Mercurial > hg > Members > kono > Proof > ZF-in-agda

comparison ordinal.agda @ 264:fee0fab14de0

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 23 Sep 2019 10:43:48 +0900
parents 2e75710a936b
children 30e419a2be24
comparison
equal deleted inserted replaced
263:2e75710a936b 264:fee0fab14de0
248 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc 248 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc
249 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc 249 ⊆→o< {x} {y} lt | tri≈ ¬a b ¬c = subst ( λ k → k o< osuc y) (sym b) <-osuc
250 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl ) 250 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with lt (ord→od y) (o<-subst c (sym diso) refl )
251 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) 251 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
252 252
253 -- ZFSubset : (A x : OD ) → OD
254 -- ZFSubset A x = record { def = λ y → def A y ∧ def x y }
255
256 -- Def : (A : OD ) → OD
257 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
258
259 Ord-lemma : (a : Ordinal) (x : OD) → _⊆_ (ord→od a) (Ord a) {x}
260 Ord-lemma a x lt = o<-subst (c<→o< lt ) refl diso
261
262 ⊆-trans : {a b c x : OD} → _⊆_ a b {x} → _⊆_ b c {x} → _⊆_ a c {x}
263 ⊆-trans a⊆b b⊆c a∋x = b⊆c (a⊆b a∋x)
264
265 _∩_ = IsZF._∩_ isZF
266
267 ord-power-lemma : {a : Ordinal} → Power (Ord a) == Def (Ord a)
268 ord-power-lemma {a} = record { eq→ = left ; eq← = right } where
269 left : {x : Ordinal} → def (Power (Ord a)) x → def (Def (Ord a)) x
270 left {x} lt = lemma1 where
271 lemma : od→ord ((Ord a) ∩ (ord→od x)) o< sup-o ( λ y → od→ord ((Ord a) ∩ (ord→od y)))
272 lemma = sup-o< { λ y → od→ord ((Ord a) ∩ (ord→od y))} {x}
273 lemma1 : x o< sup-o ( λ x → od→ord ( ZFSubset (Ord a) (ord→od x )))
274 lemma1 = {!!}
275 right : {x : Ordinal } → def (Def (Ord a)) x → def (Power (Ord a)) x
276 right {x} lt = def-subst {_} {_} {Power (Ord a)} {x} (IsZF.power← isZF (Ord a) (ord→od x) {!!}) refl diso
277
253 uncountable : (a y : Ordinal) → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od y) 278 uncountable : (a y : Ordinal) → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od y)
254 uncountable a y = TransFinite {n} {suc n} {λ z → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od z)} caseΦ caseOSuc y where 279 uncountable a y = ⊆→o< lemma where
255 caseΦ : (lx : Nat) → ((x : Ordinal) → x o< ordinal lx (Φ lx) → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od x)) 280 lemma-a : (x : OD ) → _⊆_ (ZFSubset (Ord a) (ord→od y)) (Ord a) {x}
256 → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od (record { lv = lx ; ord = Φ lx })) 281 lemma-a x lt = proj1 lt
257 caseΦ lx prev = {!!} 282 lemma : (x : OD ) → _⊆_ (Ord ( od→ord (ZFSubset (Ord a) (ord→od y)))) (Ord a) {x}
258 caseOSuc : (lx : Nat) (ox : OrdinalD lx) 283 lemma x = {!!}
259 → ((y₁ : Ordinal) → y₁ o< ordinal lx (OSuc lx ox) → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od y₁))
260 → Ord (osuc a) ∋ ZFSubset (Ord a) (ord→od (record { lv = lx ; ord = OSuc lx ox }))
261 caseOSuc lx ox prev = {!!}
262 284
263 continuum-hyphotheis : (a : Ordinal) → (x : OD) → _⊆_ (Power (Ord a)) (Ord (osuc a)) {x} 285 continuum-hyphotheis : (a : Ordinal) → (x : OD) → _⊆_ (Power (Ord a)) (Ord (osuc a)) {x}
264 continuum-hyphotheis a x = lemma2 where 286 continuum-hyphotheis a x = lemma2 where
265 lemma1 : sup-o (λ x₁ → od→ord (ZFSubset (Ord a) (ord→od x₁))) o< osuc a 287 lemma1 : sup-o (λ x₁ → od→ord (ZFSubset (Ord a) (ord→od x₁))) o< osuc a
266 lemma1 = {!!} 288 lemma1 = {!!}
267 lemma : _⊆_ (Def (Ord a)) (Ord (osuc a)) {x} 289 lemma : _⊆_ (Def (Ord a)) (Ord (osuc a)) {x}
268 lemma = o<→c< lemma1 290 lemma = o<→c< lemma1
269 lemma2 : _⊆_ (Power (Ord a)) (Ord (osuc a)) {x} 291 lemma2 : _⊆_ (Power (Ord a)) (Ord (osuc a)) {x}
270 lemma2 = {!!} 292 lemma2 = subst ( λ k → _⊆_ k (Ord (osuc a)) {x} ) (sym (==→o≡ ord-power-lemma)) lemma