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

comparison OD.agda @ 257:53b7acd63481

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move product to OD
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 30 Aug 2019 15:37:04 +0900
parents 2ea2a19f9cd6
children 63df67b6281c
comparison
equal deleted inserted replaced
255:1eba96b7ab8d 257:53b7acd63481
233 lemmay : y ≡ y' 233 lemmay : y ≡ y'
234 lemmay with lemmax 234 lemmay with lemmax
235 ... | refl with lemma4 eq -- with (x,y)≡(x,y') 235 ... | refl with lemma4 eq -- with (x,y)≡(x,y')
236 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 )) 236 ... | eq1 = lemma4 (ord→== (cong (λ k → od→ord k ) eq1 ))
237 237
238 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p 238 data ord-pair : (p : Ordinal) → Set n where
239 ppp {p} {a} d = d 239 pair : (x y : Ordinal ) → ord-pair ( od→ord ( < ord→od x , ord→od y > ) )
240
241 ZFProduct : OD
242 ZFProduct = record { def = λ x → ord-pair x }
243
244 -- open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ )
245 -- eq-pair : { x x' y y' : Ordinal } → x ≡ x' → y ≡ y' → pair x y ≅ pair x' y'
246 -- eq-pair refl refl = HE.refl
247
248 pi1 : { p : Ordinal } → ord-pair p → Ordinal
249 pi1 ( pair x y) = x
250
251 π1 : { p : OD } → ZFProduct ∋ p → OD
252 π1 lt = ord→od (pi1 lt )
253
254 pi2 : { p : Ordinal } → ord-pair p → Ordinal
255 pi2 ( pair x y ) = y
256
257 π2 : { p : OD } → ZFProduct ∋ p → OD
258 π2 lt = ord→od (pi2 lt )
259
260 op-cons : { ox oy : Ordinal } → ZFProduct ∋ < ord→od ox , ord→od oy >
261 op-cons {ox} {oy} = pair ox oy
262
263 p-cons : ( x y : OD ) → ZFProduct ∋ < x , y >
264 p-cons x y = def-subst {_} {_} {ZFProduct} {od→ord (< x , y >)} (pair (od→ord x) ( od→ord y )) refl (
265 let open ≡-Reasoning in begin
266 od→ord < ord→od (od→ord x) , ord→od (od→ord y) >
267 ≡⟨ cong₂ (λ j k → od→ord < j , k >) oiso oiso ⟩
268 od→ord < x , y >
269 ∎ )
270
271 op-iso : { op : Ordinal } → (q : ord-pair op ) → od→ord < ord→od (pi1 q) , ord→od (pi2 q) > ≡ op
272 op-iso (pair ox oy) = refl
273
274 p-iso : { x : OD } → (p : ZFProduct ∋ x ) → < π1 p , π2 p > ≡ x
275 p-iso {x} p = ord≡→≡ (op-iso p)
276
277 p-pi1 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π1 p ≡ x
278 p-pi1 {x} {y} p = proj1 ( prod-eq ( ord→== (op-iso p) ))
279
280 p-pi2 : { x y : OD } → (p : ZFProduct ∋ < x , y > ) → π2 p ≡ y
281 p-pi2 {x} {y} p = proj2 ( prod-eq ( ord→== (op-iso p)))
240 282
241 -- 283 --
242 -- Axiom of choice in intutionistic logic implies the exclude middle 284 -- Axiom of choice in intutionistic logic implies the exclude middle
243 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog 285 -- https://plato.stanford.edu/entries/axiom-choice/#AxiChoLog
244 -- 286 --
287
288 ppp : { p : Set n } { a : OD } → record { def = λ x → p } ∋ a → p
289 ppp {p} {a} d = d
290
245 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice 291 p∨¬p : ( p : Set n ) → p ∨ ( ¬ p ) -- assuming axiom of choice
246 p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } )) 292 p∨¬p p with is-o∅ ( od→ord ( record { def = λ x → p } ))
247 p∨¬p p | yes eq = case2 (¬p eq) where 293 p∨¬p p | yes eq = case2 (¬p eq) where
248 ps = record { def = λ x → p } 294 ps = record { def = λ x → p }
249 lemma : ps == od∅ → p → ⊥ 295 lemma : ps == od∅ → p → ⊥
296 subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} ) 342 subset-lemma : {A x y : OD } → ( x ∋ y → ZFSubset A x ∋ y ) ⇔ ( _⊆_ x A {y} )
297 subset-lemma {A} {x} {y} = record { 343 subset-lemma {A} {x} {y} = record {
298 proj1 = λ z lt → proj1 (z lt) 344 proj1 = λ z lt → proj1 (z lt)
299 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt } 345 ; proj2 = λ x⊆A lt → record { proj1 = x⊆A lt ; proj2 = lt }
300 } 346 }
301
302 -- Constructible Set on α
303 -- Def X = record { def = λ y → ∀ (x : OD ) → y < X ∧ y < od→ord x }
304 -- L (Φ 0) = Φ
305 -- L (OSuc lv n) = { Def ( L n ) }
306 -- L (Φ (Suc n)) = Union (L α) (α < Φ (Suc n) )
307 -- L : {n : Level} → (α : Ordinal ) → OD
308 -- L record { lv = Zero ; ord = (Φ .0) } = od∅
309 -- L record { lv = lx ; ord = (OSuc lv ox) } = Def ( L ( record { lv = lx ; ord = ox } ) )
310 -- L record { lv = (Suc lx) ; ord = (Φ (Suc lx)) } = -- Union ( L α )
311 -- cseq ( Ord (od→ord (L (record { lv = lx ; ord = Φ lx }))))
312
313 -- L0 : {n : Level} → (α : Ordinal ) → L (osuc α) ∋ L α
314 -- L1 : {n : Level} → (α β : Ordinal ) → α o< β → ∀ (x : OD ) → L α ∋ x → L β ∋ x
315 347
316 OD→ZF : ZF 348 OD→ZF : ZF
317 OD→ZF = record { 349 OD→ZF = record {
318 ZFSet = OD 350 ZFSet = OD
319 ; _∋_ = _∋_ 351 ; _∋_ = _∋_
379 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y )) 411 pair→ x y t (case2 t≡y ) = case2 (subst₂ (λ j k → j == k ) oiso oiso (o≡→== t≡y ))
380 412
381 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t 413 pair← : ( x y t : ZFSet ) → ( t == x ) ∨ ( t == y ) → (x , y) ∋ t
382 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x)) 414 pair← x y t (case1 t==x) = case1 (cong (λ k → od→ord k ) (==→o≡ t==x))
383 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y)) 415 pair← x y t (case2 t==y) = case2 (cong (λ k → od→ord k ) (==→o≡ t==y))
384
385 -- pair0 : (A B : OD ) → ((A , B) ∋ A) ∧ ((A , B) ∋ B)
386 -- proj1 (pair A B ) = omax-x (od→ord A) (od→ord B)
387 -- proj2 (pair A B ) = omax-y (od→ord A) (od→ord B)
388 416
389 empty : (x : OD ) → ¬ (od∅ ∋ x) 417 empty : (x : OD ) → ¬ (od∅ ∋ x)
390 empty x = ¬x<0 418 empty x = ¬x<0
391 419
392 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z} 420 o<→c< : {x y : Ordinal } {z : OD }→ x o< y → _⊆_ (Ord x) (Ord y) {z}