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

comparison HOD.agda @ 171:729b80df8a8a

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Jul 2019 08:34:36 +0900
parents c96f28c3c387
children 8c4d1423d7c4
comparison
equal deleted inserted replaced
170:c96f28c3c387 171:729b80df8a8a
236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α 236 -- L0 : {n : Level} → (α : Ordinal {suc n}) → L (osuc α) ∋ L α
237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x 237 -- L1 : {n : Level} → (α β : Ordinal {suc n}) → α o< β → ∀ (x : OD {suc n}) → L α ∋ x → L β ∋ x
238 238
239 -- another form of regularity 239 -- another form of regularity
240 -- 240 --
241 {-# TERMINATING #-} 241 -- {-# TERMINATING #-}
242 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m} 242 ε-induction : {n m : Level} { ψ : OD {suc n} → Set m}
243 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x ) 243 → ( {x : OD {suc n} } → ({ y : OD {suc n} } → x ∋ y → ψ y ) → ψ x )
244 → (x : OD {suc n} ) → ψ x 244 → (x : OD {suc n} ) → ψ x
245 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc) where 245 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (osuc (od→ord x)) <-osuc) where
246 ε-induction-ord : ( ox : Ordinal {suc n} ) {oy : Ordinal {suc n} } → oy o< ox → ψ (ord→od oy) 246 ε-induction-ord : ( ox : Ordinal {suc n} ) {oy : Ordinal {suc n} } → oy o< ox → ψ (ord→od oy)
250 ind {ord→od oy} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord (record { lv = lx ; ord = ox} ) (lemma y lt ))) where 250 ind {ord→od oy} ( λ {y} lt → subst (λ k → ψ k ) oiso (ε-induction-ord (record { lv = lx ; ord = ox} ) (lemma y lt ))) where
251 lemma : (y : OD) → ord→od oy ∋ y → od→ord y o< record { lv = lx ; ord = ox } 251 lemma : (y : OD) → ord→od oy ∋ y → od→ord y o< record { lv = lx ; ord = ox }
252 lemma y lt with osuc-≡< y<x 252 lemma y lt with osuc-≡< y<x
253 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso 253 lemma y lt | case1 refl = o<-subst (c<→o< lt) refl diso
254 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1 254 lemma y lt | case2 lt1 = ordtrans (o<-subst (c<→o< lt) refl diso) lt1
255 ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {record { lv = ly ; ord = oy }} (case1 (s≤s x)) with <-cmp lx ly 255 ε-induction-ord record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } {oy} =
256 ... | tri< a ¬b ¬c = ⊥-elim (lemma a x ) where 256 TransFinite {suc n} {suc n ⊔ m} {λ x → x o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od x)} lemma1 lemma2 oy where
257 lemma : {lx ly : Nat} → Suc lx ≤ ly → ly ≤ lx → ⊥ 257 lemma1 : (ly : Nat) →
258 lemma (s≤s lt1) (s≤s lt2) = lemma lt1 lt2 258 record { lv = ly ; ord = Φ ly } o< record { lv = Suc lx ; ord = Φ (Suc lx) } →
259 ... | tri≈ ¬a refl ¬c = ind {ord→od (record { lv = ly ; ord = oy })} ( λ {y} lt → subst (λ k → ψ k ) oiso ? 259 ψ (ord→od (record { lv = ly ; ord = Φ ly }))
260 ... | tri> ¬a ¬b c = ε-induction-ord record { lv = lx ; ord = (Φ lx) } (case1 c) 260 lemma1 ly lt = ind {!!}
261 lemma2 : (ly : Nat) (oy : OrdinalD ly) →
262 (record { lv = ly ; ord = oy } o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od (record { lv = ly ; ord = oy }))) →
263 record { lv = ly ; ord = OSuc ly oy } o< record { lv = Suc lx ; ord = Φ (Suc lx) } → ψ (ord→od (record { lv = ly ; ord = OSuc ly oy }))
264 lemma2 ly oy p lt = ind {!!}
261 265
262 266
263 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} 267 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
264 OD→ZF {n} = record { 268 OD→ZF {n} = record {
265 ZFSet = OD {suc n} 269 ZFSet = OD {suc n}