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

comparison src/OD.agda @ 1109:f46a16cebbab

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author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sat, 31 Dec 2022 17:56:01 +0900
parents 40345abc0949
children 607a79aef297
comparison
equal deleted inserted replaced
1108:720aff4a7fa4 1109:f46a16cebbab
293 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where 293 ε-induction {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
294 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox) 294 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
295 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso ))) 295 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
296 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy) 296 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
297 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy 297 ε-induction-ord ox {oy} lt = TransFinite {λ oy → ψ (* oy)} induction oy
298
299 ε-induction0 : { ψ : HOD → Set n}
300 → ( {x : HOD } → ({ y : HOD } → x ∋ y → ψ y ) → ψ x )
301 → (x : HOD ) → ψ x
302 ε-induction0 {ψ} ind x = subst (λ k → ψ k ) *iso (ε-induction-ord (osuc (& x)) <-osuc ) where
303 induction : (ox : Ordinal) → ((oy : Ordinal) → oy o< ox → ψ (* oy)) → ψ (* ox)
304 induction ox prev = ind ( λ {y} lt → subst (λ k → ψ k ) *iso (prev (& y) (o<-subst (c<→o< lt) refl &iso )))
305 ε-induction-ord : (ox : Ordinal) { oy : Ordinal } → oy o< ox → ψ (* oy)
306 ε-induction-ord ox {oy} lt = inOrdinal.TransFinite0 O {λ oy → ψ (* oy)} induction oy
298 307
299 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup 308 -- Open supreme upper bound leads a contradition, so we use domain restriction on sup
300 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥ 309 ¬open-sup : ( sup-o : (Ordinal → Ordinal ) → Ordinal) → ((ψ : Ordinal → Ordinal ) → (x : Ordinal) → ψ x o< sup-o ψ ) → ⊥
301 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where 310 ¬open-sup sup-o sup-o< = o<> <-osuc (sup-o< next-ord (sup-o next-ord)) where
302 next-ord : Ordinal → Ordinal 311 next-ord : Ordinal → Ordinal