{ x : ConstructibleSet } →  Set (suc n)
 _⊆_ A B {x} = A ∋ x →  B ∋ x
 
-suptraverse : (X : ConstructibleSet ) ( max : ConstructibleSet) ( ψ : ConstructibleSet  → ConstructibleSet ) → ConstructibleSet
+suptraverse : {n : Level} → (X : ConstructibleSet {n}) ( max : ConstructibleSet {n}) ( ψ : ConstructibleSet  {n} → ConstructibleSet  {n}) → ConstructibleSet {n}
 suptraverse X max ψ  = {!!} 
 
-Sup : (ψ : ConstructibleSet → ConstructibleSet )  → (X : ConstructibleSet)  → SupR (λ x a → (α a ≡ α x) ∨ (a ∋ x)) ψ X
-sup (Sup ψ X ) = suptraverse X c∅ ψ 
-smax (Sup ψ X ) = {!!} -- TransFinite {!!} {!!} {!!} {!!} {!!} 
+Sup : {n : Level } → (ψ : ConstructibleSet → ConstructibleSet )  → (X : ConstructibleSet)  → SupR (λ x a → (α a ≡ α x) ∨ (a ∋ x)) ψ X
+sup  (Sup {n} ψ X ) = suptraverse X (c∅ {n}) ψ 
+smax (Sup ψ X ) = {!!} 
 suniq (Sup ψ 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 = {!!}
-
 open import Data.Unit
 open SupR
 
-ConstructibleSet→ZF : ZF {suc (suc n)} {suc (suc n)}
-ConstructibleSet→ZF   = record { 
+ConstructibleSet→ZF : {n : Level} → ZF {suc n} {suc n}
+ConstructibleSet→ZF {n}  = record { 
     ZFSet = ConstructibleSet 
-    ; _∋_ = λ a b → Lift (suc (suc n)) ( a ∋ b )
+    ; _∋_ = _∋_ 
     ; _≈_ = _≡_ 
-    ; ∅  = c∅ 
+    ; ∅  = c∅
     ; _,_ = _,_
     ; Union = Union
     ; Power = {!!}
     ; Select = Select
-    ; Replace = Replace
+    ; Replace = {!!}
     ; infinite = {!!}
     ; isZF = {!!}
  } where
-    conv : (ConstructibleSet → Set (suc (suc n))) → ConstructibleSet → Set (suc n)
-    conv ψ x with ψ x
+    conv : {n : Level} → (ConstructibleSet {n} → Set (suc (suc n))) → ConstructibleSet → Set (suc n)
+    conv {n} ψ x with ψ x
     ... | t =  Lift ( suc n ) ⊤
-    Select : (X : ConstructibleSet) → (ConstructibleSet → Set (suc (suc n))) → ConstructibleSet
-    Select X ψ = record { α = α X ; constructible = λ x → (conv ψ) (record { α = x ; constructible = λ x → constructible X x }  ) }
-    Replace : (X : ConstructibleSet) → (ConstructibleSet → ConstructibleSet) → ConstructibleSet
-    Replace X ψ  = record { α = α (sup (Sup ψ X))  ; constructible = λ x → {!!}  }
-    _,_ : ConstructibleSet → ConstructibleSet → ConstructibleSet
-    a , b  = record { α = maxα (α a) (α b) ; constructible = λ x → {!!} }
+    Select : (X : ConstructibleSet {n}) → (ConstructibleSet  {n} → Set (suc n)) → ConstructibleSet {n}
+    Select X ψ = record { α = α X ; constructible = λ x →  {!!} } -- ψ (record { α = x ; constructible = λ x → constructible X x }  ) }
+    Replace : (X : ConstructibleSet {n} ) → (ConstructibleSet → ConstructibleSet) → ConstructibleSet
+    Replace X ψ  = record { α = α (sup supψ)  ; constructible = λ x →  {!!}   }  where
+          supψ : SupR (λ x a → (α a ≡ α x) ∨ (a ∋ x)) ψ X
+          supψ = Sup ψ X
+          repl : Ordinal {n} → Set (suc n)
+          repl x = {!!}
+    conv1 : (Ordinal {n} → Set n) → Ordinal {n} → Set n
+    conv1 ψ x with ψ 
+    ... | t =  Lift  n ⊤
+    _,_ : ConstructibleSet {n} → ConstructibleSet → ConstructibleSet
+    a , b  = record { α = maxα (α a) (α b) ; constructible = λ x → {!!} } -- ((x ≡ α a ) ∨ ( x ≡ α b )) }
     Union : ConstructibleSet → ConstructibleSet
     Union a = {!!}