view constructible-set.agda @ 24:3186bbee99dd

separte level
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 18 May 2019 16:03:10 +0900
parents 7293a151d949
children 0f3d98e97593
line wrap: on
line source

open import Level
module constructible-set where

open import zf

open import Data.Nat renaming ( zero to Zero ; suc to Suc ;  ℕ to Nat ; _⊔_ to _n⊔_ ) 

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
   ℵ_ :  (lv : Nat) → OrdinalD (Suc lv)

record Ordinal {n : Level} : Set n where
   field
     lv : Nat
     ord : OrdinalD {n} 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
   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<_ : {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 
open import Data.Empty
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 }


≡→¬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 }  →  y d< x → x d< y → ⊥
trio<>  {n} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = 
    trio<> s 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  {_} {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 : {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 : {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
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

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≤_ : {n : Level} → Ordinal → Ordinal → Set (suc n)
a o≤ b  = (a ≡ b)  ∨ ( a o< b )

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) 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 (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 : {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 (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 : {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 = Suc lx ; ord = ℵ lx } )) 
  → ( ∀ (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 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 {n : Level} : Set (suc n) where
  field
    α : Ordinal {suc n}
    constructible : Ordinal {suc n} → Set n

open ConstructibleSet

_∋_  : {n : Level} →  (ConstructibleSet {n}) → (ConstructibleSet {n} ) → Set (suc n)
a ∋ x  = ( α x o< α a ) ∧ constructible a ( α x )

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
    sup : S
    smax  : ∀ { x : S } → x ≤ X  → ψ x ≤ sup 
    suniq : {max : S} → ( ∀ { x :  S } → x ≤ X  → ψ x ≤ max ) → max ≤ sup 

open SupR

_⊆_ : {n : Level} → ( A B : ConstructibleSet  ) → ∀{ x : ConstructibleSet } →  Set (suc n)
_⊆_ A B {x} = A ∋ x →  B ∋ x

suptraverse : {n : Level} → (X : ConstructibleSet {n}) ( max : ConstructibleSet {n}) ( ψ : ConstructibleSet  {n} → ConstructibleSet  {n}) → ConstructibleSet {n}
suptraverse X max ψ  = {!!} 

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 ) = {!!}
     
open import Data.Unit
open SupR

ConstructibleSet→ZF : {n : Level} → ZF {suc n} {suc n}
ConstructibleSet→ZF {n}  = record { 
    ZFSet = ConstructibleSet 
    ; _∋_ = _∋_ 
    ; _≈_ = _≡_ 
    ; ∅  = c∅
    ; _,_ = _,_
    ; Union = Union
    ; Power = {!!}
    ; Select = Select
    ; Replace = {!!}
    ; infinite = {!!}
    ; isZF = {!!}
 } where
    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 {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 = {!!}