Mercurial > hg > Members > kono > Proof > ZF-in-agda
diff ordinal-definable.agda @ 29:fce60b99dc55
posturate OD is isomorphic to Ordinal
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
---|---|
date | Mon, 20 May 2019 18:18:43 +0900 |
parents | constructible-set.agda@f36e40d5d2c3 |
children | 3b0fdb95618e |
line wrap: on
line diff
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/ordinal-definable.agda Mon May 20 18:18:43 2019 +0900 @@ -0,0 +1,151 @@ +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 = {!!} + + + + + +