Mercurial > hg > Members > kono > Proof > ZF-in-agda
view ordinal-definable.agda @ 29:fce60b99dc55
posturate OD is isomorphic to Ordinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Mon, 20 May 2019 18:18:43 +0900 |
parents | constructible-set.agda@f36e40d5d2c3 |
children | 3b0fdb95618e |
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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 -- X' = { x ∈ X | ψ x } ∪ X , Mα = ( ∪ (β < α) Mβ ) ' -- Ordinal Definable Set -- o∋ : {n : Level} → {A : Ordinal {n}} → (OrdinalDefinable {n} A ) → (x : Ordinal {n} ) → (x o< A) → Set n -- o∋ a x x<A = def a x x<A -- TC u : Transitive Closure of OD u -- -- all elements of u or elements of elements of u, etc... -- -- TC Zero = u -- TC (suc n) = ∪ (TC n) -- -- TC u = TC ω u = ∪ ( TC n ) n ∈ ω -- -- u ∪ ( ∪ u ) ∪ ( ∪ (∪ u ) ) .... -- -- Heritic Ordinal Definable -- -- ( HOD = {x | TC x ⊆ OD } ) ⊆ OD x ∈ OD here -- record OD {n : Level} : Set (suc n) where field def : (x : Ordinal {n} ) → Set n open OD open import Data.Unit postulate od→ord : {n : Level} → OD {n} → Ordinal {n} ord→od : {n : Level} → Ordinal {n} → OD {n} ord→od x = record { def = λ y → x ≡ y } _∋_ : { n : Level } → ( a x : OD {n} ) → Set n _∋_ {n} a x = def a ( od→ord x ) _c<_ : { n : Level } → ( a x : 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 ) postulate c<→o< : {n : Level} {x y : OD {n} } → x c< y → od→ord x o< od→ord x o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od 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-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 HOD = OD od∅ : {n : Level} → HOD {n} od∅ {n} = record { def = λ _ → Lift n ⊥ } ∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) ∅1 {n} x (lift ()) HOD→ZF : {n : Level} → ZF {suc n} {suc n} HOD→ZF {n} = record { ZFSet = OD {n} ; _∋_ = λ a x → Lift (suc n) ( a ∋ x ) ; _≈_ = _≡_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } } ; isZF = isZF } where Replace : OD {n} → (OD {n} → OD {n} ) → OD {n} Replace X ψ = sup-od ψ Select : OD {n} → (OD {n} → Set (suc n) ) → OD {n} Select X ψ = record { def = λ x → select ( ord→od x ) } where select : OD {n} → Set n select x with ψ x ... | t = Lift n ⊤ _,_ : OD {n} → OD {n} → OD {n} x , y = record { def = λ z → ( (z ≡ od→ord x ) ∨ ( z ≡ od→ord y )) } Union : OD {n} → OD {n} Union x = record { def = λ y → {z : Ordinal {n}} → def x z → def (ord→od z) y } Power : OD {n} → OD {n} Power x = record { def = λ y → (z : Ordinal {n} ) → ( def x y ∧ def (ord→od z) y ) } ZFSet = OD {n} _∈_ : ( A B : ZFSet ) → Set n A ∈ B = B ∋ A _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set n _⊆_ A B {x} = A ∋ x → B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∧ (Lift (suc n) ( B ∋ x ) )) _∪_ : ( A B : ZFSet ) → ZFSet A ∪ B = Select (A , B) ( λ x → (Lift (suc n) ( A ∋ x )) ∨ (Lift (suc n) ( B ∋ x ) )) infixr 200 _∈_ infixr 230 _∩_ _∪_ infixr 220 _⊆_ isZF : IsZF (OD {n}) (λ a x → Lift (suc n) ( a ∋ x )) _≡_ od∅ _,_ Union Power Select Replace (record { def = λ x → x ≡ record { lv = Suc Zero ; ord = ℵ Zero } }) isZF = record { isEquivalence = record { refl = refl ; sym = sym ; trans = trans } ; pair = pair ; union→ = {!!} ; union← = {!!} ; empty = empty ; power→ = {!!} ; power← = {!!} ; extentionality = {!!} ; minimul = {!!} ; regularity = {!!} ; infinity∅ = {!!} ; infinity = {!!} ; selection = {!!} ; replacement = {!!} } where open _∧_ pair : (A B : OD {n} ) → Lift (suc n) ((A , B) ∋ A) ∧ Lift (suc n) ((A , B) ∋ B) proj1 (pair A B ) = lift ( case1 refl ) proj2 (pair A B ) = lift ( case2 refl ) empty : (x : OD {n} ) → ¬ Lift (suc n) (od∅ ∋ x) empty x (lift (lift ())) union→ : (X x y : OD {n} ) → Lift (suc n) (X ∋ x) → Lift (suc n) (x ∋ y) → Lift (suc n) (Union X ∋ y) union→ X x y (lift X∋x) (lift x∋y) = lift lemma where lemma : {z : Ordinal {n} } → def X z → z ≡ od→ord y lemma {z} X∋z = {!!}