Mercurial > hg > Members > kono > Proof > ZF-in-agda
changeset 109:dab56d357fa3
remove o<→c< and add otrans in OD
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 18 Jun 2019 23:40:17 +0900 |
| parents | c8b79d303867 |
| children | 1daa1d24348c |
| files | ordinal-definable.agda |
| diffstat | 1 files changed, 85 insertions(+), 61 deletions(-) [+] |
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line diff
--- a/ordinal-definable.agda Wed Jun 12 10:45:00 2019 +0900 +++ b/ordinal-definable.agda Tue Jun 18 23:40:17 2019 +0900 @@ -17,6 +17,7 @@ record OD {n : Level} : Set (suc n) where field def : (x : Ordinal {n} ) → Set n + otrans : {x y : Ordinal {n} } → def x → y o< x → def y open OD open import Data.Unit @@ -42,30 +43,42 @@ 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) } +-- Ordinal in OD ( and ZFSet ) +Ord : { n : Level } → ( a : Ordinal {n} ) → OD {n} +Ord {n} a = record { def = λ y → y o< a ; otrans = lemma } where + lemma : {x y : Ordinal} → x o< a → y o< x → y o< a + lemma {x} {y} x<a y<x = ordtrans {n} {y} {x} {a} y<x x<a + +-- od∅ : {n : Level} → OD {n} +-- od∅ {n} = record { def = λ _ → Lift n ⊥ } od∅ : {n : Level} → OD {n} -od∅ {n} = record { def = λ _ → Lift n ⊥ } +od∅ {n} = Ord 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} + c<→o< : {n : Level} {x y : OD {n} } → def y ( od→ord x ) → od→ord x o< od→ord y + -- next assumption causes ∀ x ∋ ∅ . It menas only an ordinal becomes a set + -- o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → def (ord→od y) 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 -- 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 + -- a property of supermum required in Power Set Axiom 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-lb : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → ( ∀ {x : Ordinal {n}} → ψx o< z ) → z o< osuc ( sup-o ψ ) _∋_ : { n : Level } → ( a x : OD {n} ) → Set n _∋_ {n} a x = def a ( od→ord x ) _c<_ : { n : Level } → ( x a : OD {n} ) → Set n -x c< a = od→ord x o< od→ord a +x c< a = a ∋ x postulate - o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → ord→od x c< ord→od y + o<→c< : {n : Level} {x y : Ordinal {n} } → x o< y → Ord y ∋ Ord x _c≤_ : {n : Level} → OD {n} → OD {n} → Set (suc n) a c≤ b = (a ≡ b) ∨ ( b ∋ a ) @@ -73,17 +86,14 @@ 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 --- sup-min : {n : Level } → ( ψ : Ordinal {n} → Ordinal {n}) → {z : Ordinal {n}} → ψ z o< z → sup-o ψ o< osuc z - sup-od : {n : Level } → ( OD {n} → OD {n}) → OD {n} -sup-od ψ = ord→od ( sup-o ( λ x → od→ord (ψ (ord→od x ))) ) +sup-od ψ = Ord ( 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} {_} {_} {sup-od ψ} {od→ord ( ψ x )} - {!!} refl (cong ( λ k → od→ord (ψ k) ) oiso) - -∅1 : {n : Level} → ( x : OD {n} ) → ¬ ( x c< od∅ {n} ) -∅1 {n} x = {!!} +sup-c< {n} ψ {x} = def-subst {n} {_} {_} {Ord ( 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) ) ∅3 : {n : Level} → { x : Ordinal {n}} → ( ∀(y : Ordinal {n}) → ¬ (y o< x ) ) → x ≡ o∅ {n} ∅3 {n} {x} = TransFinite {n} c2 c3 x where @@ -103,13 +113,18 @@ c3 lx (OSuc .lx x₁) d not | t | () transitive : {n : Level } { z y x : OD {suc n} } → z ∋ y → y ∋ x → z ∋ x -transitive {n} {z} {y} {x} z∋y x∋y with ordtrans {!!} {!!} +transitive {n} {z} {y} {x} z∋y x∋y with ordtrans ( c<→o< {suc n} {x} {y} x∋y ) ( c<→o< {suc n} {y} {z} z∋y ) ... | t = lemma0 (lemma t) where lemma : ( od→ord x ) o< ( od→ord z ) → def ( ord→od ( od→ord z )) ( od→ord x) - lemma xo<z = {!!} + lemma xo<z = {!!} -- o<→c< xo<z lemma0 : def ( ord→od ( od→ord z )) ( od→ord x) → def z (od→ord x) lemma0 dz = def-subst {suc n} { ord→od ( od→ord z )} { od→ord x} dz (oiso) refl +record Minimumo {n : Level } (x : Ordinal {n}) : Set (suc n) where + field + mino : Ordinal {n} + min<x : mino o< x + ∅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< @@ -150,11 +165,11 @@ o<→o> : {n : Level} → { x y : OD {n} } → (x == y) → (od→ord x ) o< ( od→ord y) → ⊥ o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case1 lt) with yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl ) -... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!} +... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) ... | () o<→o> {n} {x} {y} record { eq→ = xy ; eq← = yx } (case2 lt) with yx (def-subst {n} {ord→od (od→ord y)} {od→ord x} {!!} oiso refl ) -... | oyx with o<¬≡ (od→ord x) (od→ord x) refl {!!} +... | oyx with o<¬≡ (od→ord x) (od→ord x) refl (c<→o< oyx ) ... | () ==→o≡ : {n : Level} → { x y : Ordinal {suc n} } → ord→od x == ord→od y → x ≡ y @@ -163,12 +178,12 @@ ==→o≡ {n} {x} {y} eq | tri≈ ¬a b ¬c = b ==→o≡ {n} {x} {y} eq | tri> ¬a ¬b c = ⊥-elim ( o<→o> (eq-sym eq) (o<-subst c (sym ord-iso) (sym ord-iso ))) -≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (record { def = λ z → z o< x } ) +≡-def : {n : Level} → { x : Ordinal {suc n} } → x ≡ od→ord (Ord x) ≡-def {n} {x} = ==→o≡ {n} (subst (λ k → ord→od x == k ) (sym oiso) lemma ) where lemma : ord→od x == record { def = λ z → z o< x } eq→ lemma {w} z = subst₂ (λ k j → k o< j ) diso refl (subst (λ k → (od→ord ( ord→od w)) o< k ) diso t ) where t : (od→ord ( ord→od w)) o< (od→ord (ord→od x)) - t = {!!} + t = c<→o< {suc n} {ord→od w} {ord→od x} (def-subst {suc n} {_} {_} {ord→od x} {_} z refl (sym diso)) eq← lemma {w} z = def-subst {suc n} {_} {_} {ord→od x} {w} {!!} refl refl od≡-def : {n : Level} → { x : Ordinal {suc n} } → ord→od x ≡ record { def = λ z → z o< x } @@ -186,12 +201,12 @@ ∋→o< : {n : Level} → { a x : OD {suc n} } → a ∋ x → od→ord x o< od→ord a ∋→o< {n} {a} {x} lt = t where t : (od→ord x) o< (od→ord a) - t = {!!} + t = c<→o< {suc n} {x} {a} lt o<∋→ : {n : Level} → { a x : OD {suc n} } → od→ord x o< od→ord a → a ∋ x o<∋→ {n} {a} {x} lt = subst₂ (λ k j → def k j ) oiso refl t where t : def (ord→od (od→ord a)) (od→ord x) - t = {!!} + t = {!!} -- o<→c< {suc n} {od→ord x} {od→ord a} lt o∅≡od∅ : {n : Level} → ord→od (o∅ {suc n}) ≡ od∅ {suc n} o∅≡od∅ {n} with trio< {n} (o∅ {suc n}) (od→ord (od∅ {suc n} )) @@ -202,14 +217,24 @@ o∅≡od∅ {n} | tri≈ ¬a b ¬c = trans (cong (λ k → ord→od k ) b ) oiso o∅≡od∅ {n} | tri> ¬a ¬b c = ⊥-elim (¬x<0 c) +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) + + o<→¬== : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (x == y ) o<→¬== {n} {x} {y} lt eq = o<→o> eq lt o<→¬c> : {n : Level} → { x y : OD {n} } → (od→ord x ) o< ( od→ord y) → ¬ (y c< x ) -o<→¬c> {n} {x} {y} olt clt = o<> olt {!!} where +o<→¬c> {n} {x} {y} olt clt = o<> olt (c<→o< clt ) where o≡→¬c< : {n : Level} → { x y : OD {n} } → (od→ord x ) ≡ ( od→ord y) → ¬ x c< y -o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) lt +o≡→¬c< {n} {x} {y} oeq lt = o<¬≡ (od→ord x) (od→ord y) (orefl oeq ) (c<→o< lt) tri-c< : {n : Level} → Trichotomous _==_ (_c<_ {suc n}) tri-c< {n} x y with trio< {n} (od→ord x) (od→ord y) @@ -220,24 +245,29 @@ c<> : {n : Level } { x y : OD {suc n}} → x c< y → y c< x → ⊥ c<> {n} {x} {y} x<y y<x with tri-c< x y c<> {n} {x} {y} x<y y<x | tri< a ¬b ¬c = ¬c y<x -c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b x<y +c<> {n} {x} {y} x<y y<x | tri≈ ¬a b ¬c = o<→o> b ( c<→o< x<y ) c<> {n} {x} {y} x<y y<x | tri> ¬a ¬b c = ¬a x<y +∅0 : {n : Level} → record { def = λ x → Lift n ⊥ ; otrans = λ () } == od∅ {n} +eq→ ∅0 {w} (lift ()) +eq← ∅0 {w} (case1 ()) +eq← ∅0 {w} (case2 ()) + ∅< : {n : Level} → { x y : OD {n} } → def x (od→ord y ) → ¬ ( x == od∅ {n} ) -∅< {n} {x} {y} d eq with eq→ eq d +∅< {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} {x} {x} {!!} {!!} +∅6 {n} {x} x∋x = c<> {n} {x} {x} x∋x x∋x 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-∋ : {n : Level} → ( x y : OD {suc n} ) → Dec ( x ∋ y ) is-∋ {n} x y with tri-c< x y -is-∋ {n} x y | tri< a ¬b ¬c = no {!!} -is-∋ {n} x y | tri≈ ¬a b ¬c = no {!!} -is-∋ {n} x y | tri> ¬a ¬b c = yes {!!} +is-∋ {n} x y | tri< a ¬b ¬c = no ¬c +is-∋ {n} x y | tri≈ ¬a b ¬c = no ¬c +is-∋ {n} x y | tri> ¬a ¬b c = yes c is-o∅ : {n : Level} → ( x : Ordinal {suc n} ) → Dec ( x ≡ o∅ {suc n} ) is-o∅ {n} record { lv = Zero ; ord = (Φ .0) } = yes refl @@ -246,16 +276,6 @@ open _∧_ -¬∅=→∅∈ : {n : Level} → { x : OD {suc n} } → ¬ ( x == od∅ {suc n} ) → x ∋ od∅ {suc n} -¬∅=→∅∈ {n} {x} ne = def-subst (lemma (od→ord x) (subst (λ k → ¬ (k == od∅ {suc n} )) (sym oiso) ne )) oiso refl where - lemma : (ox : Ordinal {suc n}) → ¬ (ord→od ox == od∅ {suc n} ) → ord→od ox ∋ od∅ {suc n} - lemma ox ne with is-o∅ ox - lemma ox ne | yes refl with ne ( ord→== lemma1 ) where - lemma1 : od→ord (ord→od o∅) ≡ od→ord od∅ - lemma1 = cong ( λ k → od→ord k ) o∅≡od∅ - lemma o∅ ne | yes refl | () - lemma ox ne | no ¬p = subst ( λ k → def (ord→od ox) (od→ord k) ) o∅≡od∅ {!!} - -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) -- postulate f-extensionality : { n : Level} → Relation.Binary.PropositionalEquality.Extensionality (suc n) (suc (suc n)) @@ -265,7 +285,7 @@ -- 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 } +ZFSubset A x = record { def = λ y → def A y ∧ def x y ; otrans = {!!} } 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 )) ) ) @@ -275,7 +295,7 @@ 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 }) )) } + record { def = λ y → osuc y o< (od→ord (L {n} (record { lv = lx ; ord = Φ lx }) )) ; otrans = {!!} } OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} OD→ZF {n} = record { @@ -294,11 +314,11 @@ 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 )) } + Select X ψ = record { def = λ x → ( def X x ∧ ψ ( Ord x )) ; otrans = {!!} } _,_ : OD {suc n} → OD {suc n} → OD {suc n} - x , y = record { def = λ z → z o< (omax (od→ord x) (od→ord y)) } + x , y = Ord (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) } + Union U = record { def = λ y → osuc y o< (od→ord U) ; otrans = {!!} } -- power : ∀ X ∃ A ∀ t ( t ∈ A ↔ ( ∀ {x} → t ∋ x → X ∋ x ) Power : OD {suc n} → OD {suc n} Power A = Def A @@ -313,6 +333,7 @@ -- A ∪ B = Select (A , B) ( λ x → (A ∋ x) ∨ ( B ∋ x ) ) ｛_｝ : ZFSet → ZFSet ｛ x ｝ = ( x , x ) + infixr 200 _∈_ -- infixr 230 _∩_ _∪_ infixr 220 _⊆_ @@ -335,16 +356,18 @@ ; replacement = replacement } where open _∧_ + open Minimumo 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 () + empty x (case1 ()) + empty x (case2 ()) --- --- ZFSubset A x = record { def = λ y → def A y ∧ def x y } subset of A --- 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 + -- if Power A ∋ t, from a propertiy of minimum 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 -- @@ -353,9 +376,11 @@ minsup : OD minsup = ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) lemma-t : csuc minsup ∋ t - lemma-t = {!!} + lemma-t = {!!} -- o<→c< (o<-subst (sup-lb (o<-subst (c<→o< P∋t) refl diso )) refl refl ) lemma-s : ZFSubset A ( ord→od ( sup-x (λ x → od→ord ( ZFSubset A (ord→od x))))) ∋ x - lemma-s = {!!} + lemma-s with osuc-≡< ( o<-subst (c<→o< lemma-t ) refl diso ) + lemma-s | case1 eq = def-subst ( ==-def-r (o≡→== eq) (subst (λ k → def k (od→ord x)) (sym oiso) t∋x ) ) oiso refl + lemma-s | case2 lt = transitive {n} {minsup} {t} {x} (def-subst {!!} oiso refl ) 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 @@ -377,49 +402,49 @@ lemma {oz} {ooz} lt = def-subst {suc n} {ord→od ooz} {!!} refl refl union→ : (X z u : OD) → (X ∋ u) ∧ (u ∋ z) → Union X ∋ z union→ X y u xx with trio< ( od→ord u ) ( osuc ( od→ord y )) - union→ X y u xx | tri< a ¬b ¬c with osuc-< a {!!} + union→ X y u xx | tri< a ¬b ¬c with osuc-< a (c<→o< (proj2 xx)) union→ X y u xx | tri< a ¬b ¬c | () - union→ X y u xx | tri≈ ¬a b ¬c = lemma b {!!} where + union→ X y u xx | tri≈ ¬a b ¬c = lemma b (c<→o< (proj1 xx )) where lemma : {oX ou ooy : Ordinal {suc n}} → ou ≡ ooy → ou o< oX → ooy o< oX lemma refl lt = lt - union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c {!!} + union→ X y u xx | tri> ¬a ¬b c = ordtrans {suc n} {osuc ( od→ord y )} {od→ord u} {od→ord X} c ( c<→o< (proj1 xx )) union← : (X z : OD) (X∋z : Union X ∋ z) → (X ∋ csuc z) ∧ (csuc z ∋ z ) union← X z X∋z = record { proj1 = def-subst {suc n} {_} {_} {X} {od→ord (csuc z )} {!!} oiso (sym diso) ; proj2 = union-lemma-u X∋z } 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 } + 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) {!!} } } 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 minimul : (x : OD {suc n} ) → ¬ (x == od∅ )→ OD {suc n} - minimul x not = od∅ + minimul x not = {!!} 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 ) = ¬∅=→∅∈ not - proj2 (regularity x not ) = record { eq→ = reg ; eq← = λ () } where + proj1 (regularity 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 proj1 t - ... | x∈∅ = x∈∅ + ... | x∈∅ = {!!} 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 xx-union : {x : OD {suc n}} → (x , x) ≡ record { def = λ z → z o< osuc (od→ord x) } - xx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) (omxx (od→ord x)) + xx-union {x} = cong ( λ k → Ord k ) (omxx (od→ord x)) xxx-union : {x : OD {suc n}} → (x , (x , x)) ≡ record { def = λ z → z o< osuc (osuc (od→ord x))} - xxx-union {x} = cong ( λ k → record { def = λ z → z o< k } ) lemma where + xxx-union {x} = cong ( λ k → Ord k ) lemma where lemma1 : {x : OD {suc n}} → od→ord x o< od→ord (x , x) - lemma1 {x} = {!!} + lemma1 {x} = c<→o< ( proj1 (pair x x ) ) lemma2 : {x : OD {suc n}} → od→ord (x , x) ≡ osuc (od→ord x) lemma2 = trans ( cong ( λ k → od→ord k ) xx-union ) (sym ≡-def) lemma : {x : OD {suc n}} → omax (od→ord x) (od→ord (x , x)) ≡ osuc (osuc (od→ord x)) lemma {x} = trans ( sym ( omax< (od→ord x) (od→ord (x , x)) lemma1 ) ) ( cong ( λ k → osuc k ) lemma2 ) uxxx-union : {x : OD {suc n}} → Union (x , (x , x)) ≡ record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } - uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k } ) lemma where + uxxx-union {x} = cong ( λ k → record { def = λ z → osuc z o< k ; otrans = {!!} } ) lemma where lemma : od→ord (x , (x , x)) ≡ osuc (osuc (od→ord x)) lemma = trans ( cong ( λ k → od→ord k ) xxx-union ) (sym ≡-def ) uxxx-2 : {x : OD {suc n}} → record { def = λ z → osuc z o< osuc (osuc (od→ord x)) } == record { def = λ z → z o< osuc (od→ord x) } @@ -461,4 +486,3 @@ -- 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 = ¬∅=→∅∈ {!!} -