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

comparison HOD.agda @ 173:e6e1bdbda450

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 19 Jul 2019 15:28:20 +0900
parents 8c4d1423d7c4
children ad7a6185b6d5
comparison
equal deleted inserted replaced
172:8c4d1423d7c4 173:e6e1bdbda450
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 (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where 245 ε-induction {n} {m} {ψ} ind x = subst (λ k → ψ k ) oiso (ε-induction-ord (lv (osuc (od→ord x))) (ord (osuc (od→ord x))) <-osuc) where
246 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly } 246 ε-induction-ord : (lx : Nat) ( ox : OrdinalD {suc n} lx ) {ly : Nat} {oy : OrdinalD {suc n} ly }
283 lemma z lt | case2 lz=ly with <-cmp lx ly 283 lemma z lt | case2 lz=ly with <-cmp lx ly
284 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen 284 lemma z lt | case2 lz=ly | tri< a ¬b ¬c = ⊥-elim ( lemma0 y<sox a) -- can't happen
285 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- z(b) 285 lemma z lt | case2 lz=ly | tri> ¬a ¬b c with d<→lv lz=ly -- z(b)
286 ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c ))) 286 ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord lx (Φ lx) {_} {ord (od→ord z)} (case1 (subst (λ k → k < lx ) (trans (sym lemma1)(sym eq) ) c )))
287 lemma z lt | case2 lz=ly | tri≈ ¬a refl ¬c with d<→lv lz=ly -- z(c) 287 lemma z lt | case2 lz=ly | tri≈ ¬a refl ¬c with d<→lv lz=ly -- z(c)
288 ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord (Suc lx) (Φ (Suc lx)) {_} {ord (od→ord z)} 288 ... | eq = subst (λ k → ψ k ) oiso (ε-induction-ord lx
289 (case1 (subst (λ k → k < Suc lx) (trans (sym lemma1) (sym eq)) lemma2 ))) 289 (ox lz=ly -- ord (od→ord z) d< ord (od→ord (ord→od (record { lv = lx ; ord = oy })))
290 (subst (λ k → lv (od→ord z) ≡ k ) lemma1 eq) ) {_} {ord (od→ord z)} (case2 {!!})) where
291 ox : {lx lz : Nat} → {oy : OrdinalD {suc n} lz} {oz : OrdinalD {suc n} lx} → {!!} d< {!!} → lz ≡ lx → OrdinalD {suc n} lx
292 ox = ?
290 293
291 294
292 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n} 295 OD→ZF : {n : Level} → ZF {suc (suc n)} {suc n}
293 OD→ZF {n} = record { 296 OD→ZF {n} = record {
294 ZFSet = OD {suc n} 297 ZFSet = OD {suc n}