Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 34:c9ad0d97ce41
fix oridinal
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 22 May 2019 11:52:49 +0900 |
| parents | 2b853472cb24 |
| children | 88b77cecaeba |
| files | ordinal-definable.agda ordinal.agda zf.agda |
| diffstat | 3 files changed, 33 insertions(+), 44 deletions(-) [+] |
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--- a/ordinal-definable.agda Tue May 21 18:17:24 2019 +0900 +++ b/ordinal-definable.agda Wed May 22 11:52:49 2019 +0900 @@ -15,30 +15,8 @@ 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 @@ -95,10 +73,11 @@ ... | t with t (case2 Φ< ) c3 lx (Φ .lx) d not | t | () c3 lx (OSuc .lx x₁) d not with not ( record { lv = lx ; ord = OSuc lx x₁ } ) - ... | t with t (case2 (s< {!!} ) ) --- x d< OSuc lx x is bad on ℵ case + ... | t with t (case2 (s< s<refl ) ) c3 lx (OSuc .lx x₁) d not | t | () - c3 .(Suc lv) (ℵ lv) not = {!!} + c3 (Suc lx) (ℵ lx) d not with not ( record { lv = Suc lx ; ord = OSuc (Suc lx) (Φ (Suc lx)) } ) + ... | t with t (case2 (s< (ℵΦ< {_} {_} {Φ (Suc lx)}))) + c3 .(Suc lx) (ℵ lx) d not | t | () ∅2 : {n : Level} → od→ord ( od∅ {n} ) ≡ o∅ {n} ∅2 {n} = {!!}
--- a/ordinal.agda Tue May 21 18:17:24 2019 +0900 +++ b/ordinal.agda Wed May 22 11:52:49 2019 +0900 @@ -1,3 +1,4 @@ +{-# OPTIONS --allow-unsolved-metas #-} open import Level module ordinal where @@ -17,11 +18,16 @@ lv : Nat ord : OrdinalD {n} lv +data ¬ℵ {n : Level} {lx : Nat } : ( x : OrdinalD {n} lx ) → Set where + ¬ℵΦ : ¬ℵ (Φ lx) + ¬ℵs : {x : OrdinalD {n} lx } → ¬ℵ x → ¬ℵ (OSuc lx x) + 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) + ℵ< : {lx : Nat} → {x : OrdinalD {n} (Suc lx) } → ¬ℵ x → OSuc (Suc lx) x d< (ℵ lx) + ℵs< : {lx : Nat} → (ℵ lx) d< OSuc (Suc lx) (ℵ lx) open Ordinal @@ -39,13 +45,19 @@ o∅ : {n : Level} → Ordinal {n} o∅ = record { lv = Zero ; ord = Φ Zero } +s<refl : {n : Level } {lx : Nat } { x : OrdinalD {n} lx } → x d< OSuc lx x +s<refl {n} {lv} {Φ lv} = Φ< +s<refl {n} {lv} {OSuc lv x} = s< s<refl +s<refl {n} {Suc lv} {ℵ lv} = ℵs< + ≡→¬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} {lx} {.(OSuc lx _)} {.(OSuc lx _)} (s< s) (s< t) = trio<> s t +trio<> {_} {.(Suc _)} {.(OSuc (Suc _) (ℵ _))} {.(ℵ _)} ℵs< (ℵ< {_} {.(ℵ _)} ()) +trio<> {_} {.(Suc _)} {.(ℵ _)} {.(OSuc (Suc _) (ℵ _))} (ℵ< ()) ℵs< trio<≡ : {n : Level} → {lx : Nat} {x : OrdinalD {n} lx } { y : OrdinalD lx } → x ≡ y → x d< y → ⊥ trio<≡ refl = ≡→¬d< @@ -62,9 +74,9 @@ 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 {_} {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 {_} {.(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< @@ -74,15 +86,22 @@ d<→lv Φ< = refl d<→lv (s< lt) = refl d<→lv ℵΦ< = refl -d<→lv ℵ< = refl +d<→lv (ℵ< _) = refl +d<→lv ℵs< = 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 {_} {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< () +orddtrans {_} {Suc lx} {.(OSuc (Suc lx) _)} {.(OSuc (Suc lx) (Φ (Suc lx)))} {.(ℵ lx)} (s< ()) (ℵ< ¬ℵΦ) +orddtrans {_} {Suc lx} {OSuc (Suc lx) x} {OSuc (Suc lx) (OSuc (Suc lx) y)} {.(ℵ lx)} (s< x<y) (ℵ< (¬ℵs nℵ)) = ℵ< lemma where + lemma : ¬ℵ x + lemma = {!!} +orddtrans ℵΦ< ℵs< = {!!} +orddtrans (ℵ< ¬ℵΦ) ℵs< = {!!} +orddtrans (ℵ< (¬ℵs nℵ)) ℵs< = {!!} +orddtrans ℵs< (s< ℵs<) = {!!} +orddtrans ℵs< (ℵ< ()) max : (x y : Nat) → Nat max Zero Zero = Zero
--- a/zf.agda Tue May 21 18:17:24 2019 +0900 +++ b/zf.agda Wed May 22 11:52:49 2019 +0900 @@ -33,15 +33,6 @@ infixr 140 _∨_ infixr 150 _⇔_ -record Func {n m : Level } (ZFSet : Set n) (_≈_ : Rel ZFSet m) : Set (n ⊔ suc m) where - field - rel : Rel ZFSet m - dom : ( y : ZFSet ) → ∀ { x : ZFSet } → rel x y - fun-eq : { x y z : ZFSet } → ( rel x y ∧ rel x z ) → y ≈ z - -open Func - - record IsZF {n m : Level } (ZFSet : Set n) (_∋_ : ( A x : ZFSet ) → Set m)