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

comparison src/zorn.agda @ 1067:074b6a506b1b

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ic case
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 13 Dec 2022 00:14:32 +0900
parents 86f6cc26e315
children f24f4de4d459
comparison
equal deleted inserted replaced
1066:86f6cc26e315 1067:074b6a506b1b
1334 uz02 = ordtrans<-≤ u<x (o<→≤ (osucc a<b)) 1334 uz02 = ordtrans<-≤ u<x (o<→≤ (osucc a<b))
1335 uz03 : ZChain.supf (pzc (pic<x (proj2 ib))) u ≡ u 1335 uz03 : ZChain.supf (pzc (pic<x (proj2 ib))) u ≡ u
1336 uz03 = trans (sym ( zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ u<x) )) su=u 1336 uz03 = trans (sym ( zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ u<x) )) su=u
1337 uz04 : FClosure A f (ZChain.supf (pzc (pic<x (proj2 ib))) u) (& a) 1337 uz04 : FClosure A f (ZChain.supf (pzc (pic<x (proj2 ib))) u) (& a)
1338 uz04 = subst (λ k → FClosure A f k (& a)) ( zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ u<x) ) fc 1338 uz04 = subst (λ k → FClosure A f k (& a)) ( zeq _ _ (o<→≤ (osucc a<b)) (o<→≤ u<x) ) fc
1339 ... | tri≈ ¬a ia=ib ¬c = uz01 where 1339 ... | tri≈ ¬a ia=ib ¬c = ? where
1340 uz01 : Tri (a < b) (a ≡ b) (b < a ) 1340 uz01 : Tri (* (& a ) < * (& b)) (* (& a) ≡ * (& b)) (* (& b) < * (& a ))
1341 uz01 with pchainx⊆chain ia 1341 uz01 with proj2 ia | proj2 ib
1342 ... | ⟪ ai , ch-init fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ib))) ⟪ ai , ch-init fc ⟫ (pchainx⊆chain ib) 1342 ... | ic-init fca | ic-init fcb = fcn-cmp y f mf fca fcb
1343 ... | ⟪ ai , ch-is-sup u u<x su=u fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ib))) ⟪ ai , ch-is-sup u uz02 uz03 uz04 ⟫ (pchainx⊆chain ib) where 1343 ... | ic-isup ia ia<x sa<x fca | ic-init fcb = ?
1344 uz02 : u o< osuc (IChain-i (proj2 ib)) 1344 ... | ic-init fca | ic-isup ib ib<x sb<x fcb = ?
1345 uz02 with osuc-≡< (ZChain.supf-mono (pzc (pic<x (proj2 ia))) (o<→≤ u<x)) 1345 ... | ic-isup ia ia<x sa<x fca | ic-isup ib ib<x sb<x fcb = fcn-cmp ? f mf ? ?
1346 ... | case1 eq = ⊥-elim ( o<¬≡ (trans (sym su=u) eq) (subst (λ k → u o< k) ? u<x) )
1347 ... | case2 lt = ZChain.supf-inject (pzc (pic<x (proj2 ia))) ?
1348 uz03 : ZChain.supf (pzc (pic<x (proj2 ib))) u ≡ u
1349 uz03 = trans (zeq _ _ ? ?) su=u
1350 uz04 : FClosure A f (ZChain.supf (pzc (pic<x (proj2 ib))) u) (& a)
1351 uz04 = subst (λ k → FClosure A f k (& a)) ? fc
1352 ... | tri> ¬a ¬b ib<ia = uz01 where 1346 ... | tri> ¬a ¬b ib<ia = uz01 where
1353 uz01 : Tri (a < b) (a ≡ b) (b < a ) 1347 uz01 : Tri (a < b) (a ≡ b) (b < a )
1354 uz01 with pchainx⊆chain ib 1348 uz01 with pchainx⊆chain ib
1355 ... | ⟪ bi , ch-init fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainx⊆chain ia) ⟪ bi , ch-init fc ⟫ 1349 ... | ⟪ bi , ch-init fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainx⊆chain ia) ⟪ bi , ch-init fc ⟫
1356 ... | ⟪ bi , ch-is-sup u u<x su=u fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainx⊆chain ia) ⟪ bi , ch-is-sup u uz02 uz03 uz04 ⟫ where 1350 ... | ⟪ bi , ch-is-sup u u<x su=u fc ⟫ = ZChain.f-total (pzc (pic<x (proj2 ia))) (pchainx⊆chain ia) ⟪ bi , ch-is-sup u uz02 uz03 uz04 ⟫ where