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

comparison Ordinals.agda @ 340:639fbb6284d8

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 13 Jul 2020 09:26:34 +0900
parents feb0fcc430a9
children 27d2933c4bd7
comparison
equal deleted inserted replaced
339:feb0fcc430a9 340:639fbb6284d8
226 next< {x} {y} {z} x<nz y<nx | tri≈ ¬a b ¬c = ⊥-elim ((proj2 (proj2 next-limit)) (next z) x<nz (subst (λ k → k o< next x) b y<nx) 226 next< {x} {y} {z} x<nz y<nx | tri≈ ¬a b ¬c = ⊥-elim ((proj2 (proj2 next-limit)) (next z) x<nz (subst (λ k → k o< next x) b y<nx)
227 (λ w nz=ow → o<¬≡ nz=ow (subst₂ (λ j k → j o< k ) (sym nz=ow) nz=ow (proj1 (proj2 next-limit) w (subst (λ k → w o< k ) (sym nz=ow) <-osuc) )))) 227 (λ w nz=ow → o<¬≡ nz=ow (subst₂ (λ j k → j o< k ) (sym nz=ow) nz=ow (proj1 (proj2 next-limit) w (subst (λ k → w o< k ) (sym nz=ow) <-osuc) ))))
228 next< {x} {y} {z} x<nz y<nx | tri> ¬a ¬b c = ⊥-elim (proj2 (proj2 next-limit) (next z) x<nz (ordtrans c y<nx ) 228 next< {x} {y} {z} x<nz y<nx | tri> ¬a ¬b c = ⊥-elim (proj2 (proj2 next-limit) (next z) x<nz (ordtrans c y<nx )
229 (λ w nz=ow → o<¬≡ (sym nz=ow) (proj1 (proj2 next-limit) _ (subst (λ k → w o< k ) (sym nz=ow) <-osuc )))) 229 (λ w nz=ow → o<¬≡ (sym nz=ow) (proj1 (proj2 next-limit) _ (subst (λ k → w o< k ) (sym nz=ow) <-osuc ))))
230 230
231 nexto=n : {x y : Ordinal} → x o< next (osuc y) → x o< next y
232 nexto=n {x} {y} x<noy = next< (proj1 (proj2 next-limit) _ (proj1 next-limit)) x<noy
233
231 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where 234 record OrdinalSubset (maxordinal : Ordinal) : Set (suc n) where
232 field 235 field
233 os→ : (x : Ordinal) → x o< maxordinal → Ordinal 236 os→ : (x : Ordinal) → x o< maxordinal → Ordinal
234 os← : Ordinal → Ordinal 237 os← : Ordinal → Ordinal
235 os←limit : (x : Ordinal) → os← x o< maxordinal 238 os←limit : (x : Ordinal) → os← x o< maxordinal