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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Tue, 18 Jun 2019 22:01:15 +0900 |
parents | 745bee73b444 |
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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 -- Ordinal Definable Set record OD {n : Level} : Set (suc n) where field def : (x : Ordinal {n} ) → Set n open OD open import Data.Unit open Ordinal record _==_ {n : Level} ( a b : OD {n} ) : Set n where field eq→ : ∀ { x : Ordinal {n} } → def a x → def b x eq← : ∀ { x : Ordinal {n} } → def b x → def a x id : {n : Level} {A : Set n} → A → A id x = x eq-refl : {n : Level} { x : OD {n} } → x == x eq-refl {n} {x} = record { eq→ = id ; eq← = id } open _==_ eq-sym : {n : Level} { x y : OD {n} } → x == y → y == x eq-sym eq = record { eq→ = eq← eq ; eq← = eq→ eq } eq-trans : {n : Level} { x y z : OD {n} } → x == y → y == z → x == z eq-trans x=y y=z = record { eq→ = λ t → eq→ y=z ( eq→ x=y t) ; eq← = λ t → eq← x=y ( eq← y=z t) } od∅ : {n : Level} → OD {n} od∅ {n} = record { def = λ x → x o< o∅ } postulate -- OD can be iso to a subset of Ordinal ( by means of Godel Set ) od→ord : {n : Level} → OD {n} → Ordinal {n} ord→od : {n : Level} → Ordinal {n} → OD {n} oiso : {n : Level} {x : OD {n}} → ord→od ( od→ord x ) ≡ x diso : {n : Level} {x : Ordinal {n}} → od→ord ( ord→od x ) ≡ x -- supermum as Replacement Axiom sup-o : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} sup-o< : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → ∀ {x : Ordinal {n}} → ψ x o< sup-o ψ -- a contra-position of minimality of supermum sup-x : {n : Level } → ( Ordinal {n} → Ordinal {n}) → Ordinal {n} sup-lb : {n : Level } → { ψ : Ordinal {n} → Ordinal {n}} → {z : Ordinal {n}} → z o< sup-o ψ → z o< osuc (ψ (sup-x ψ)) -- sup-min : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → {z : Ordinal {n}} → ψ z o< z → sup-o ψ o< osuc z minimul : {n : Level } → (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} x∋minimul : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → def x ( od→ord ( minimul x ne ) ) minimul-1 : {n : Level } → (x : OD {suc n} ) → ( ne : ¬ (x == od∅ ) ) → (y : OD {suc n}) → ¬ ( def (minimul x ne) (od→ord y)) ∧ (def x (od→ord y) ) _∋_ : { n : Level } → ( a x : OD {n} ) → Set n _∋_ {n} a x = def a ( od→ord x ) Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} Ord {n} a = record { def = λ y → y o< a } _c<_ : { n : Level } → ( x a : Ordinal {n} ) → Set n _c<_ {n} x a = Ord {n} a ∋ Ord x postulate c<→o< : { n : Level } → { x a : Ordinal {n} } → Ord a ∋ Ord x → x o< a o<→c< : { n : Level } → { x a : Ordinal {n} } → x o< a → Ord a ∋ Ord x ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → Ord x == Ord y → x ≡ y ==→o≡ {n} {x} {y} eq with trio< {n} x y ==→o≡ {n} {x} {y} eq | tri< a ¬b ¬c with eq← eq {x} a ... | t = ⊥-elim ( o<¬≡ x x refl t ) ==→o≡ {n} {x} {y} eq | tri≈ ¬a refl ¬c = refl ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c with eq→ eq {y} c ... | t = ⊥-elim ( o<¬≡ y y refl t ) ∅∨ : { n : Level } → { x y : Ordinal {suc n} } → ( Ord {suc n} x == Ord y ) ∨ ( ¬ ( Ord x == Ord y ) ) ∅∨ {n} {x} {y} with trio< x y ∅∨ {n} {x} {y} | tri< a ¬b ¬c = case2 ( λ eq → ¬b ( ==→o≡ eq ) ) ∅∨ {n} {x} {y} | tri≈ ¬a refl ¬c = case1 ( record { eq→ = id ; eq← = id } ) ∅∨ {n} {x} {y} | tri> ¬a ¬b c = case2 ( λ eq → ¬b ( ==→o≡ eq ) ) -- ¬x∋x' : { n : Level } → { x : Ordinal {n} } → ¬ ( record { def = λ y → y o< x } ∋ record { def = λ y → y o< x } ) -- ¬x∋x' {n} {record { lv = Zero ; ord = ord }} (case1 ()) -- ¬x∋x' {n} {record { lv = Suc lx ; ord = Φ .(Suc lx) }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = lx ; ord = Φ lx }} (case1 {!!}) -- ¬x∋x' {n} {record { lv = Suc lx ; ord = OSuc (Suc lx) ox }} (case1 (s≤s x)) = ¬x∋x' {n} {record { lv = Suc lx ; ord = ox}} (case1 {!!}) -- ¬x∋x' {n} {record { lv = lv ; ord = Φ (lv) }} (case2 ()) -- ¬x∋x' {n} {record { lv = lv ; ord = OSuc (lv) ox }} (case2 x) = -- ¬x∋x' {n} {record { lv = lv ; ord = ox }} (case2 {!!}) -- ¬x∋x : { n : Level } → { x : OD {n} } → ¬ x ∋ x -- ¬x∋x = {!!} oc-lemma : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → def (record { def = λ y → y o< oa }) oa → ⊥ oc-lemma {n} {x} {oa} lt = o<¬≡ oa oa refl lt -- oc-lemma1 : { n : Level } → { x : OD {n} } → { oa : Ordinal {n} } → od→ord (record { def = λ y → y o< oa }) o< oa → ⊥ -- oc-lemma1 {n} {x} {oa} lt = ¬x∋x' {n} lt -- lt : def (record { def = λ y → y o< oa }) (record { def = λ y → y o< oa }) -- this one cannot be proved because if we have this OD and Ordinal has one to one corespondence -- oc-lemma2 : { n : Level } → { x a : OD {n} } → { oa : Ordinal {n} } → oa o< od→ord (record { def = λ y → y o< oa }) → ⊥ -- this is not allowed in our case. ( avoid one-to-one of Ord and OD ) -- Ord=ord→od : {n : Level} → { x : Ordinal {n} } → Ord x ≡ ord→od x _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) a c≤ b = (a ≡ b) ∨ ( b ∋ a ) def-subst : {n : Level } {Z : OD {n}} {X : Ordinal {n} }{z : OD {n}} {x : Ordinal {n} }→ def Z X → Z ≡ z → X ≡ x → def z x def-subst df refl refl = df o<-def : {n : Level } {x y : Ordinal {n} } → x o< y → def (record { def = λ x → x o< y }) x o<-def x<y = x<y def-o< : {n : Level } {x y : Ordinal {n} } → def (record { def = λ x → x o< y }) x → x o< y def-o< x<y = x<y sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} sup-od ψ = record { def = λ y → y o< ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) } sup-c< : {n : Level } → ( ψ : OD {n} → OD {n}) → ∀ {x : OD {n}} → def ( sup-od ψ ) (od→ord ( ψ x )) sup-c< {n} ψ {x} = def-subst {n} {_} {_} {record { def = λ y → y o< ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) }} {od→ord ( ψ x )} lemma refl (cong ( λ k → od→ord (ψ k) ) oiso) where lemma : od→ord (ψ (ord→od (od→ord x))) o< sup-o (λ x → od→ord (ψ (ord→od x))) lemma = subst₂ (λ j k → j o< k ) refl diso (o<-subst sup-o< refl (sym diso) ) ∅0 : {n : Level} → record { def = λ x → Lift n ⊥ } == od∅ {n} eq→ ∅0 {w} (lift ()) eq← ∅0 {w} (case1 ()) eq← ∅0 {w} (case2 ()) ∅1 : {n : Level} → ( x : Ordinal {n} ) → ¬ ( x c< o∅ {n} ) ∅1 {n} record { lv = Zero ; ord = (Φ .0) } (case1 ()) ∅1 {n} record { lv = Zero ; ord = (Φ .0) } (case2 ()) ∅1 {n} record { lv = Zero ; ord = (OSuc .0 ox) } (case1 ()) ∅1 {n} record { lv = Zero ; ord = (OSuc .0 ox) } (case2 ()) ∅1 {n} record { lv = (Suc lx) ; ord = (Φ .(Suc lx)) } (case1 ()) ∅1 {n} record { lv = (Suc lx) ; ord = (Φ .(Suc lx)) } (case2 ()) ∅1 {n} record { lv = (Suc lx) ; ord = (OSuc .(Suc lx) ox) } (case1 ()) ∅1 {n} record { lv = (Suc lx) ; ord = (OSuc .(Suc lx) ox) } (case2 ()) ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} ∅3 {n} {x} = TransFinite {n} c2 c3 x where c0 : Nat → Ordinal {n} → Set n c0 lx x = (∀(y : Ordinal {n}) → ¬ (y o< x)) → x ≡ o∅ {n} c2 : (lx : Nat) → c0 lx (record { lv = lx ; ord = Φ lx } ) c2 Zero not = refl c2 (Suc lx) not with not ( record { lv = lx ; ord = Φ lx } ) ... | t with t (case1 ≤-refl ) c2 (Suc lx) not | t | () c3 : (lx : Nat) (x₁ : OrdinalD lx) → c0 lx (record { lv = lx ; ord = x₁ }) → c0 lx (record { lv = lx ; ord = OSuc lx x₁ }) c3 lx (Φ .lx) d not with not ( record { lv = lx ; ord = Φ lx } ) ... | 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< s<refl ) ) c3 lx (OSuc .lx x₁) d not | t | () transitive-Ord : {n : Level } { z y x : Ordinal {suc n} } → Ord z ∋ Ord y → Ord y ∋ Ord x → Ord z ∋ Ord x transitive-Ord {n} {z} {y} {x} z∋y x∋y = o<→c< ( ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) ) ∅5 : {n : Level} → { x : Ordinal {n} } → ¬ ( x ≡ o∅ {n} ) → o∅ {n} o< x ∅5 {n} {record { lv = Zero ; ord = (Φ .0) }} not = ⊥-elim (not refl) ∅5 {n} {record { lv = Zero ; ord = (OSuc .0 ord) }} not = case2 Φ< ∅5 {n} {record { lv = (Suc lv) ; ord = ord }} not = case1 (s≤s z≤n) ord-iso : {n : Level} {y : Ordinal {n} } → record { lv = lv (od→ord (ord→od y)) ; ord = ord (od→ord (ord→od y)) } ≡ record { lv = lv y ; ord = ord y } ord-iso = cong ( λ k → record { lv = lv k ; ord = ord k } ) diso -- avoiding lv != Zero error orefl : {n : Level} → { x : OD {n} } → { y : Ordinal {n} } → od→ord x ≡ y → od→ord x ≡ y orefl refl = refl ==-iso : {n : Level} → { x y : OD {n} } → ord→od (od→ord x) == ord→od (od→ord y) → x == y ==-iso {n} {x} {y} eq = record { eq→ = λ d → lemma ( eq→ eq (def-subst d (sym oiso) refl )) ; eq← = λ d → lemma ( eq← eq (def-subst d (sym oiso) refl )) } where lemma : {x : OD {n} } {z : Ordinal {n}} → def (ord→od (od→ord x)) z → def x z lemma {x} {z} d = def-subst d oiso refl =-iso : {n : Level } {x y : OD {suc n} } → (x == y) ≡ (ord→od (od→ord x) == y) =-iso {_} {_} {y} = cong ( λ k → k == y ) (sym oiso) ord→== : {n : Level} → { x y : OD {n} } → od→ord x ≡ od→ord y → x == y ord→== {n} {x} {y} eq = ==-iso (lemma (od→ord x) (od→ord y) (orefl eq)) where lemma : ( ox oy : Ordinal {n} ) → ox ≡ oy → (ord→od ox) == (ord→od oy) lemma ox ox refl = eq-refl o≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → ord→od x == ord→od y o≡→== {n} {x} {.x} refl = eq-refl O≡→== : {n : Level} → { x y : Ordinal {n} } → x ≡ y → Ord x == Ord y O≡→== {n} {x} {.x} refl = eq-refl >→¬< : {x y : Nat } → (x < y ) → ¬ ( y < x ) >→¬< (s≤s x<y) (s≤s y<x) = >→¬< x<y y<x c≤-refl : {n : Level} → ( x : OD {n} ) → x c≤ x c≤-refl x = case1 refl o<→o> : {n : Level} → { x y : Ordinal {suc n} } → (Ord x == Ord y) → x o< y → ⊥ o<→o> {n} {x} {y} eq lt with ==→o≡ {n} eq ... | refl = o<¬≡ _ _ refl lt ==-def-r : {n : Level } {x y z : Ordinal {suc n} } → (Ord x == Ord y) → def (Ord x) z → def (Ord y) z ==-def-r {n} {x} {y} {z} eq z>x = eq→ eq z>x o<→¬== : {n : Level} → { x y : Ordinal {suc n} } → x o< y → ¬ (Ord x == Ord y ) o<→¬== {n} {x} {y} lt eq = o<→o> {n} eq lt ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) ∅< {n} {x} {y} d eq with eq→ (eq-trans eq (eq-sym ∅0) ) d ∅< {n} {x} {y} d eq | lift () -- ∅6 : {n : Level} → { x : OD {suc n} } → ¬ ( x ∋ x ) -- no Russel paradox -- ∅6 {n} {x} x∋x = c<> {n} {{!!}} {{!!}} {!!} {!!} def-iso : {n : Level} {A B : OD {n}} {x y : Ordinal {n}} → x ≡ y → (def A y → def B y) → def A x → def B x def-iso refl t = t is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl is-o∅ {n} record { lv = Zero ; ord = (OSuc .0 ord₁) } = no ( λ () ) is-o∅ {n} record { lv = (Suc lv₁) ; ord = ord } = no (λ()) open _∧_ ord-od∅ : {n : Level} → o∅ {suc n} ≡ od→ord (Ord (o∅ {suc n})) ord-od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (Ord (o∅ {suc n}))) ord-od∅ {n} | tri< a ¬b ¬c = ⊥-elim (lemma a) where lemma : o∅ {suc n } o< (od→ord (Ord (o∅ {suc n}))) → ⊥ lemma lt with o<→c< lt lemma lt | t = o<¬≡ _ _ refl t ord-od∅ {n} | tri≈ ¬a b ¬c = b ord-od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) ¬∅=→∅∈ : {n : Level} → { x : Ordinal {suc n} } → ¬ ( Ord x == od∅ {suc n} ) → Ord x ∋ od∅ {suc n} ¬∅=→∅∈ {n} {x} ne with is-o∅ x ¬∅=→∅∈ {n} {x} ne | yes refl = ⊥-elim ( ne (eq-sym (eq-refl) )) ¬∅=→∅∈ {n} {x} ne | no ¬p = o<-subst (∅5 ¬p) ord-od∅ refl -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) csuc : {n : Level} → OD {suc n} → OD {suc n} csuc x = ord→od ( osuc ( od→ord x )) -- Power Set of X ( or constructible by λ y → def X (od→ord y ) ZFSubset : {n : Level} → (A x : OD {suc n} ) → OD {suc n} ZFSubset A x = record { def = λ y → def A y ∧ def x y } Def : {n : Level} → (A : OD {suc n}) → OD {suc n} Def {n} A = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) -- Constructible Set on α L : {n : Level} → (α : Ordinal {suc n}) → OD {suc n} L {n} record { lv = Zero ; ord = (Φ .0) } = od∅ L {n} record { lv = lx ; ord = (OSuc lv ox) } = Def ( L {n} ( record { lv = lx ; ord = ox } ) ) L {n} record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α ) record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) } Ordsuc : {n : Level} → Ordinal {suc n} → OD {suc n} Ordsuc x = record { def = λ y → y o< osuc x } OrdSubset : {n : Level} →(A x : Ordinal {suc n} ) → OD {suc n} OrdSubset A x = record { def = λ y → (y o< A) ∧ (y o< x ) } OrdDef : {n : Level} → (A : Ordinal {suc n}) → OD {suc n} OrdDef A = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( OrdSubset A x))) } omega : {n : Level} → Ordinal {n} omega = record { lv = Suc Zero ; ord = Φ 1 } OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} OD→ZF {n} = record { ZFSet = OD {suc n} ; _∋_ = _∋_ ; _≈_ = _==_ ; ∅ = od∅ ; _,_ = _,_ ; Union = Union ; Power = Power ; Select = Select ; Replace = Replace ; infinite = Ord omega ; isZF = isZF } where Replace : OD {suc n} → (OD {suc n} → OD {suc n} ) → OD {suc n} Replace X ψ = sup-od ψ Select : OD {suc n} → (OD {suc n} → Set (suc n) ) → OD {suc n} Select X ψ = record { def = λ x → ( def X x ∧ ψ ( ord→od x )) } _,_ : OD {suc n} → OD {suc n} → OD {suc n} x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } Union : OD {suc n} → OD {suc n} Union U = record { def = λ y → osuc y o< (od→ord U) } -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) Power : OD {suc n} → OD {suc n} Power A = Def A ZFSet = OD {suc n} _∈_ : ( A B : ZFSet ) → Set (suc n) A ∈ B = B ∋ A _⊆_ : ( A B : ZFSet ) → ∀{ x : ZFSet } → Set (suc n) _⊆_ A B {x} = A ∋ x → B ∋ x _∩_ : ( A B : ZFSet ) → ZFSet A ∩ B = Select (A , B) ( λ x → ( A ∋ x ) ∧ (B ∋ x) ) -- _∪_ : ( A B : ZFSet ) → ZFSet -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) {_} : ZFSet → ZFSet { x } = ( x , x ) infixr 200 _∈_ -- infixr 230 _∩_ _∪_ infixr 220 _⊆_ isZF : IsZF (OD {suc n}) _∋_ _==_ od∅ _,_ Union Power Select Replace (Ord omega) isZF = record { isEquivalence = record { refl = eq-refl ; sym = eq-sym; trans = eq-trans } ; pair = pair ; union-u = λ _ z _ → csuc z ; union→ = union→ ; union← = union← ; empty = empty ; power→ = power→ ; power← = power← ; extensionality = extensionality ; minimul = minimul ; regularity = regularity ; infinity∅ = infinity∅ ; infinity = λ _ → infinity ; selection = λ {ψ} {X} {y} → selection {ψ} {X} {y} ; replacement = replacement } where open _∧_ pair : (A B : OD {suc n} ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B) proj1 (pair A B ) = omax-x {n} (od→ord A) (od→ord B) proj2 (pair A B ) = omax-y {n} (od→ord A) (od→ord B) empty : (x : OD {suc n} ) → ¬ (od∅ ∋ x) empty x (case1 ()) empty x (case2 ()) power→Ord : (A t : Ordinal) → OrdDef {suc n} A ∋ (Ord t) → {x : OD} → Ord t ∋ x → Ord A ∋ x power→Ord A t P∋t {x} t∋x = proj1 lemma-s where minsup : OD minsup = OrdSubset A ( sup-x (λ x → od→ord ( OrdSubset A x))) lemma-t : Ord (sup-x (λ x → od→ord ( OrdSubset A x))) ∋ Ord t lemma-t = {!!} -- sup-lb P∋t = (od→ord (OrdSubset A (ord→od (sup-x (λ x₁ → od→ord (OrdSubset A (ord→od x₁))))))) lemma-s : OrdSubset A ( sup-x (λ x → od→ord ( OrdSubset A x))) ∋ x lemma-s with osuc-≡< {suc n} ( o<-subst (c<→o< {!!} ) refl diso ) lemma-s | case1 eq = {!!} lemma-s | case2 lt = {!!} power←Ord : (A t : Ordinal) → ({x : OD} → (Ord t ∋ x → Ord A ∋ x)) → OrdDef {suc n} A ∋ Ord t power←Ord = {!!} --- --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A --- Def {n} A = record { def = λ y → y o< ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )))) } --- = Power X = ord→od ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) Power X is a sup of all subset of A -- -- if Power A ∋ t, from a minimulity of sup, there is osuc ZFSubset A ∋ t -- then ZFSubset A ≡ t or ZFSubset A ∋ t. In the former case ZFSubset A ∋ x implies A ∋ x -- In case of later, ZFSubset A ∋ t and t ∋ x implies ZFSubset A ∋ x by transitivity -- power→ : (A t : OD) → Power A ∋ t → {x : OD} → t ∋ x → A ∋ x power→ A t P∋t {x} t∋x = {!!} -- -- we have t ∋ x → A ∋ x means t is a subset of A, that is ZFSubset A t == t -- Power A is a sup of ZFSubset A t, so Power A ∋ t -- power← : (A t : OD) → ({x : OD} → (t ∋ x → A ∋ x)) → Power A ∋ t power← A t t→A = {!!} union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ = {!!} union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) union← = {!!} ψiso : {ψ : OD {suc n} → Set (suc n)} {x y : OD {suc n}} → ψ x → x ≡ y → ψ y ψiso {ψ} t refl = t selection : {ψ : OD → Set (suc n)} {X y : OD} → ((X ∋ y) ∧ ψ y) ⇔ (Select X ψ ∋ y) selection {ψ} {X} {y} = record { proj1 = λ cond → record { proj1 = proj1 cond ; proj2 = ψiso {ψ} (proj2 cond) (sym oiso) } ; proj2 = λ select → record { proj1 = proj1 select ; proj2 = ψiso {ψ} (proj2 select) oiso } } replacement : {ψ : OD → OD} (X x : OD) → Replace X ψ ∋ ψ x replacement {ψ} X x = sup-c< ψ {x} ∅-iso : {x : OD} → ¬ (x == od∅) → ¬ ((ord→od (od→ord x)) == od∅) ∅-iso {x} neq = subst (λ k → ¬ k) (=-iso {n} ) neq regularity : (x : OD) (not : ¬ (x == od∅)) → (x ∋ minimul x not) ∧ (Select (minimul x not) (λ x₁ → (minimul x not ∋ x₁) ∧ (x ∋ x₁)) == od∅) proj1 (regularity x not ) = x∋minimul x not proj2 (regularity x not ) = record { eq→ = reg ; eq← = {!!} } where reg : {y : Ordinal} → def (Select (minimul x not) (λ x₂ → (minimul x not ∋ x₂) ∧ (x ∋ x₂))) y → def od∅ y reg {y} t with minimul-1 x not (ord→od y) (proj2 t ) ... | t1 = {!!} extensionality : {A B : OD {suc n}} → ((z : OD) → (A ∋ z) ⇔ (B ∋ z)) → A == B eq→ (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {A} {B} (sym diso) (proj1 (eq (ord→od x))) d eq← (extensionality {A} {B} eq ) {x} d = def-iso {suc n} {B} {A} (sym diso) (proj2 (eq (ord→od x))) d infinite : OD {suc n} infinite = Ord omega infinity∅ : Ord omega ∋ od∅ {suc n} infinity∅ = {!!} infinite∋x : (x : OD) → infinite ∋ x → od→ord x o< omega infinite∋x x lt = subst (λ k → od→ord x o< k ) diso t where t : od→ord x o< od→ord (ord→od (omega)) t = {!!} infinite∋uxxx : (x : OD) → osuc (od→ord x) o< omega → infinite ∋ Union (x , (x , x )) infinite∋uxxx x lt = {!!} where t : od→ord (Union (x , (x , x))) o< od→ord (ord→od (omega)) t = subst (λ k → od→ord (Union (x , (x , x))) o< k ) (sym diso ) ( subst ( λ k → k o< omega ) {!!} lt ) infinity : (x : OD) → infinite ∋ x → infinite ∋ Union (x , (x , x )) infinity x lt = infinite∋uxxx x ( lemma (od→ord x) (infinite∋x x lt )) where lemma : (ox : Ordinal {suc n} ) → ox o< omega → osuc ox o< omega lemma record { lv = Zero ; ord = (Φ .0) } (case1 (s≤s x)) = case1 (s≤s z≤n) lemma record { lv = Zero ; ord = (OSuc .0 ord₁) } (case1 (s≤s x)) = case1 (s≤s z≤n) lemma record { lv = (Suc lv₁) ; ord = (Φ .(Suc lv₁)) } (case1 (s≤s ())) lemma record { lv = (Suc lv₁) ; ord = (OSuc .(Suc lv₁) ord₁) } (case1 (s≤s ())) lemma record { lv = 1 ; ord = (Φ 1) } (case2 c2) with d<→lv c2 lemma record { lv = (Suc Zero) ; ord = (Φ .1) } (case2 ()) | refl -- ∀ X [ ∅ ∉ X → (∃ f : X → ⋃ X ) → ∀ A ∈ X ( f ( A ) ∈ A ) ] -- this form is no good since X is a transitive set -- ∀ z [ ∀ x ( x ∈ z → ¬ ( x ≈ ∅ ) ) ∧ ∀ x ∀ y ( x , y ∈ z ∧ ¬ ( x ≈ y ) → x ∩ y ≈ ∅ ) → ∃ u ∀ x ( x ∈ z → ∃ t ( u ∩ x) ≈ { t }) ] record Choice (z : OD {suc n}) : Set (suc (suc n)) where field u : {x : OD {suc n}} ( x∈z : x ∈ z ) → OD {suc n} t : {x : OD {suc n}} ( x∈z : x ∈ z ) → (x : OD {suc n} ) → OD {suc n} choice : { x : OD {suc n} } → ( x∈z : x ∈ z ) → ( u x∈z ∩ x) == { t x∈z x } -- choice : {x : OD {suc n}} ( x ∈ z → ¬ ( x ≈ ∅ ) ) → -- axiom-of-choice : { X : OD } → ( ¬x∅ : ¬ ( X == od∅ ) ) → { A : OD } → (A∈X : A ∈ X ) → choice ¬x∅ A∈X ∈ A -- axiom-of-choice {X} nx {A} lt = ¬∅=→∅∈ {!!}