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

comparison ordinal.agda @ 271:2169d948159b

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fix incl
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 30 Dec 2019 23:45:59 +0900
parents 30e419a2be24
children 6f10c47e4e7a
comparison
equal deleted inserted replaced
270:fc3d4bc1dc5e 271:2169d948159b
237 237
238 import OD 238 import OD
239 -- open inOrdinal C-Ordinal 239 -- open inOrdinal C-Ordinal
240 open OD (C-Ordinal {n}) 240 open OD (C-Ordinal {n})
241 open OD.OD 241 open OD.OD
242 242 open _⊆_
243 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z} 243
244 o<→c< lt lt1 = ordtrans lt1 lt 244 o<→c< : {x y : Ordinal } → x o< y → Ord x ⊆ Ord y
245 245 o<→c< lt = record { incl = λ lt1 → ordtrans lt1 lt }
246 ⊆→o< : {x y : Ordinal } → (∀ (z : OD) → _⊆_ (Ord x) (Ord y) {z} ) → x o< osuc y 246
247 ⊆→o< : {x y : Ordinal } → Ord x ⊆ Ord y → x o< osuc y
247 ⊆→o< {x} {y} lt with trio< x y 248 ⊆→o< {x} {y} lt with trio< x y
248 ⊆→o< {x} {y} lt | tri< a ¬b ¬c = ordtrans a <-osuc 249 ⊆→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 250 ⊆→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 ) 251 ⊆→o< {x} {y} lt | tri> ¬a ¬b c with incl lt (o<-subst c (sym diso) refl )
251 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl )) 252 ... | ttt = ⊥-elim ( o<¬≡ refl (o<-subst ttt diso refl ))
252 253
253 -- ZFSubset : (A x : OD ) → OD 254 -- ZFSubset : (A x : OD ) → OD
254 -- ZFSubset A x = record { def = λ y → def A y ∧ def x y } 255 -- ZFSubset A x = record { def = λ y → def A y ∧ def x y }
255 256
256 -- Def : (A : OD ) → OD 257 -- Def : (A : OD ) → OD
257 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) ) 258 -- Def A = Ord ( sup-o ( λ x → od→ord ( ZFSubset A (ord→od x )) ) )
258 259
259 Ord-lemma : (a : Ordinal) (x : OD) → _⊆_ (ord→od a) (Ord a) {x} 260 Ord-lemma : (a : Ordinal) → ord→od a ⊆ Ord a
260 Ord-lemma a x lt = o<-subst (c<→o< lt ) refl diso 261 Ord-lemma a = record { incl = λ lt → o<-subst (c<→o< lt ) refl diso }
261 262
262 ⊆-trans : {a b c x : OD} → _⊆_ a b {x} → _⊆_ b c {x} → _⊆_ a c {x} 263 ⊆-trans : {a b c x : OD} → a ⊆ b → b ⊆ c → a ⊆ c
263 ⊆-trans a⊆b b⊆c a∋x = b⊆c (a⊆b a∋x) 264 ⊆-trans a⊆b b⊆c = record { incl = λ a∋x → incl b⊆c (incl a⊆b a∋x) }
264 265
265 _∩_ = IsZF._∩_ isZF 266 _∩_ = IsZF._∩_ isZF
266 267
267 -- 268 --
268 -- ord-power-lemma : {a : Ordinal} → Power (Ord a) == Def (Ord a) 269 -- ord-power-lemma : {a : Ordinal} → Power (Ord a) == Def (Ord a)