Mercurial > hg > Members > kono > Proof > ZF-in-agda
view src/VL.agda @ 1458:171c3f3cdc6b
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author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Sat, 26 Aug 2023 08:37:08 +0900 |
parents | 47d3cc596d68 |
children |
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open import Level open import Ordinals module VL {n : Level } (O : Ordinals {n}) where open import logic import OD open import Relation.Nullary open import Relation.Binary open import Data.Empty open import Relation.Binary open import Relation.Binary.Core open import Relation.Binary.PropositionalEquality open import Data.Nat renaming ( zero to Zero ; suc to Suc ; ℕ to Nat ; _⊔_ to _n⊔_ ) import BAlgebra open BAlgebra O open inOrdinal O import OrdUtil import ODUtil open Ordinals.Ordinals O open Ordinals.IsOrdinals isOrdinal -- open Ordinals.IsNext isNext open OrdUtil O open ODUtil O open OD O open OD.OD open ODAxiom odAxiom -- import ODC open _∧_ open _∨_ open Bool open HOD -- The cumulative hierarchy -- V 0 := ∅ -- V α + 1 := P ( V α ) -- V α := ⋃ { V β | β < α } V1 : (x : Ordinal ) → ( ( y : Ordinal) → y o< x → HOD ) → HOD V1 x V0 with trio< x o∅ V1 x V0 | tri< a ¬b ¬c = ⊥-elim ( ¬x<0 a) V1 x V0 | tri≈ ¬a refl ¬c = Ord o∅ V1 x V0 | tri> ¬a ¬b c with Oprev-p x V1 x V0 | tri> ¬a ¬b c | yes p = Power ( V0 (Oprev.oprev p ) (subst (λ k → _ o< k) (Oprev.oprev=x p) <-osuc )) V1 x V0 | tri> ¬a ¬b c | no ¬p = record { od = record { def = λ y → (y o< x ) ∧ ((lt : y o< x ) → odef (V0 y lt) x ) } ; odmax = x; <odmax = λ {x} lt → proj1 lt } record VOrd (x : Ordinal) : Set n where field β : Ordinal ov : odef (TransFinite V1 β) x V : OD V = record { def = λ x → VOrd x } record Vn : Set n where field x : Ordinal β : Ordinal ov : odef (TransFinite V1 β) x Vn∅ : Vn Vn∅ = record { x = o∅ ; β = o∅ ; ov = ? } vsuc : Vn → Vn vsuc v = ? v< : Vn → Vn → Set n v< x y = ? IsVOrd : IsOrdinals Vn Vn∅ vsuc ? IsVOrd = ?